Übersicht der Quellen

 
     
 
 
Anforderungen  |   Konzepte  |   Entwurf  |   Entwicklung  |   Qualitätssicherung  |   Lebenszyklus  |   Steuerung
 
 
 
 

Benutzer

Quelle  chap71.html   Sprache: HTML

 

 products/Sources/formale Sprachen/GAP/doc/ref/chap71.html


<":=""java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 71: Character Tables</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel=
<div classchlinkprevnexttop">nbsp<"chap0html> Book</&  <href=chap0.contents"[Contents<a>nbsp; nbsp;a "chap70h>Previous]/>nbsp; n< "."java.lang.StringIndexOutOfBoundsException: Range [234, 226) out of bounds for length 253
< typetextjavascript>)<script
</head>
<body class="divclass="ontSect>span class=tocline>span class="nocss"> </span><a href="chap71.html#X7B9FCBBC7B95F91B">71.1 <span class="Heading">Some Remarks about Character Theory in <strong class="pkg">GAP</strong></span></a>


<div class="chlinktop">java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap70.html">[java.lang.StringIndexOutOfBoundsException: Index 172 out of bounds for length 7

<p id="mathjaxlink" class="pcenter"><a href="chap71_mj.html">[MathJax on]</a></p>
<p><a id="X7B7A9EE881E01C10" name="X7B7A9EE881E01C10"></a></p>
<divclass=ChapSects"< ="hap71.tml#77A9EE881E01C10><class"Heading>java.lang.StringIndexOutOfBoundsException: Range [98, 97) out of bounds for length 115
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7B9FCBBC7B95F91B">71.1 <span class="Heading">Some Remarks about Character/>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a spanclass"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 151
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X8701174D86B586AF">71.3 <span class="Heading">Creating Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7FCA7A7A822BDA33">71.3-1 <span class="Heading">CharacterTable</span></a>
</span>
<span class="ContSS"><br /=C"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 142
</java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 151
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X85DB8AE7786A2DB5">71.3-3 CharacterTableRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7DBEF4BF87F10CD6">71.3-4 SupportedCharacterTableInfo</a></span>
<""< >span">&<spanahref"java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 147
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X789FAC077AEF088A">71.4 <span class="Heading">Character Table Categories</span></a>
<>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  span =ContSS"br />span=">nbsp;n;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 140
spanclass"ContSS>br /=nocss">nbsp&;/>a "hap71html>.- java.lang.StringIndexOutOfBoundsException: Range [133, 131) out of bounds for length 142
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7FA867637EBB30F9">71.4-3 NearlyCharacterTablesFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X829C4B6E83998F40">71.5 <span class="Heading">Conventions for  class"ContSS"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 153
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X793E0EBF84B07313">71.6 <span class="Heading">The Interface between Character Tables and Groups</span></a>
</span>
<div class="ContSSBlock">
<java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
<es</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X84DC12AA804C8085">71.6-3 IdentificationOfConjugacyClasses</a></span>
<panclass=""ContSS">br /> class="nocss">nbsp; </pan><a href="chap71.html#X8788C6C7829C1ADE>71.- CharacterTableWithStoredGroup/<span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X790019E87CFDDB98">71.6-5 CompatibleConjugacyClasses</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7CADCBC9824CB624">71.7 <span class="Heading">Operators for Character Tables</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7F9D58208241D35E">71.8 <span java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 7
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X81FEFF768134481A">71.8-1 <span class="Heading">CharacterDegrees</="h#81".- <panclass=HeadingGroup ApplicabletoTablesspan<>
<>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X873B3CC57E9A5492/>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X8549899A7DE206BA">71.8-3 <span class="Heading">LinearCharacters</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html">.-SizesConjugacyClasses<a><span>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X81EFD9FE804AC6EE">71.8-5 <span class="Heading">Group Operations Applicable to Character Tables</span></a>
</span>
</div></div>
<div class/>
</span>
<div class="ContSSBlock">
< java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 150
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7CF7907F790A5DE6">71.9-2 SizesCentralizers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7D9D2A45879A6A62">71.9-3 SizesConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7C2753DE8094F4BA">71.9-4 AutomorphismsOfTable</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [104, 5) out of bounds for length 170
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X804CFD597C795801">71.9-6 <span class="Heading">Class Names and Character Names</span></a>
</span>
<span class="ContSS"><br span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7919E2897BE8234A">71.9-10 InverseClasses</a></span>
</span>
<span class="ContSS"><br /><span class="nocss"> spanclass""></< nocssn;/< =chap71X7>java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 132
<span class="ContSS"><br /><span class="nocss">  </span><a href=chap71.tmlX7919E2897BE8234A71.9- InverseClasses/></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X87FF547981456932">71.9-11 RealClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7ABB007C799F7C49">71.9-12 ClassOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7F863B15804E0835">71.9-13 ClassRoots</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><</pan>
</span>
<div class="<span class="ContSS"><b ><pan class="nocss">  </span><a href="chap71.html#X850C7D947B3DBFA2">71.10-1 ClassPositionsOfNormalSubgroups</a></span>
<span="ContSS">br /<span  class="nocss">  </span><a href="chap71.html#X850C7D947B3DBFA2">71.10-1 ClassPositionsOfNormalSubgroups</a></span>
<=C"br/<">&java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 146
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7A6B1F8A84A495DC">71.10-3 ClassPositionsOfCentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7D53F60785AB22B1">71.10-4 ClassPositionsOfDirectProductDecompositions</a></span>
<span class="</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X86ABB2E179D7F6E1">71.10-6 ClassPositionsOfElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7D2A55A584F955DB">71.10-7 ClassPositionsOfFittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X79AEFC4384769B72">71.10-8 ClassPositionsOfLowerCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X86065D217A36CD9B">71.10-9 ClassPositionsOfUpperCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X877FDE8A84A9F52C">71.10-10 ClassPositionsOfSolvableRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71."><br /><span class="nocss">  </span><a href="chap71.html#8392DD5B813250A4>.- //span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7BBE7EBA7A64A6B0">71.10-12 ClassPositionsOfPCore</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7FCF905D7FD7CC40">71.10-13 ClassPositionsOfNormalClosure</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X8733F0EA801785D4">71.11 <span class="Heading">Operations Concerning Blocks</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7ACB9306804F4E3F">71.11-1 PrimeBlocks</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7E80E35985275F35">71.11-2 SameBlock</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a span class="ContSS"><br /><span class="nocss">  </span><a href">71.- java.lang.StringIndexOutOfBoundsException: Range [136, 133) out of bounds for length 144
<span class="ContSS"><br /><span class="nocss"> </div>/div>
<class=ContSS> /<panclass=n"&; /java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 155
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X873211618402ACF7">71.12 <span class="Heading">Other Operations for Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span =ontSS>< >span class="nocss">  </span><a href="chap71.html#X8123650E817926FC">71.12-2 IsInternallyConsistent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X8123650E817926FC">71.12-2 IsInternallyConsistent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7A0CBD1884276882">71.12-3 IsPSolvableCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X82F523E8784B3752">71.12-4 IsClassFusionOfNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7FD3D3047DE6381E">71.12-5 Indicator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X83AE05BF8085B3C8">71.12-6 NrPolyhedralSubgroups<span=< /span class""&&;span=chap71.#java.lang.StringIndexOutOfBoundsException: Range [106, 104) out of bounds for length 148
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71<><div>
<pan ContSS">>span "nocss>nbspnbsp;</span>< "chap71htmljava.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 148
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X809E67E57D4933B3">71.12-9 MatClassMultCoeffsCharTable</a></span>
</div></div>
<div class="ContSect">java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 7
<span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7D45224B86D802E5">71.13-1 ViewObj</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 133
<.- a></>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X85E883A87A190AA4">71.13-4 DisplayOptions</a></span>
<span class="ContSS"><br<panclass=ContSS<br/>span=nocss"&; /> "hap71.tml#7C3A4E5283B240BE"71203 haracterTableFactorGroup</a</span>
</>div
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X79BC08C6846718D9">71.14 <span class="Heading">Computing the Irreducible Characters of a Group</span></a>
</</span=""br /<class="ocss>;&/>< =.tml#java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 154
<"ContSS">< / class"ocss">  </span><a href="chap71.html#X83E71B1F7FA70134">71.20-7 CharacterValueWreathSymmetric</a></span>
< href="chap71.html#X7ED39DB680BFEA96">7114- />/java.lang.StringIndexOutOfBoundsException: Range [142, 141) out of bounds for length 142
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7E81BCD686561DF0">71.14-2 IrrConlon</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7BF15729839203FC">71.14-3 IrrBaumClausen</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7F29C5447B5DC102">71.14-4 IrreducibleRepresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X8493ED7A86FFCB8A">71.14-5 IrreducibleRepresentationsDixon</a></span>
</div></div>
<div class="ContSect"><span class="tocline" class""><>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 141
</span>
<divclass="java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
< "ontSS"<br>< =n">&bsp<span><a href="chap71.html#X87E82F8085745523">71.15-1 IrreducibleModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7D0BD5337D1C069B">71.15-2 AbsolutelyIrreducibleModules/span>
<pan ="< <""nbsp;&<java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 138
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X86CDA4007A5EF704">71.16 <span class="Headingspan classContSS< /<spann&&;/>a ".html"/<java.lang.StringIndexOutOfBoundsException: Range [143, 142) out of bounds for length 143
<>
</div>
< ">span "toclinespan ""&/a ".X7C083207868066C1">. < =Heading>AdvancedMethodsforDixonSchneider java.lang.StringIndexOutOfBoundsException: Range [195, 194) out of bounds for length 205
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> &java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 12
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7E33C03E7BDDC9B0">71.17-2 DixonInit</a></span>
<class=ContSS>br/>span nocss"&&bsp;/pan><ahref"chap71html#"71.173 ixontinI</>/
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X87ABE0B081DAD476">71.17-4 DixonSplit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7BFD4C1A821731FB">71.17-5 BestSplittingMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7C85B56C80BFA2E3">71.17-6 DxIncludeIrreducibles</spanclass=java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 157
<span class="ContSS"><br /><span class="nocss">  </span><a href/
<span class<3>71<span class="Heading">Character Tables</span></h3>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> <p>This chapterdescribes operations for <em>character tables of finite groups</>./>

</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X782B5E37848786BC">71.19java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7C38C5067941D496">71.20 <span class="Heading">Constructing Character Tables from Others</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br>.3spanclass=Heading">Creating Character Tables</span></h4>
<span class="ContSS"><br /><span class="nocss"
<><span">nbsp;nbsp</< "hap71htmljava.lang.StringIndexOutOfBoundsException: Range [106, 104) out of bounds for length 150
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X85BE46F784B83938">71.20-4 CharacterTableIsoclinic</a></span>
<span class="ContSS"><br /><span class="nocss"li
<span class=ContSS"<br ><span class=""&&;span>ahref"java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 154
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X83E71B1F7FA70134">71.20-7 CharacterValueWreathSymmetric</a><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<
<div class="ContSSBlock">
<span class="ContSS"><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">  </java.lang.StringIndexOutOfBoundsException: Range [0, 65) out of bounds for length 5
< =ContSS"br/>>span class="nocss">  </span><a href="chap71.html#X7E3DE0A47E62BE6B">71.21-3 CharacterTableWithSortedClasses</a></span>
<span class="ContSS"><br /><span class="p>Theoperation <code class="unc"CharacterTable</ode (<a href"hap71.html#X7FCA7A7A822BDA33"><span class="RefLink">71.3-1</span></a>) can be used for the cases 1. to 3.</p>
<span class="ContSS"><br /><span class="nocss">  </span><a href
</div></div>
<div class="ContSect"><span class="tocline"><
</span
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X84353BB884AF0365">71.22-1 MatrixAutomorphisms</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 143
<spanclass="ContSS"<br/>span class="">&&bsp;/><a =chap71.tml#D721E3D7AA319F5"71223TransformingPermutations</a>/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X849731AA7EC9FA73">71.22-4 TransformingPermutationsCharacterTables</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X8117D940835B0B47">71.22-5 FamiliesOfRows</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X81272CEE79F13E7B">71.23 <span class="Heading">Storing Normal Subgroup Information</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7E66174C7C7A8C0C">71.23-1 NormalSubgroupClassesInfo</a<>Called a group <var class=Arg"></, <code class="func">CharacterTable</code> calls the attribute <code class="func">OrdinaryCharacterTable</code> (<a href="chap71.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>). Called with first argumentjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X7C2A87E085111090">71.23-2 ClassPositionsOfNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X87E7391F7F92377C">71.23-3 NormalSubgroupClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X79D451F0808EB252">71.23-4 FactorGroupNormalSubgroupClasses</a></span>
</div></div>
</div>

<h3The why <ode class=">java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 774

<p>This chapter describes operations for <em>character tables of finite groups</em>.</p>

<p>Operations for <em>characters</em> (or, more general, <em>class functions</em>) are described in Chapter <a href="chap72.html#X7C91D0D17850E564"><span class="RefLink">72</span></a>.</p>

<p>For a description of the <strong class="pkg">GAP</strong> Library of Character Tables, see the separate manual for the <strong class="pkg">GAP</strong> package <strong class="pkg">CTblLib</strong>.</p>

<p>Several examples in this chapter require the <strong class="pkg">GAP</strong> Character Table Library to java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib" );</span>
trueclass""table =func"width"100%>tr<td class="tdleft"><code class="func">&#8227; BrauerTable</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
</pre></div>

<p><a id="X7B9FCBBC7B95F91B" name="X7B9FCBBC7B95F91B"></a></p>

<h4>71.1 <span class="Heading">Some Remarks about Character Theory in <strong class="pkg">GAP</strong></span></h4>

<p>It seems to be necessary to state some basic facts –and maybe >Called with an ordinary character table <var class="Arg">ordtbl</var><varclass="">/, code ="unc"BrauerTable<code> returnsits < class="Arg"p</>modularcharacter   strong class="pkg">GAP</strong> can compute this table, and <code class="keyw">fail</code> otherwise.</p>

<p>Intuitively, <em>characters<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>At this stage one could ask where there is a<uljava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<p>For computations with characters of a finite group <span class="SimpleMath">G</span> with <span class="SimpleMath">n</span> conjugacy classes, we fix an ordering java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>As a consequence, also other information involving characters is given implicitly. For example, we can talk about the kernel of a character not as a group but as a list of classes (more exactly: a list of their positions according to the chosen java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>So typical calculations with characters involve loops over lists of character values. For example, the scalar product of two characters <span class="SimpleMath">χ</span>, <span class="SimpleMath">ψ</span> of <span class="SimpleMath">G</span> given by</p>

<p class="pcenter">[ χ, ψ ] = ( ∑_{g ∈ G} χ(g) ψ(g^{-1}) ) / |G|</p>

<p>can be written as</p>


<p> default for a group anda  delegatesto < =func>code> for the ordinary character table of this group. The default method for <var class="Arg">ordtbl</var> uses the attribute <code class="func">ComputedBrauerTables</code> for storing the computed Brauer table at position <var class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

                          * ComplexConjugate( psi[i] ) ) / Size( t );
</pre></div>

<p>where <code class="code">t</code> is the character table of <span class="SimpleMath">G</span>, and <code class="code">chi</code>, <code class="code">psi</code> are the lists of values of <span class="SimpleMath">χ

<class=func"table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableRegular</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>

<p><strong class="pkg">GAP</strong> uses character <em>tables</em> to store information such as class lengths, element orders, the irreducible characters of <span class="SimpleMath">G</span> etc. in a consistent way; java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Note that the character table of a group <span class="SimpleMath">G</span> in <java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 0

<p>span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );</span>

<p>So class functions, power maps, and fusion maps are represented by lists in <strong class="pkg">GAP</strong>. If they are plain lists then java.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 35

<p>One can construct class function objects that store their underlying character table and other attribute values (see Chapter <a href="chap72.html#X7C91D0D17850E564"><span class="

<< =namejava.lang.StringIndexOutOfBoundsException: Range [55, 52) out of bounds for length 62

<>. span class="Heading"HistoryCharacterTheory Stuff />h4

<p<strong class="kg">AP</strong>provides functions for dealing with group characters since the version <strong class="pkg">GAP</strong> 3.1, which was released in March 1992. The reason for adding this branch of mathematics to the topics of <strong <span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104

<p>For integrating the existing CharacterTable( "Sym(4)" )

<p>The development of character theory code for <strong class="pkg">GAP</strong> has been supported by several DFG grants, in java.lang.StringIndexOutOfBoundsException: Range [1, 136) out of bounds for length 41

<p>The major contributions can be listed as follows.</p>


<ul>
<li><p>The arithmetic for the cyclotomics data typeh5713- SupportedCharacterTableInfoh5

</li>
ractertableswerewritten byThomas Breuer and Götz Pfeiffer.</p>

</li>
<li><p>The lattice related functions, such<code ="">SupportedCharacterTableInfo/>is a list that contains at position <span class="SimpleMath">3i-2</span> an attribute getter function, at position <span class="SimpleMath">3i-1</span> the name of this attribute, and at position java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
<li><p>Functions for computing possible class fusions, possible power maps, and table automorphisms were written by Thomas Breuer (see <a href="chapBib.html#biBBre91">[Bre91]</a>).</p>

</li>
<li><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
<li><p>Functions for computing character tables from groups were java.lang.StringIndexOutOfBoundsException: Range [0, 72) out of bounds for length 0

</li>
<li><p>Functions for dealing with Clifford matrices were written by Ute Schiffer (see <a href="chapBib.html<h5>71.3-5 ConvertToCharacterTable</h5>

</li>
<li>><>Functions for were  Thomas Breuer and Erzsébet Horváth.</p>

<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
</ul>
<p>Since then, the code has been maintained and extended further by Alexander Hulpkep>Let <var class=record<var arecord. code =func"</   ">record</>into acomponentobject (  href"chap79html#E223484649E5A><class="RefLink">79.2</span></a>) representing a character table. The values of those components of <var class="Arg">record</var> whose names occur in <code class="func">SupportedCharacterTableInfo</code> (<a href="chap71.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>) correspond to attribute values of the returned character table. All other components of 

<p>Currently <strong class="pkg">GAP</strong> does not provide special functionality for computing Brauer character tables, but there is an interface to the <strong class="pkg">MOC</strong> system (see <a href="chapBib.html#biBHJLP92">[HJLP]</a>), and the <strong class="pkg">GAP</strong> Character Table Library contains many known Brauer character tables.</p>

<p><a id="X8701174D86B586AF" name="X8701174D86B586AF"></a></p>

<h4>71.3 <span class="Heading">Creating Character Tables</span></h4>

<p>There are in general five different ways to get a character table in <strong class="pkg">GAP</strong>. You can</p>

<ol>
<li><p>compute the table from a group,</p>

</li>
<li><p>read a file that contains the table data,</p>

</li>
<li><p>construct the table using generic formulae,</p>

</i>
<li><p>derive it from known character tablesspan class=ContSS">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
< partialinformation  conjugacy ,powermaps ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
</ol>
<>In1. java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<java.lang.StringIndexOutOfBoundsException: Range [56, 2) out of bounds for length 178

<p><a id

<h5>71.3-1 <span class="Heading"div "unc>"w"trclass"java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 225

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CharacterTable</div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBrauerTable</code><var>)/tdtdclass=tdright"(&;nbsp;)/d></tr/java.lang.StringIndexOutOfBoundsException: Index 211 out of bounds for length 132
<"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 253
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CharacterTable</code>( <var class="Arg">name</var>[, <var class="Arg">param</var>] )</tdclass=""> <java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 144
< second argument a prime <var class="Arg">p</var>, <code class="func">CharacterTable</code> calls the operation <code class="func">BrauerTable</code> java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

<p>=ContSS" ><class"ocss"&;</><a href=chap71.#7">.4- IrrBaumClausen/>/panjava.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 139

<pspan class="ContSS"><br /><span class="nocss">  </span><a href="chap71.html#X8493ED7A86FFCB8A">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The reason why <code class="func">CharacterTable</code> does not compute the irreducible characters is java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 0

<p>For class=C<>

<p>If the divclass">java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 185

<p><a id="X8476B25A79D7A7FC" name="X8476B25A79D7A7FC"></a></p>

<>713-=java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 205

<<spa ContSSbr/><"nocss  />a =chap71.tml#C398F2680C8616B">71.17-1 DixonRecord</a></span>
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<div "unc">table &t< >java.lang.StringIndexOutOfBoundsException: Range [77, 75) out of bounds for length 116
<divclass"unc">class
<p  class="ContSS< njava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 144

<p>The <var class="Argbpan class"ontSS"</<class=nocss


<ul>
<>p>The groupisvar "rg

</java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<i<the  <class=A"p/>subgroup

</li>
<li><p>the 4-2 InfoCharacterTable

<
< the <strong class"><strong charactertable library for which also the <var <p>is the info class (see <a href="chap7.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>) for computations with character tables.<

</li>
</ljava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<>The method < "java.lang.StringIndexOutOfBoundsException: Range [18, 16) out of bounds for length 204

<p>So if one wants to install a new method for computing Brauer tables then it is sufficient to install< "ontSS"</>spanclass=">nbsp;nbsp;/><a href"hap71htmlX806E55A58397B11B71.-5""3<<)/pjava.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145

<>The <code ="keyw">mod<code operator  a character table and  aprime (see <a href="chap71.html#X7CADCBC9824CB624"><span class="RefLink">71java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>pjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

<h5>71.3-3 CharacterTableRegular</h5>

<="table "uncwidth=<> "">#8227;<> <class=A"tbl</> < class""</> pan ="br/java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 156
<p>For

<p>li<>Most div</div>


< class="example"><pre>
<GAPprompt>&span>span class="GAPinput">g:= SymmetricGroup( 4 );</span>
ym[ 
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );;  HasIrr( tbl );</span>
false
<span class="GAPprompt">gapspan<span class
BrauerTable Sym [1 .. 4]) )
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<"gapgt;<pan>< ""tblmod2 BrauerTable( ,2 ;/>r/<class"> /<a="chap71htmlX"71.-2 java.lang.StringIndexOutOfBoundsException: Range [155, 144) out of bounds for length 155
true
<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = BrauerTable( java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
true
<span class="GAPprompt">gap></span> <span class="GAPinput">libtbl:= CharacterTable( "M" );</span>
CharacterTable( "M" )
<span class="GAPprompt">gap></spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
BrauerTable( "M", 2 )
<span class="GAPinput">rauerTable
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">ass/
CharacterTable( "Sym(4)" )
< =GAPprompt>gt<span <panclass="GAPinput>omputedBrauerTables(  an java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 166
[ , BrauerTable( Sym( [ 1 .. 4 ] ), 2 ) ]
<pre</>

<p><a id="X7DBEF4BF87F10CD6" name="span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib" );</span>

<h5>71.3-4 SupportedCharacterTableInfo</h5>

<div class="func"><
<dCharacterTableInfo>  list contains at position <span class="SimpleMath">3i-2</span> an

<>hen(ordinary or ) java.lang.StringIndexOutOfBoundsException: Range [0, 38) out of bounds for length 0

<p>New java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0

<p><a id="X8195BC057B1DFAD5" name="X8195BC057B1DFAD5"></a></p>

<h5>713-ConvertToCharacterTable/h5java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

<div class="func"><table class=as/>
<div class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p></code> converts <var Arg"</> java.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 63

<>  in< classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><code class="func">ConvertToCharacterTableNC

<p>An java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0

<p><a id="X789FAC077AEF088A" name="X789FAC077AEF088A"></a></p>

<h4>71.4 <span class="Heading">Character Table Categories</span></h4>

<p><a id="<p>For an ordinary character table <var class="Arg">tbl</var> with known underlying group <span class="SimpleMath">G</span>, <code java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 0

<>71canjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 366

<><aid"F8AB7CB7A46002Fjava.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 62
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 225
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 224
< =f"<class"unc ="%">tr>< class""<code "func>#8227;IsBrauerTable</java.lang.StringIndexOutOfBoundsException: Range [121, 120) out of bounds for length 222
<div class="func"><table class="func" width<pan"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 125
<> <> <GAPinput:(g)   #thestoresthejava.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 124

<ajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

dcode ="IsBrauerTablecode   tables provide enough informationtocompute power maps and irreducible characters (and in the case of Brauer tables to get java.lang.StringIndexOutOfBoundsException: Index 254 out of bounds for length 4

<p>The character table objects in <code class="func">IsCharacterTable</code> are

<p>The <strong class="pkg">GAP</strong> objects in < ,1 ,3, java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17


<div class=
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );;</span>
<=GAPprompt>&t</>span=GAPinput"bl= CharacterTable g);  := 2</pan>>
CharacterTable( Sym( [ 1 .. 4 ] ) )
BrauerTable Sym([1 .  ],  java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<span class="GAPprompt">gap></span> <span class="GAPinput">
true
true
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0
true
false
<span class="GAPprompt">gap></span>p fordealingwith Clifford java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 0
true
false
<span>java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 361
false
false
</pre></div>

<p><a id="X7C6F3D947E5188D1" name="X7C6F3D947E5188D1"></a></p>

<h5>71.4-2 InfoCharacterTable</h5>

<div class=
<p>is the info class (see <divclass= in generalfive different ways to get a character table in <strong class="pkg">GAP</strong>. You can</p>

<="X7FA867637EBB30F9"></a></p>

<h571.- java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class="func" 100%tr>tdjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<s in this family (see <a href=

<p><a id="X829C4B6E83998F40" name="java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 5

<h4>/>

<p>The following few java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0


<>
<li><p>The class of the <em>identity element<<ijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

</li>
<li><p>The <em>trivial character<p><a id="X7FCA7A7A822BDA33" name="X7FCA7A7A822BDA33"></a></p>

</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<> that take character  as an argument andworkwithcharacters expect

</li>
<li><p>For a stored class fusion, the image table is denoted by its <code class="func">whose div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTable</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>

</li>
</java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
<>a X793"name=793java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 62

<h4>71.>reason why < ""<code>java.lang.StringIndexOutOfBoundsException: Range [1, 63) out of bounds for length 5

<p>For a character table with underlying group (java.lang.StringIndexOutOfBoundsException: Range [0, 51) out of bounds for length 26

<>Character  class="GAPpromptgap&</>span=GAPinputtblCharacterTable"A5 ;/span>

<p>However, given a group <span class="SimpleMath

<>Tasks may be delegatedfrom a  itscharacter table or vice versa only if these three attribute values are stored in the character table.</p>

<>a="=F4826A82B667AF"a><java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h5>71.6-1 UnderlyingGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func[  
<p>For an ordinary li><p>The group is <var class="Arg">p</var>-solvable (see <code class="func">IsPSolvable</java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 35

<p>Brauer tables do not store the underlying group, they access it via

<p><a id="X849A38F887F6EC86" name="X849A38F887F6EC86"></a></p>

<h5>71.6-2 ConjugacyClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227;java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 17
<Fora  var"">tbl/> with known underlying group <java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 514

<p>This ordering need <em>not</em> coincide with>The < class="<code>>operator  character table and a prime (see <a href="chap71.html#X7CADCBC9824CB624"><span class="RefLink">71.7</span></a>) delegates to <code class="func">BrauerTable</code>.</p>

<p>The class of java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0

<p>If <var class="Arg">tbl</var> was constructed from <span class="SimpleMath">G</span> then the conjugacy classes have been stored at the same time when <span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Brauer tables do not store

<p><a id="X84DC12AA804C8085" name="X84DC12AA804C8085"></a></p>

<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 48

<div class=((  [1 .4])  )
<> java.lang.StringIndexOutOfBoundsException: Range [35, 28) out of bounds for length 562


 =example">pre>>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );;</span>
<=>&;span> <span class=">strong "ark>"">classArg"tbl</>mod varclass""p<ar<ode<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4
<span( g, x ) );;java.lang.StringIndexOutOfBoundsException: Range [1, 122) out of bounds for length 4
<span class="GAPprompt">gap></span> <span class="GAPinput">SetConjugacyClasses( g, ccl );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );;   # the table stores already the values</BrauerTable(""  )
<span class="GAPprompt">gap></span> <span class="GAPinput">HasConjugacyClasses( tbl );  HasUnderlyingGroup( < class"GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "Symmetric", 4 );</span>
true
true
<java.lang.StringIndexOutOfBoundsException: Range [77, 5) out of bounds for length 96
true
<span class="GAPprompt">gap&  BrauerTable( ([1 . ] ) 2 )java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
true
<span class="
[ 5, 1, 2, 3, 4 ]
</pre></div>

<p><a id="X8788C6C7829C1ADE" name="X8788C6C7829C1ADE"></a

<h5>71.6-4 CharacterTableWithStoredGroup</h5>

le=func"width100%%"<tr<java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 295
<p>Let <var class="Arg""G</var>be agroup and <var class="Arg">tbl</var> a character table of (a group isomorphic to) <var class="Arg">G</var>, such that <var class="Arg">G</var> does not store its <code class="func">OrdinaryCharacterTable</code> (<a href="chap71.html#X8011EEB684848039"><span class="RefLink">71.8-4</span></a>) value. <code class="func">CharacterTableWithStoredGroup</code> calls <code class="func">CompatibleConjugacyClasses</code> (<a href="chap71.html#X790019E87CFDDB98"><span class="RefLink">71.6-5</span></a>), trying to identify the classes of <var class="Arg">G</var> with the columns of <var class="Arg">tbl</var>.</p>

< thisidentification is unique up automorphismsof <ar class=A"tbl/java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 995

<p>Otherwise, i.e., if <strong<p>Several <em>for </em   alsofor  tables./>

<>fpresentasthe< ""info/var java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 297

<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 491

<p><a id="X790019E87CFDDB98" name="X790019E87CFDDB98"></a></p>

<h5>71.6-5 CompatibleConjugacyClasses</h5>

<class""table "" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CompatibleConjugacyClasses</code>( [<var class="Arg">G</var>, <var class="Arg">ccl<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Ifthe arguments <var  class=="Arg"G<var>and<var class=Arg></var>are present then <var class="Arg">ccl</var> must be a list of the conjugacy classes of the group <var class="Arg">G</var>, and <var class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<> <ar =>/ar> the argument  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The usual approach is that one first calls <code class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p>The following invariants are used.</p>

<ol>
<lip>lement orders (see <code class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
< class="SizesConjugacyClasses<<code> (< href=" class">pre>

</java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<li><p>power maps (see <code class="func">PowerMap</code> (<a href="chap73.html#X781FAA497E3B4D1A<spanclass=">&<span spanclass">((4 ,2 );/>

<lijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<><psymmetries ofthe  (see  =f">AutomorphismsOfTable/> <href="html#java.lang.StringIndexOutOfBoundsException: Range [124, 122) out of bounds for length 170

</li>
</ol>
< whose components describe additional information for the class identification. The following components are supported.</p>


<l>
<dt><strong class="Mark"><code class="code">natchar</code> </java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 12
<dd><p>if <span class="SimpleMath">G</span> is a permutation group or matrix group then the value of this component is regarded as the list java.lang.StringIndexOutOfBoundsException: Range [0, 142) out of bounds for length 0

<dd
<dt><strong class="Mark"><code class="code">bijection</codediv class=func"<table " width100%<trtdclass=tdleft"<code class="func">‣ Irr</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<dd><p>a list describing a partial bijection; the <span class="SimpleMath">i</span>-th entry, if bound, is the position of the <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNearlyCharacterTable</code>( class="tdright">( category )</td></tr></table></div>

</dd>
</dl>

<java.lang.StringIndexOutOfBoundsException: Range [19, 4) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AlternatingGroup( 5 );</span>
Alt   .  ])
<span class="GAPprompt">gap></span> <span class="div class="func"><table class="func" width="100%"><tr><tdtdleft"< ""&8227; java.lang.StringIndexOutOfBoundsException: Range [115, 114) out of bounds for length 222
CharacterTable( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">HasUnderlyingGroup( tbl );  HasOrdinaryCharacterTable( g );</span>
false
false
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibleConjugacyClasses( tbl );   # unique identification</span>
[
<span class="GAPprompt">gap></span> <span class="GAPinput">new:= CharacterTableWithStoredGrouppThe ordering ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
racterTable(Alt( [1 . 5 ]] ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 96
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HasConjugacyClasses( new );  HasUnderlyingGroup( new );</span>
true
true
<span class="GAPprompt">gap java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
[ 1, 2, 3, 4, 5 ]
<span class="GAPprompt">gap></spanclass="">gt;/span span class"APinput"tbl: CharacterTable java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 69
<span class="GAPprompt">ap></pan>span class="GAPinput
[ [ 17, 18 ],pan class="(Sym(   . )
</pre></div>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 4

<h4>71.7 <span class="Heading">Operators for Character

<>Thefollowinginfix operators java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 5


<dl>
<dt><strong class="Mark"><code class="code"><var class="Arg">tbl1</var> * <var class="Arg">tbl2</var></code></strong></dt>
<>oftwojava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 197

</dd>
<dt><strong class=div class="func"><table class="span class="GAPprompt">gap></spaspan class=">rr(CharacterTable(>  s hrefchap7.tmljava.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 146
<dd><p>the table of the factor group modulo the normal subgroup spanned by thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">tbl</var> mod  Character BrauerTable "  
<>p</var>-odularBrauercharacter table corresponding to the ordinary character table <var class="Arg">tbl</var> (see <code class="func">BrauerTable</code> (<a href="chap71.html#X8476B25A79D7A7FC"><span class="RefLink">71.3-2</span></a>)),</p>

</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">tbl</var>.<var class="Arg">name</varpa id=X829C4B6E83998F40" name="X829C4B6E83998F40"></a></p>
<> position theclasswithname< =Arg>ame/var varclass=Arg"</var (ee ode class""lassNames</code> (<a href="chap71.html#X804CFD597C795801"><span class="RefLink">71.9-6</span></a>)).</p>

</dd>
<dl>
<p><a id="X7F9D58208241D35E" name="X7F9D58208241D35E"></a></p>

<h4>71.8 <span class="Heading">Attributes and Properties for Groups and Character Tables</span></h4>

<p>Several <em>attributes 

<p>These are first those /i>

<< ="">&/pan <class="LinearCharacters (4))<span>

<p><a id="X81FEFF768134481A" nameCharacter (([1.  ] ,[1,-,1 , -

<h5>71.8-1 <pre/>

<divjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>Inthe java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 592

<p>(The defaultclassf"java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 235

<p>For solvable groups, the default pCharacter tables constructed from groups know these values upon construction, and for character tables constructed without groups, these values are usually not known and cannot be computed from the table.</p>


<div
<span class="GAPprompt"pSince Brauer charactertablesare constructed  ordinary  valuefor< class">< java.lang.StringIndexOutOfBoundsException: Range [136, 135) out of bounds for length 231
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 112
[ [ 1, 1 ], [ 2, 1 ] ]
<span class="GAPprompt">gap></span><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 1, ,] 41,[5  java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 118
[ [ 1, 1 ], [ 2, 2 ], [ 4, 1 ] ]
<><pre/divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p" name="X873B3CC57E9A5492"</</java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h5>71.8-2 <span class="Heading">Irr</span></h5>

<div<ivclass=""< "unc "00%>trtd class">class">8227;/ode( <varclass"></td class="drightjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div classdiv class=func>tableclass=func"=100%"tr<class=tdleft>codeclass=func"ãCommutatorLengthcode>( <var class=t</> <td>td class="dright>& f"table 100<>td =t"code"&8227;ConjugacyClasses<code(<var =A"><>)/dtdjava.lang.StringIndexOutOfBoundsException: Range [178, 177) out of bounds for length 226
<p>Called with a group<varclass==Arg>/var> < class="func>Irr</ returns the  characters theordinary character table of <var class="Arg">G</var>. Called with a group <var class="Arg">G</var> and a prime integer <var class="Arg">p</var>, <code class="func">Irr</code> returns the irreducible characters of 

<p>For a character pIf <ar"rg>tbl</var was constructed from <span class="impleMathG</span> then the conjugacy classes have  stored at the same time when <span class="SimpleMath">G</span> was stored. If <span class="SimpleMath">G</span> and <var class="Arg">tbl</var> have been java.lang.StringIndexOutOfBoundsException: Index 287 out of bounds for length 0

<p>(If java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<> thatthe <>7163 </h5>

<p ordering   entriesinattribute< class=><code>of java.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 287


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup
 java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 67
     ] <class"GAPprompt&;/ span class=GAPinput">: List(repres, x & ConjugacyClass(<="" ="unc width="100%><tr>td ""< =""&8227;IsMonomial/code(varclasstbl/<r> )</td><td class="tdright">( propertynbsp;)</td></tr></table></div>
    [ 3, -1, -1, 0, 1 ] ),
  Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 2, 0, 2, -1, 0 ] )
    , Character( CharacterTable( Sym( [ 1 .. 4 ] ) ),
     ,-0- 
  Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] )
 ]
<span java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
[ ((([1.    ,  ] ),
  Character( BrauerTable( Sym( [ 1 < class""table= 100"<>< classt"<class=#;java.lang.StringIndexOutOfBoundsException: Range [119, 117) out of bounds for length 225
< "APprompt"gap;/pan <spanclass=Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 99
[ haracter CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ),
  ( CharacterTable(A5 )
    [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] 0 ">codeclass="">&8227 Size</code> < class"rg"tbl/var>)/java.lang.StringIndexOutOfBoundsException: Range [147, 146) out of bounds for length 214
  Character CharacterTable A5 ),
    [3,-,0, -E5^(5^,-()(5)^,
  Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 (java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 59
< class">gapg;<span>  <span class=">
[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ),
   Character BrauerTable A", 2),
    [ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ),
  Character( BrauerTable( "A5", 2 ),
    [ 2, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ),
  Character( BrauerTable( "A5", 2 ), < class"APprompt>/pan  ="GAPinput>(tables  );>
</pre></div>

<  ,,60java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

<h5>71.8-3 <spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 100

<functable=funcwidth=100%">tr>td class"< ="gap&t;<> =GAPinput>(,IsCyclic );</span>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Range [0, 85) out of bounds for length 29
<>codeclass=func></ returns thelinear(.. degree < class=SimpleMath1/span>)  in the<code class></code (a href=chap71.html#X873B3CC57E9A5492<span class="RefLink">71.8-</>/a> listof  var ="G     =><var  In  ,<java.lang.StringIndexOutOfBoundsException: Range [378, 377) out of bounds for length 520

<p>For a character table <var class="Arg">tbl</var> with underlying group, <code class="func">LinearCharacters</code> may delegate to the group. For a group <var class="Arg">G</var>, <code class="func">LinearCharacters</code> may delegate to its character table only if the irreducibles are already stored there.</p>

<p>The ordering of linear span class="GAPprompt">gap> class=>(  )</span


< =example<pre
<spanclass="APprompt">gap></span> <span class="GAPinput">LinearCharacters( SymmetricGroup( 4 ) );</span>
  java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 70
  Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1
     ] ]
</pre>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X8011EEB684848039" name="X8011EEB684848039"></a>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>71.8-4 <span class="Heading">OrdinaryCharacterTable G">g;/>""< <ajava.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 167

=>tableclass=func ="java.lang.StringIndexOutOfBoundsException: Range [13, 5) out of bounds for length 104
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color: green'>3, ,5 ]
<p><code class="

<Since java.lang.StringIndexOutOfBoundsException: Range [33, 25) out of bounds for length 231


<div class="example"><pre>
< "APprompt>java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 115
aracterTable Sym( 1. 4 ]))
<span class="GAPprompt">gap></span<li>
 A52java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
true
</predd>>ifspan ="SimpleMath"G a group then  ofjava.lang.StringIndexOutOfBoundsException: Range [115, 105) out of bounds for length 400

<p><a id="X81EFD9FE804AC6EE" name="java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 24

<h5>71.8-5 <span class="Heading">Group Operations Applicable to Character Tables</span></h5>

<iv class=func>< class=f" =100"<trtd> ">&8227;> <<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianInvariants</code>( <var class
< class==func>tableclass="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Range [0, 64) out of bounds for length 24
< class="=func width="00%"<r class=tdleft"<java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 218
iv "func"<class" "%><tr class">codeclass="func">‣ IsAbelian</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func">java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
< class"func"><table=func width=100"<tr><put>: AlternatingGroup( 5)<spanjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
<="nc"<table ==func"width"%>tr>td class="tdleft">codeclass=func>&8227 IsElementaryAbelian<code> var ""tbl/ )/>td class=tdright&bspproperty<class="&;/ span java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 135
<div class="panclass==GAPprompt">gap></span> <span class="GAPinput">HasUnderlyingGroup( tbl );  HasOrdinaryCharacterTable( g );</span>
<spanclass="APprompt"gap&<<"A2AD83BC58A0 =X7995>/>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 220
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPerfect</code>( <var class="Arg">tbl</var> )</CharacterTable( Alt( [ 1 .. 5 ] ) )
<div class="func"><table class="func" width="100%"><tr><td class="tdleft<pan class="GAPprompt">gap></span> <span class="GAPinput">HasConjugacyClasses( new );  HasUnderlyingGroup( new );</span>
=func" width="100%"tr><tdclass="left>="func">#; IsSimple/(<var"Arg"<var>)/><class"tdright"(nbsp&bsp;<td<tr<table<divjava.lang.StringIndexOutOfBoundsException: Index 217 out of bounds for length 217
<div="func"<table class="func" width
<div class="func"><table class="func" width="100%"><< class=GAPprompt"gap&;/pan> <panclass="GAPinput">CompatibleConjugacyClasses( CharacterTable( "J2" ) );</span>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSupersolvable</code>( <var class="Arg">tbl</var> )</td><td
<class=func"< = =100">trtdclass"dleft"<class=func&#8227;IsomorphismTypeInfoFiniteSimpleGroup<code>( <var class="Arg">tbl<var> )</td>td class="tdright">(nbsp&;)<td>/></><>
<div class="func"><table class="func" width="<h4>717  <panclass="Heading"Operatorsfor<>alist of orders of representatives of conjugacy classes of the character table <var class="Arg">tbl</var>, in the same ordering as the conjugacy classes of <var class="Arg">tbl</var>.</p>
<div class="func"><table class="func" width="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< tables and meanthesamefor  table as for its underlying group; see Chapter <a href="chap39.html#X8716635F7951801B"><span class="RefLink">39</span></a> for the definitions. The operations are mainly useful for selecting character tables with certain properties, also for character tables without access to a group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tables:= [ CharacterTable( CyclicGroup( 3 ) ),</span>
<span class="GAPprompt">
<span class"APprompt>></pan> < ="">CharacterTable AlternatingGroup 5)),/>
<span class="GAPprompt">></span> <span class="GAPinput">               =func"#;SizesCentralisers</ode>(var class=Arg"><var )/>td java.lang.StringIndexOutOfBoundsException: Range [170, 169) out of bounds for length 227
<panclass="Pprompt>p></span> span ="GAPinput"List(tables );</pan>
[ [
<span class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<divclass="xample">pre
<span class="GAPprompt">gap></span> <span class="GAPinput"><span class="GAPprompt">gap></span> <span class="GAPinput">tbl"A5 );</spanjava.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
[ 3, 12, 30, 60 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(dt<strong class="Mark><codeclass"code"<var class=""tbl</var> mod <var class="Arg></ar<></strong></dt>
[ true, false, false, false ]
<pan =APprompt"gapgt;pan><span  class=">>( tables,IsAlmostSimple;/span>
[ false, false, true, false ]
<span class="GAPprompt">gap></span> <spanh5>.93SizesConjugacyClasses>
[ true, false, false, false ]
<span class="GAPpromptss=GAPinput">List , IsFinite )<span>
[ true, true, true, true ]

[ true, true, false, false ]
<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ true, false, false, false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List/ivclass=="xample"<pre>
[ false, false, true, true ]
<span class"GAPprompt">gapgt;</span> <pan"APinput">izesConjugacyClasses( tbl );</span>
[ false, false, true, true ]
<span class="h4>71.8 <span class>Attributes and Properties Groups Character Tables/>/>
<>Several <em><a id="X7C2753DE8094F4BA name="X7C2753DE8094F4BA></>/>
<span class="GAPprompt">gap></span> <span class="GAPinput
,false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsSupersolvable );</span>
[ true, , false, ]
< =""gap&t<span <span"List( tables, NrConjugacyClasses)<span>
[ 3, 5, 5, 9 ]<<aid="X81FEFF768134481A" name"X81FEFF768134481A"<a>/>
<class"GAPprompt"gap&</<span""List   rA"bl>   of   ="/>  (see also  class=f"TableAutomorphisms>(a=".#8082DD827C673138"<java.lang.StringIndexOutOfBoundsException: Range [301, 300) out of bounds for length 342
[3,24, 60,120 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
<spandiv class"func">class="unc" width"100"<<tdclass=tdleft<an =GAPprompt>&;/ span"APinput"tbl= ( "",8);</span>
java.lang.StringIndexOutOfBoundsException: Range [43, 3) out of bounds for length 69
 )
<span class="GAPprompt">gapspan class=GAPprompt>ap&;/spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 4
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup>></
rec 1  1 , 2]
  shortname :=/pre<divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<span class="GAPprompt">gap></span> <span class="GAPinput">div class="xample"<>aid=X7F58A82F7D88000A" name="X7F58A82F7D88000A"></a></p>
rec( name := "B(3,3) = O(7,3)", parameter := [<h5>.95 spanclass"<h5>71.9-5 <span class="Heading
  shortname := "O7(3)" )
<span class="APprompt"gap>/span><span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "A8" ) );</span>
ec divclass=="unc"<table class="unc width="00%><tr<td class=tdleft"><code class="func">‣ UnderlyingCharacteristic</code>( <var class="Arg">psi</var> )</td><td class="tdright">( attribute )</<p>For an ordinary character table <var class="Arg">tbl</var>, the result is <code class="code">0</code>, for a <span class="SimpleMath">p</span>-modular Brauer table <var class="Arg">tbl</var>, it is <span class=""rg"></> isequal to ofits underlyingcharacter table</>
  parameter := 8, series := "A", shortname := "A8" )
spanclass=="APprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "L3(4)" ) );</span>
rec name : "(2, =L(3,), parameter = [ 3,4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shortname := "L3(4)" )
</re></iv>

<"></java.lang.StringIndexOutOfBoundsException: Range [62, 57) out of bounds for length 62

<>71.9 <spanclass="eading"">Attributes and Properties only for Character Tables</span></h4>

<p>Thejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<p>The other attributes introduced in this section apply only to character tables, not to groups.</p>

<p><a id="X86F455DA7A9C30EE" name="X86F455DA7A9C30EE"></a></p>

<h5>71.9-1 OrdersClassRepresentatives</h5>

h5719-< eading>java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 76
<p>is a list of orders of representatives of conjugacy classes of the character table <var class="div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassNames</code>( <var class="Arg">tbl</var>[, <var class="java.lang.StringIndexOutOfBoundsException: Index 164 out of bounds for length 0


<div class="example
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
<spanclass=""gap&t</pan span ="APinput>( ;/>
[ 1, 2, 3, 5, 5 ]
</><div>

<p><a id="X7CF7907F790A5DE6" name="X7CF7907F790A5DE6"></a></p>

<h5>71.9-2 SizesCentralizers</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classpThe  the class java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 227
<p>is a list that stores at position <span class="SimpleMath">i</span> the size of the centralizer of any element in the <span class="SimpleMath">i</span>-th conjugacyCharacter( CharacterTable(Sym( [1. 4 ]) ,[2, 0 2 1  ]


<div class="example"><pre>
< "");span
<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 1, 1 ] ),
[60 4, 3 , 5]
</pre</>

<p><a id="X7D9D2A45879A6A62" name="X7D9D2A45879A6A62"></a></p>

<5>1<

<class="">able ""width=100%>tr>classtdleft> class">#8227SizesConjugacyClasses<code> var class=Arg"tbl/td<td class="tdright">( attribute )</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>is a list that stores at position <span java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0


<div class="example"><pre>
spanGAPpromptgapgt;<span <spanclass=GAPinput>bl:=CharacterTable A5 ;<span>
< ="GAPprompt>&t;/span><class=G">SizesConjugacyClasses)<spanjava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
[ 1, 15, 20, 12, 12 ]
</p>It on< class""var>whatthese  are so there  nodefaultto compute andcharacter ./pjava.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 140

<="X7C2753DE8094F4BA>/</p>

<h5>71.9-4 AutomorphismsOfTable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AutomorphismsOfTable</code>( <var class="Arg">tbl</var> )</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<=func>table=func""%><<"tdleft>code =func>#8227 </code>( var ="<var)/tdtd ="dright"(nbspattributenbsp;)</td></tr></table></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span <p>s astring  thatidentifies the character  <var class="Arg">tbl</var> in the current <strong class="pkg">GAP</strong> session. It is used mainly for class fusions into <var class="Arg">tbl</var> that are stored on other character tables. For character tables without group, the identifier is also used to print the table; this is the case for library tables, but also for tables that are constructed as direct products, factors etc. involving java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">AutomorphismsOfTable( tbl );</span>
Group([ (4,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">OrdersClassRepresentatives( tbl );</span>
[ 1, 4, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SizesConjugacyClasses( tbl );</span>
[ 1, 2, 1, 2, 2 ]
</pre></div>

<p><a id="X7F58A82F7D88000A" name="X7F58A82F7D88000A"></a></p>

<h5>71.9-5 <span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 113

<div class="func"><table class="func" width="100%"><tr><td</>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classp< id"7932C35180C80953 name"7932"<a>/p
<p>For an ordinary character table <var class="Arg">tbl</var>, the result is <code class="code">0</code>, for a <span class="SimpleMath">p</span>-modular Brauer table <var class

<p>The underlying characteristic must be stored when the table is constructed, there is no method to compute it.</p>

<p>We <p>is a muta stringwith informationabout the character table <var class="Arg">tbl</var>. There is no default method to create an info text.</p>


<div class="example"><pre>
<pan ">>/< GAPinput"tbl  "5 );<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingCharacteristic( tbl );</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingCharacteristic( tbl mod 17 );</span>
s classGAPprompt>&;<<panclass=">PrintInfoText(  CharacterTable "") ,"");</panjava.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
</pre></div>

<p><a id="X804CFD597C795801" name="X804CFD597C795801"></a></p>

<h5>71.9-6 <span class="Heading">Class Names and Character Names</span></

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ClassNames</code>( <var class="Arg">tbl</var>[, <var class="Arg">"ATLAS"</var>] )</td><td class="tdright">( attribute )</td></tr><<div ="func>#;InverseClasses<code> <class=Arg"tbl</var )/java.lang.StringIndexOutOfBoundsException: Range [158, 156) out of bounds for length 224
< classfunc"width=100"<><d ="tdleft"><code class="func">&#8227; CharacterNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
>classfunc"<>  < =func"</>   , one each  or character ,ofGAPprompt>gt<>>spanclass=GAPinput>nverseClasses CharacterTable"java.lang.StringIndexOutOfBoundsException: Range [100, 97) out of bounds for length 110

<p>The default method for

<p>The default method for <code class="func">CharacterNames</code> returns the list <code class="code">[ "X.1", "X.2", ... ]</code>, whose length is java.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 12

<p>The position of the class with java.lang.StringIndexOutOfBoundsException: Range [0, 38) out of bounds for length 28

<"code">"ATLAS"</code>, theclass names returned obey conventionused in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 5]</a>. If one is interested in <q>relative><ode =func"LinearCharacters/code returns thejava.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 520


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
<pan class="GAPprompt">gap></span> <span class="GAPinput">ClassNames( tbl );</span>
[ "1a",2,a 5a,5b"]
<span class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
2
</pre></div>

<> ="java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

<h5>//

<class=func>java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 225
<div class="func"><table class="func" widthh5>.12<h5
<p>The values of these attributes are lists div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code <divclass="func">table =""width=100%"<tr><d "><code class=f"&8227; OrdinaryCharacterTable>(< class"Arg">"<)>java.lang.StringIndexOutOfBoundsException: Range [167, 166) out of bounds for length 230

<p>It depends on <var class="Arg">tbl</var> what these parameters are, so there is no defaultp< class""OrdinaryCharacterTablecode theordinary charactertable groupvar""Gvar>  theBrauercharacter  var ></var>,respectively<p>

<p>For example, 

<p>If <var class="Arg">tbl</var> is a <span class="SimpleMath">p</span>-modular Brauer table such that class parameters are stored in the =GAPprompt>&java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0

<p><a id="BrauerTable( "A5", 2 )

<h5>71.9-8 Identifier</h5>

<div class="func"><table class="func" width="100%"><tr>
<p>is a string that identifies the character table <var class="Arg">tbl</var> in the current <strong class="pkg">GAP</strong> session. It is used mainly for class fusions

<p>The default method for ordinary tables constructs <div class="func"><table class="func" width="100%"><tr><td="tdleft">code class=func"&divclass="func>f=%><java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 220

<p>The default method for Brauer tables returns the concatenation of the identifier of the ordinary table, the string <code class="code">"mod"</code>, and the (string of the) underlying characteristic.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Identifier( CharacterTable( "A5" ) );</span>
"
span =>apg;/span<span ">:=CharacterTable( Group( ( ));</>
<span class="GAPprompt">gap>span class=GAPprompt>gap></span> <span class="GAPinput">ClassRoots( CharacterTable( "Cyclic", 6 ) );</span>
"CT8"
"CT8mod2"
</pre></div>

<p><a id="X7932C35180C80953" name="X7932C35180C80953"></a></p>

<h5>71.9-9 InfoText</h5>

< class">table ="unc width=%"<>tdclass="tdleft"><code class="func">‣ InfoText</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>is a mutable string with information about the character table <var class="Arg">tbl</var>. There is no default method to java.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 62

<p>This attribute is used mainly for library tables (see the manual of the <strong class="pkg">GAP</strong> Character Table Library). Usual parts of the information are the origin of the table, tests it has passed (<code class="code">1.o.r.</code> for the test of orthogonalityjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<iv "example"><pre>
classGjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 119
p><a ="X850C7D947B3DBFA2" name="X850C7D947B3DBFA2"></a></p>
</>/div

<p><a id="X7919E2897BE8234A" name="X7919E2897BE8234A"></a></p>

<h5>71.9-10 InverseClasses</h5>

<divclass=func"<class=f"width=100">tr< class"dleft"<code=func>8227; <code(var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a character table <var class="Arg">tbl</var>, <code class="func">InverseClasses</code> returns the list mappingdivclass=func"<tableclass=" "00%tr>td java.lang.StringIndexOutOfBoundsException: Range [66, 64) out of bounds for length 251


<div class="div =""< "java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 228
<span class="GAPprompt">gap></span> <span class="GAPinput">InverseClasses( CharacterTable( "A5" ) );</span>
[ 1, 2, 3, 4, 5 ]
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 117
[1, 3 2  ]
</pre></div>

<p><a id="X87FF547981456932" name="X87FF547981456932"></a></p>

<<iv =example>java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

< "unc"<funcwidth="100">tr<td class"tdleft"><codeclass=func"ãRealClasses</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<>For  character table <var class="Arg">tbl</var>, <code classtable <var class"Arg"tbl/var>, < class="unc"RealClassescode returns  strictlysorted list of positions ofspan class=GAPprompt"gap&t/&gt;<spanclass=GAPinput"ClassPositionsOfNormalSubgroups )</>

<p>An element <span class="SimpleMath">x</span> is <em>real</em> iff it is conjugate to its inverse <span class="SimpleMath


<div class="example"><pre>
< class=G">ap>/span><span class="GAPinput>ealClasses CharacterTable "A5" ));/span
[,[,,30 
<class">&t;/span> <span class="GAPinput">RealClasses( CharacterTable( "Cyclic", 3 ) );</span>
[ 1 ]
</pre></div>

<p><a id=s="gt<span>> span "GAPinput"List tables, IsCyclic );</span>

<h5>71.9-12 ClassOrbit</h5>

< java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 26
<p>is the list of positions of those 

<p>This<><a id="7" "7>/a</>


 classexample>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassOrbit( CharacterTable( "A5" ), 4 <div cl="func>table width"100%"<tr<class=tdleft>< =func"&#8227;ClassPositionsOfCentre<code>(< ""java.lang.StringIndexOutOfBoundsException: Range [156, 155) out of bounds for length 235
[ 4, 5 ]
</pre></div>

<p><a id="X7F863B15804E0835" name="X7F863B15804E0835"></a></p>

<5>71.-3 ClassRootsh5

<div class=func> "uncwidth100%"><td=java.lang.StringIndexOutOfBoundsException: Range [74, 72) out of bounds for length 220
<p>For a character table <var class="Arg 3,24 60,120]

pThisinformation  the power mapsof varclass=A>tbl<var.<p


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassRoots( CharacterTable( "A5" ) );</span>
[[2,3,4 5 ][ ],[  ,[], [ ]]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassRoots( CharacterTable( "Cyclic", 6 ) );</java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
</prepLet <varclass"rgvarbeordinary charactertable  group spanclass=>( name :=(,)  (7)", : [3  ,java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 69

<p><a id="rec( name := "A(8) ~ A(3,2) = L(4,2) ~ D(3,2) = O+(6,2)",

<h4>71.10 <span class="Heading">Normal Subgroups Represented by Lists of Class Positions</span></h4>

p following attributes foracharacter table <var class="Arg">tbl</var> correspond to attributes for the group <>correspondscodeclass"funcDerivedSubgroup<code> <a  href=".html"<spanclass<re<div>

<p><a id="X850C7D947B3DBFA2" name="X850C7D947B3DBFA2"></a></p>

h57110- /h5>

<div class="func"> attributesjava.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 101
<div class="func"><table class="func" java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ClassPositionsOfMinimalNormalSubgroups</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></table><
>   ="/> < >class">3919-<span, <java.lang.StringIndexOutOfBoundsException: Range [149, 148) out of bounds for length 490

pThe oftheresult are sortedaccording increasinglength.(So   orderrespects partial java.lang.StringIndexOutOfBoundsException: Range [136, 135) out of bounds for length 170


<classexample<>
< classjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 111
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalSubgroups( tbls4 );</span>
[ [ 1 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1 .. 5 ] ]
</prespan class=">&;</><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 118

<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>71.10-2 ClassPositionsOfAgemo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>corresponds to <code class="func">Agemo</code> (<a href="chap39.html#X83DB33747F069ACC"><span class="RefLink">39.14-2</span></a>) for the


<examplepre
<anclass="&t;/pan><span class=G"tbls4 CharacterTable( Symmetric, 4;<span
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[,3 4 ] ]
</pre></div>

<p><a id="X7A6B1F8A84A495DC" name=<pan class=GAPprompt"gapgt;</pan <span class=GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>

<5>71.103 ClassPositionsOfCentre<//5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="funch5>.- />
<div class="func">div ="func"< class="unc width="100"<tr>< class=="tdleft ="ãSizesConjugacyClasses<code>(<var java.lang.StringIndexOutOfBoundsException: Range [142, 141) out of bounds for length 231
<p>corresponds pis alistthat stores at >71.10-8 ClassPositionsOfLowerCentralSeries


<div class="example"><pre>
< "APprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "<pan class="APpromptjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 98
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfCentrepre>div>
[ 1, 3 ]
</pre></div>

<p><a id="X7D53F60785AB22B1" name="X7D53F60785AB22B1"></a></p>

<h5>71.10-4 ClassPositionsOfDirectProductDecompositions</h5>

<class"unc"<table=fjava.lang.StringIndexOutOfBoundsException: Range [44, 36) out of bounds for length 288
<p>Let <var class="Arg">tbl</var> be the span class="GAPprompt">gap></span> <spanGAPinput">ClassPositionsOfLowerCentralSeries( tbld8 );</span>

<p><a div=example

<h5>71.10-5 <span class="GAPprompt">gap></span>"> )/

"100>java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 244
<corresponds < class="unc"></code<a href"hap39.X7CC17CF179ED7EF2"><spanclass=RefLink>3912-/><a>  theof   < ="ordtbl/var><p>


<div class="example"><pre<iv class"func"<table class="func" width"100"<tr>tdtdleft"<codeclass="func">‣ ClassPositionsOfUpperCentralSeries</code>( <var class="Arg 1,2, ,2 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfDerivedSubgroup( tbld8 );</span>
[ 1, 3 ]
<pre/

<p><a id="span class="GAPprompt>gapgt</span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>

h5>.10-6ClassPositionsOfElementaryAbelianSeries>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ClassPositionsOfElementaryAbelianSeries</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( <p>For an ordinary character table <var class="Arg">tbl</var>, the result is <code class="code">0</code>, for a <span class="SimpleMath">p</span>-modular Brauer table <var class="Arg">tbl</var>, it is <span class="SimpleMunderlyingjava.lang.StringIndexOutOfBoundsException: Range [358, 357) out of bounds for length 368
<p>corresponds to <code class="func">ElementaryAbelianSeries</code> (<a href="chap39.html#X83F057E5791944D6"><span class="RefLink">39.17-9</span></a>) for the group of the ordinary character table <var class="Arg">ordtbl</var>.</p>


java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 26
<""><spanspan ="tbls4(Symmetric,4);<span
<span class="GAPprompt">gap></span> <span class="GAPinput">tbla5:= CharacterTable( "A5" );;</java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 1
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfElementaryAbelianSeries( tbls4 );</span>
[ [ 1 .. 5 ], [ 1, 3, 4 ], [ 1, 3 ], [ 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfElementaryAbelianSeries( tbla5 );</span>
fail
<<>

<p><a id="X7D2A55A584F955DB" name="<div class=func"<tableclass"unc"="100"<><java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 254

<h5>71.10-7 ClassPositionsOfFittingSubgroup</h5>
pcode=>java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 300
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 131 out of bounds for length 49
<> (<a =h>< ="39125/>/>)    of the   table  class="ordtbl.<p


<class=examplejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput
< class"GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfFittingSubgroup( tbls4 );</span>
 ,3java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
</pre></div>

<p><a id="X79AEFC4384769B72" name="X79AEFC4384769B72"></a></p>

<h5>71.10-8 ClassPositionsOfLowerCentralSeries</h5>

<div class="func"><table span class="GAPprompt">gap/ span class"APinput"tbl.2a;/>
<p>corresponds to <code class="func">LowerCentralSeriesOfGroup</code> (<a href="chap39.html#X879D55A67DB42676"><span class="RefLink">39.17-11</span></a>) for the group of<pre></div>


<div class="example"><pre>
<class""gap;<span spanclass"GAPinput">tbls4:= CharacterTable
< "GAPprompt">gap>/span> span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CharacterParameters</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute
[  1 ..5 ,[ 1,,4]java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
<span class="GAPprompt"< class"">gap&t/> <panclass=GAPinput"ClassPositionsOfSupersolvableResiduumjava.lang.StringIndexOutOfBoundsException: Range [116, 106) out of bounds for length 116
[ [ 1 .. 5 ], [ 1, 3 ], [ 1 ] ]
</pre></div>

<< =X86065D217A36CD9B=java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<h5>7110-2 ClassPositionsOfPCore>

< =func>tableclass width="100"<trtdclass=">code class="func">‣ ClassPositionsOfUpperCentralSeries</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
p tocodeclass""UpperCentralSeriesOfGroup>< ".#8428592E8773CD7B""span class=RefLink"39.1712/pan/a)  thegroup  the ordinary  table< classArgordtbl/>./p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:= CharacterTable( "Dihedral", 8 );;</span>
< "APprompt"gap&t</>[ ]
[<pre<div
<spanclass==GAPprompt
[ [ 1, 3 ], [ 1<><a id"FCF905D7FD7CC40" name="X7FCF905D7FD7CC40"
</pre></div>

<p><a id="X877FDE8A84A9F52C" name="X877FDE8A84A9F52C"></a></p>

<h5>71.10-10 ClassPositionsOfSolvableRadical</h5>

< =func"< "func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassPositionsOfSolvableRadical</code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<>to<=java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 223


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfSolvableRadical( CharacterTable( "2.A5" ) );</span>
[ 1, 2 ]
</pre></div>

<p><a id="X8392DD5B813250A4" name="X8392DD5B813250A4"></a></p>

<>.-11</h5


<p>corresponds to <code class="func">SupersolvableResiduum</code> (<<>For an ordinary java.lang.StringIndexOutOfBoundsException: Range [34, 28) out of bounds for length 200


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( "Symmetric", 4 );;</span>
<class="APprompt">gapgt;<span span class="GAPinput"dt<rong "rk>code class=odeblock/ode>/strong>><dt>
[1,3]
<pre</>

<p><a id="X7BBE7EBA7A64A6B0" name="X7BBE7EBA7A64A6B0"></a></p>

<p"name="87F547981456932"></a></p>

<div e class="func width>d>
<> to<


<div class="example"><pre>
<pan class="GAPprompt">g;/ <class="APinput>tbls4:=( Symmetric" 4 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput<t><strongclass=Mjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 73
[,3]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfPCore( tbls4, 3 );</span>
[ 1 ]
</pre></div>

<p><a id="X7FCF905D7FD7CC40" name="X7FCF905D7FD7CC40"></a></p>

<h5>71.10-13 ClassPositionsOfNormalClosure</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ClassPositionsOfNormalClosure</code><dd>
<isthe list ass""<<> and<code class="ode>centralcharacters/code>  used by <code class="func">SameBlock</code> (<a href="chap71.html#X7E80E35985275F35"><span class="RefLink">71.11-2</span></a>).</p>


< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
<, 4 );;</span>
<span classpThis  is computed fromthe maps ofvar =Arg><var>.<</java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
[ 1, 3, 4 ]
<//>

<p><a id="X8733F0EA801785D4" name="X8733F0EA801785D4"></a></p>

<h4>71.11 <span class="Heading">Operations/pre>/divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p><a id="X7ACB9306804F4E3F" name="X7ACB9306804F4E3F"></a></p>

<h5>71.11-1 PrimeBlocks</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimeBlocks"</span>the  of<divexample">>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PrimeBlocksOp</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>an ordinary character table<varclass=Arg"ordtbl</ara  primeinteger <varclass="Arg">p</var>, <code class="func">PrimeBlocks</code> returns a record with the following components.</p>


<dl>
<dt><strong class="Mark"><code class="code">block</code></strong></dt>
<dd><p>a list, the value <span class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</dd>
 M<="<>/trong>/dt>
<>>  containing positionspan""ispan thedefect  the<span class"impleMath">i</span>-th block,</p>

</dd>
<dtp>code class=func"SameBlock</code> returns <code class="keyw">true</code> if <var class="Arg">omega1</var> and <var class="Arg">omega2</var> are equal modulo any maximal ideal in the ring of complex algebraic integers containing the ideal spanned by <var class="Arg">p</var>, and <code class="keyw">false</code> otherwise.</p>
<dd><p>a list containing at position <span class="SimpleMath">i</span> the height of the <span class="SimpleMath">i</span>-th irreducible character of <var class="Arg">ordtbl</var> in its block,</p>

</dd>
<dt><strong class="Mark"><span class="GAPprompt">gap>=>java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 48
<dd><p>a list of class positions such that only the restriction to thesefalse

</dd>
<dt>p< ="7FF4CE4A7A272F88="<a<p>
<dd><p

</dd>
<dl>
<p>The components <code class="code">relevant</code> and <code class="code">centralcharacters</code> are used by <code class="func">div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlocksInfo</code>( <var class="Arg"(nbspattribute )/></>/><java.lang.StringIndexOutOfBoundsException: Index 223 out of bounds for length 223

<p>If <code class="func">InfoCharacterTable</code> (<a href="chap71.html#X7C6F3D947E5188D1"><span class="RefLink">71.4-2</span></a>) has

<p>The default method uses the attribute <code class="func">ComputedPrimeBlockss</code> for storing the computed value java.lang.StringIndexOutOfBoundsException: Range [0, 121) out of bounds for length 0

<p>Two ordinary irreducible characters <span class="SimpleMath">χ, ψ</span><l>

<p>For <span class="SimpleMath">|G| = p^a m</span> where <span class="SimpleMath">p</span> does not divide <span java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 35

<p>The <em>height</em> of a character <span class="SimpleMath">χ</span> in the<t>strong class"Mark"<class=codejava.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 0


<div class="example"><pre>
 cterTable("3( );/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pbl:= PrimeBlocks( tbl, 2 );</span>
rec( block := [ 1, 1, 1, 1, 1, 2 ],
  centralcharacter := [ [ ,, 56,, 24 ], [ ,, -7,, 3 ] ],
  defect := [ 3, 0 ], height := [ 0, 0, 0, 1, 0, 0 ],
relevant  =3,])
</pre></div>

<p><a id="X7E80E35985275F35" name="X7E80E35985275F35"></a></p>

<h5>71.11-2 SameBlock</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SameBlock</code>( <var class="Arg">p</var>d<> listof positions ordinarycharactersintheblock whose restrictionto< class=Arg></>ismaximally independent  to <codeclass"code"( OrdinaryCharacterTable varclass=""modtbl/var))/><p
<p>Let <var class="Arg">p</var> be a prime integer, <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 26

<p><code class="func">SameBlock</code> returns <code class="keyw">true</code> if <var class="Arg">omega1</var> and <var class="Arg">omega2</var> are equal modulo </


<div class="example"><pre>
<span java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
< class=GAPprompt>&;/span <dt< class="">codeclass=code><</trong></>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SameBlock( 2, omega[1], omega[6], pbl.relevant );</span
false
</pre></div>

<p><a id="X7FF4CE4A7A272F88" name="X7FF4CE4A7A272F88"></a></p>

<h5>71.11-3 BlocksInfo</h5>

<java.lang.StringIndexOutOfBoundsException: Range [111, 4) out of bounds for length 223
<p>For a Brauer character table <var class="Arg">modtbl</var>, the valuedefect : 3 modchars: [3],


<dl>
<dt="code"/>/><>
<>p java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 0

</dd>
<dt><strong class="Mark"><code class="code">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<>thelist ofpositionsoftheordinarycharacters belongtothe block,relativeto code="ode>( OrdinaryCharacterTable  class="Arg">modtbl</var> ) )</code>,</p>

</dd>
<</codestrong</
<dd><p>the list of java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of bounds for length 0

</d
</dl>
<p>Optional components are</p>


<dl>
<dt><strong class="Mark"><code class="code">basicset</code></strong></dt>
<dd><p>a list of positions of ordinary characters in the block whose restriction to <var class="Arg">modtbldiv =func"<table class==func width=100"<>td class=tdleft"><ode "unc>#8227 ClassPositionsOfElementaryAbelianSeries<code>( <var class="Arg">ordtbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>

</dd>
<dt><strong class="Mark"><code class="code">decmat</code></strong></dt>
<dd><p>the decomposition matrix of the block, it is stored java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0

<dd>
<dt><strong class="Mark"><code class="code">decinv</code></strong></dt>
<dd><p>inverse of the decomposition matrix of the block, restricted to the ordinary characters described by <code class="code">basicset</code>,</p>

</dd>
<dt><strong class="Mark"><code class="code">brauertree</codespan ="GAPprompt">gapgt</<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 110
<dd><p>a list that describes the Brauer tree ofBrauerTable("L3(2", 2 java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

</dd>
<dl

<ivclass="">pre
<span class="GAPprompt">gap></span> <span class="GAPinput">BlocksInfo( CharacterTable( "L3(2)" ) mod 2 );</span>
[ rec( basicset := [ 1, 2, 3 ],
      decinv := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],[  1 0 0 ] [ 0,1,0],[ 0, , [0,1 1], [1,1,1 ]]
      defect := 3, modchars := [ 1, 2, 3 ],
      :[1 2, 3,4,5 ]),
  rec( basicset := [ 6 ], decinv := [ [ 1 ] ], defect := 0,
      modchars := [ 4 ], ordchars := [ 6 ] ) ]
</pre></div>

<p><a id="X84701640811D2345" name=o5>71.

<h5>71.11-4 DecompositionMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DecompositionMatrix</code>( <var class="Arg">modtbl</var>[, <var class="Arg">blocknr</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var

Calledargument ">/> returns decomposition matrix of < class=Arg"modtbl/> java.lang.StringIndexOutOfBoundsException: Range [157, 147) out of bounds for length 358

<

<pAn ordinary java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p><code class="span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfLowerCentralSeries( tbls4 );</span


<div class="example"><pre>
<span class="GAPprompt">gap/pre>/div>
BrauerTable( "L3(2)", 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">DecompositionMatrix( modtbl );</span>
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 1, 0 ],
  [ 1, 1, 1, 0 ], [ 0, 0, 0, 1 ] ]

[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ], [ 0, 1, 1 ], [ 1, 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DecompositionMatrix( modtbl, 2 );</span>
[ [ 1 ] ]
</pre></div>

span class=java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 18

<h5>71.11-5 LaTeXStringDecompositionMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LaTeXStringDecompositionMatrix</code
<pp><a=X877FDE8A84A9F52C =X877FDE8A84A9F52C>p>

<p>The optional argument

<p>The optional components <code class="code">nrows</code> and <code class="code">ncols</code> denote the maximal number of rows and columns per array;p> to class=f"<code> (a java.lang.StringIndexOutOfBoundsException: Range [0, 68) out of bounds for length 2

<p>If the component <code class="code">decmat</code> is bound in <var class="Arg">options</var> java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 2

<p>For∓\varphi_{


<div10 </h5
<span class="GAPprompt">gap></span> <span class="GAPinput">modtbl:= CharacterTable( "L3(2)" ) mod 2;;</span>
<span class=<hi_{amp amp;.&mp  \
\[
\begin{array}{r|rrr} \hline
 & {\tt Y}_{1}
 & {\tt YGpan=""gapgt; span=GAPinput>: CharacterTable"ymmetric,4);</pan>
 & {\tt Y}_{3}
 \rule[-7pt]{0pt}{20pt} \\ \hline
{java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
{ _2 &; .  & 1 a;.java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
{\tt X}_{3} & . & . & 1 \\
{ X_{}∓.&1  1\\
{\tt X}_{5} & 1 & 1 & 1 \rule[-7pt]{0pt}{5pt} \\
\line
\end{array}
\]
<span class="GAPprompt">gap></span> <span class="GAPinput">options:= rec( phi:= "\\varphi", chi:= "\\chi" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print(<panclass=GAPprompt>>span><span class"GAPinput">:=CharacterTable",4 );/
\[
\begin{array}{r|rrr} \hline
 & \varphi_}
 & \varphi_{2}
 & \varphi_{3}
 \[-pt0pt}20\\hline
\chi_{1} & 1 & . & . \rule[0pt]{0pt}{13pt} \\
\chi_{2} & . & 1 & . \\
\hi_3}& . & . &1\java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
\chi_{4} & . & 1 & 1 \\
\chi_{5} & 1 & 1 & 1 \rule[-7pt]{0pt}{5pt} \\
\hline
\end{array}
\
</pre></div>

<p><a id="X873211618402ACF7" name="X873211618402ACF7"></a></p>

<h4>71.12 <span class="Heading">Other Operations for

<p>In the following, we list operations for character tables that are not attributes.</p>

<p>

<h5>71.12-1 Index</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Index</code>( <var class="<i<p>code class="func">SizesCentralizers</code> (<a href="chap71.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>) and <code class="func">Irr</code> (<a href="chap71.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>).</p>
<p>For two character p>For an ordinary character class=Argordtbl<>  prime integer <varclass""p/>,<code class=func">rimeBlocks<code>returns a record with the following components.</p>

>a="X8123650E817926FC"nameX8123650E817926FC><a<p<dl

java.lang.StringIndexOutOfBoundsException: Index 179 out of bounds for length 39

<div class="func"><table classdt< =M>< "ode><<></>
""code< "hap12htmlXF6C5C3287E8B816"<java.lang.StringIndexOutOfBoundsException: Range [168, 167) out of bounds for length 276


<ul>
<li><p><code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), <code class="func">SizesCentralizers</code> (<a href="chap71.html#X7CF7907F790A5DE6"><span class="RefLink">71.9-2</span></a>), and <code classjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

</li>
</ < hrefjava.lang.StringIndexOutOfBoundsException: Range [207, 206) out of bounds for length 276

</li>
<li><p><code class="func">ComputedPowerMaps</code> (<a href="chap73.html#X781FAA497E3B4D1A<li>

</li>
<li><p><code class="func">SizesCentralizers</code> (<</d

<
<li><p><code class="func">Irr</code> (<a href="chap71.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) (first orthogonality relation).</p>

</li>
</ul>
<p>For a <em>Brauer</em> table <var class="Arg">tbl</var>, <code class="func">IsInternallyConsistent</code> checks the consistency of the following attribute values (if stored).</p>


<ul>
<<<<p>Two ordinary < class"SimpleMath"χψ/>   group< ="impleMath"G/> are said to   thesame span""p/>-<><em if the    centralcharacters <classSimpleMath>ωχ _/span s code=funcCentralCharacter/>< href=".X7DD8FDCF7FB7834A><span class=""72817span<a)  thenatural  epimorphism  < class">→R/M/>java.lang.StringIndexOutOfBoundsException: Range [494, 493) out of bounds for length 938

</li>
<></code <a href=c.htmlX86F455DA7A9C30EE classR>.-1span>/)<pjava.lang.StringIndexOutOfBoundsException: Index 276 out of bounds for length 276

</java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<lip>class=""ComputedPowerMaps<>(a ="hap73html#">spanclass"efLink>73.1-1</span><a>)  code class="func">OrdersClassRepresentatives</code> (<a href="chap71.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>).</p>

</li>
<li><p><code class="func">Irr</code> (<a href="chap71.html#span class="GAPprompt">gap></span> <span class="GAPinput">tbl:() );>

</li>
</ul>
<p>If no inconsistency occurs, <code p><code class="func">IsPSolvableCharacterTable  ordinarycharacter <class""tbl<varcorresponds class"">sPSolvable</> < ".#"><   :=[3 0]  :  ,0 0, 1 0  ,

<p><a id="X7A0CBD1884276882" name="X7A0CBD1884276882"></a></p>

<123 h5h5

<span=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 105
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPSolubleCharacterTable</code>( <var class="Arg">gt  ="java.lang.StringIndexOutOfBoundsException: Range [94, 87) out of bounds for length 105
<div class="func"< class="<code>returns< =k><  var =">/>  "rg"/>  any  
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&
< class=func>table ="unc"width=100"<tr>td class="tdleft"><code class="func<div ="example">>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ComputedIsPSolubleCharacterTables</code>( <var class="Arg">tbl</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 190 out of bounds for length 0
<p><code class="func">sPSolvableCharacterTable<</code> forthe  character table var class"">tbl<varcorrespondsto code class"unc"IsPSolvable/>  =chap39.tmlX81130F9A7CFCF6BF>span class=R>39.5-26<span>/) for  group codeclass""></code  <""></> <java.lang.StringIndexOutOfBoundsException: Range [466, 465) out of bounds for length 603

<p>The default method uses the attribute <code class="func">ComputedIsPSolvableCharacterTables</code> for storing the computed value at position <var class="Arg">p</var>, and calls the operation <code class="func"><iv="example"<


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Sz(8)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSolvableCharacterTable( tbl, 2 );</span>
false
<spanclass="APprompt>gap></span> <span class="GAPinput">List(fus, map -> IsClassFusionOfNormalSubgroup(tblc2, map, tbld8));</span>
true
</pre></div>

<p><a id="div class="func"><table class"width=100"<tr>java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 223

<71.2-4 IsClassFusionOfNormalSubgroup>

<iv =func"table class="func" width="100%"><tr><td class="tdleft"><code class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>For two ordinary character tables <var class="Arg">tbl</var> and <var class="Arg">subtbl</var> of a group <span class="SimpleMath">G</span> and its subgroup<l>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tblc2:= CharacterTable( "Cyclic", 2 );;</span>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 110
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( tblc2, tbld8 );</span>
[ [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ]
<span(tblc2, map, tbld8));</span>
[ true, false, false ]
</pre></div>

<>a id"X7FD3D3047DE6381E"dtstrong class=Mark"< class=code"modchars<code<strong<dtjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

<h571.125 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"
<span>>< "":CharacterTableL() ;<span
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ComputedIndicators</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">tbl</var> is an ordinary character table then <code class="func">Indicator</code> returns the list of <var class="Arg">n</var>-th Frobenius-Schur indicators of dt>strong =Mark"><code class=""basicset/code><

=></>   groupspanclass=SimpleMath></  givenby spanclass=S">_(χ( _G}χg^n)  / G<span></>

<p>If <var class="Arg">tbl</var> is a Brauer table in characteristic <span class="SimpleMath">≠ 2</span> and <span class="SimpleMath"><var

method theattribute< ""ComputedIndicators<codestoringthecomputedvalueat < =A"n/var>  calls operation <code class="unc></ forcomputing yet known.p


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L3(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Indicator( tbl, 2 );</span>
[ 1, 0, 0, 1, 1, 1 ]
</pre></div>

<Innonzero characteristic<pan class="SimpleMath"p<span FrobeniusSchur indicator is defined only for irreducible characters. For odd <span class="SimpleMath">p</span>, the is java.lang.StringIndexOutOfBoundsException: Range [234, 233) out of bounds for length 479


<ul>
<li><p>java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0

<li>
<li><p>The indicator is <span class="SimpleMath">0</span> if and only if the character is not real-valued.</p>

</li>
<li><p>Real characters outside the principal block (the <span java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 58

<>
<li><p>By <a href="chapBib.html#biBGW95">[GW95, Lemma 1.2]</a>, any real java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 59

</li>
</ul>
<p>For each <span class="SimpleMath">2</span>-modular Brauer characters where these conditions are not sufficient to determine the indicatorjava.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 0

<p><a id="X83AE05BF8085B3C8" name="X83AE05BF8085B3C8"></a></p>

<h5>71pjava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 358

<div class="func"><table class="func" width="100%"><tr><td class="tdleft
<returns number isomorphismjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>According to <a href="chapBib.html#biBNPP84">[NPP84, p. 233]</a>, the number of polyhedral

<p>The classes <var class="Arg">c1</var>, <var class="Arg">c2</var> and <var class="Arg">c3</var> in the parameter list must be ordered according to the order of the elements in these classes. If elements in class <var class="Arg">c1</var> and <var class="Arg">c2</var> do not generate a polyhedral group then <code class="keyw">fail</code> is returned.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrPolyhedralSubgroups( tbl, 2, 2, 4 );</span>
rec( number := 21, type := "D8" )
</pre></div>

<p><a id="X7E2EA9FE7D3062D3" name="X7E2EA9FE7D3062D3"></a></p>

<h5>71.12-7 ClassMultiplicationCoefficient</h5>

<func>table=func ""< "">code=>#227 java.lang.StringIndexOutOfBoundsException: Range [132, 131) out of bounds for length 318
<p>returns the class span class="GAPprompt">gap></span"java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 110

< classjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 560

<p/pre></div>

<p class="pcenter">K_i K_j = ∑_k c_ijk K_k .</p>

<p>Given the character table of a finite group <span class="SimpleMath">G</span>, whose   [1,1,0]  , ,0, ]]

<p [ [ 1, 0 0 ,[0 ,0]  0,0 1 ], 0 , 1  1, ]

<>On]

<p><a id="X7A19F56C7FD5EFC7" namep> =java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

<h5>71.12-8 ClassStructureCharTable</h5>

<div class="func">Theclass  span=>{jk}</span>ofjava.lang.StringIndexOutOfBoundsException: Range [109, 107) out of bounds for length 560
<>socalledclassstructureof theclasses list< class="">classes/>, for charactertable<varclass=Arg>tbl/> ofthegroup <span class="SimpleMath">G</span>. The length of <var class="Arg">classes</var> must be at least 2.</p>

<<>The optional argumentvar="">ptions</must  arecordjava.lang.StringIndexOutOfBoundsException: Range [91, 90) out of bounds for length 717

<p><span class="SimpleMath">n(C_1, C_2, ..., C_n)</span> is computed using the<p> the   the group algebra of <span class="SimpleMath">G</span>, these numbers are found as coefficients of the decomposition of the product of two class sums <span class="SimpleMath">K_i</span> and <span class="SimpleMath">K_j</span> into class sums:</p>

<p class="pcenter">n(C_1, C_2, ..., C_n) = |C_1| |C_2| ⋯ |C_n| / |G| ⋅ ∑_{χ ∈ Irr(G)} χ(g_1) χ(g_2) ⋯ χ(g_n) / χ(1)^{n-2} .</p>

<p><a id="X809E67E57D4933B3" name="X809E67E57D4933B3"></a></p>

<h5>71.12-9 MatClassMultCoeffsCharTable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MatClassMultCoeffsCharTable</code>( <var class="Arg">tbl</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<.html#X7E2EA9FE7D3062D3"><span class="RefLink">71.12-7</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L3(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassMultiplicationCoefficient( tbl, 2, 2, 4 );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassStructureCharTable( tbl, [ 2, 2, 4 ] );</span>
168
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassStructureCharTable( tbl, [ 2, 2, 2, 4 ] );</span>
1848
<span class="GAPprompt">gap></span> <span class="GAPinput">MatClassMultCoeffsCharTable( tbl, 2 );</span>
[ [ 0, 1, 0, 0, 0, 0 ], [ 21, 4, 3, 4, 0, 0 ], [ 0, 8, 6, 8, 7, 7 ],
  [<p>For those character tables from the <strong class="pkg">GAP</strong> table library that belong to the <strong"
</pre></div>

<p><a id="X7C1941F17BE9FC21" name="X7C1941F17BE9FC21"></a></p>

<h4>71.13 <span class="Heading">Printing Character Tables</span></h4>

<p><a id="X7D45224B86D802E5" name="X7D45224B86D802E5"></a></p>

<h5>71.13-1 ViewObj</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ViewObj</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p<>Let<span class"{\tt X}_{1} & 1 & amp;.\ulept]{t}13pt}\

<p><a id="X836554207C678D41" pspan class"SimpleMath">nC_1,C_2,.. C_n)<span>isusingthe formula>

<h5>71

<div class\line
<p>The default <code class=\ndarray}}

<p><a id="X7B41F36478C47364" name="X7B41F36478C47364"></a></p>

1.-Display/h5java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

="table f =100%>tr><d class=tdleft"<="">#227Display>(<>)tdtdclass=tdright>&;method/><<table<div
<p>There are various ways to customize the <code class="func">Display</code> (<a href="chap6.html#X83A5C59278E13248"><span b{rray}| \

<p>The default <code class="func">Display</code> (<a href="chap6.html#X83A5C59278E13248"><span class="RefLink" "gt<<class">: (L32;/>

<</code> (chap6html#"<java.lang.StringIndexOutOfBoundsException: Range [109, 108) out of bounds for length 969

<p>An interactive variant of <code class="func">Display</code> (<a href="chap6.html#X83A5C59278E13248"><span class="RefLink">6.3-6</span></ahi_{}∓ &;.a;1\

>java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 566

<p>The first lines of the output describe the order of the centralizer of an element of the class factorized into its prime divisors.</p>

<p>The next line gives the name of each class. If no class names are stored on <var class="Arg">tbl</var>, <code class="func">ClassNames</code> (<a href="chap71.html#X804CFD597C795801"><span class="RefLink"<><id="X873211618402ACF7" name="X873211618402ACF7"></a></p>

<p>Preceded by a name <code class="code">P<var class="Arg">n</var></code>, the next lines show <pIn thefollowing,welist operations for    notattributes<pjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89

<p>Every ambiguous or unknown (see Chapter <a h5>.- <h5

<>Irrationalcharacter values are notprintedexplicitly because  the  of their printed representation might disturb the layout. Instead of that every irrational value is indicated by a name, which is a string of at least one capital letter.</p>

<p>Once a name for an irrational value is found, it is used all over the printed table. Moreover the complex conjugate (see <code class="func">ComplexConjugate</code> (<a href="chap18.html#X7BE001A0811CD599"><span class="RefLink">18.5-2<<p<=X836554207java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

hejava.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 483

< record <ar class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 4
<li><p>as an optional argument in the call to <code class="func">Display</code> (<a href<< <li>

</li>
<li><p>as the value of the attribute <code class="func">DisplayOptions</code> (<a href="chap71.java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 0

</li>
<li><p>as the value<iv class="unc=func"SizesCentralizerscode>(a "chap71..htmlhtmlX7CF7907F790A5DE6"< class">1.9-<span>/> and<class=func"Irrcode>< hrefchap71htmlX873B3CC57E9A5492"class==RefLink">71.2/a)</pjava.lang.StringIndexOutOfBoundsException: Index 253 out of bounds for length 253

</li>
>java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 109

</li>
</ol>
<>inthisorderprecedence<p>

<p>The following components of <var class="Arg">arec</var> are supported.</p>


<dl>
<dt><strong class="Mark"><code class="code">centralizers</code></strong></dt>
<dd><p><code class="keyw">false</code> to suppress the printing of the orders of the centralizers, or the string <code class="code">"ATLAS"</code> to force the printing of non-factorized centralizer orders in a style similar to that used in the <strong class="pkg">Atlas</strong> of Finite</li>

</dd>
<dt><strong class="Mark"><code class="code">characterField</code></strong></dt>
<dd><p><code class="keyw">true</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</dd>
<dt><strong class="Mark"><code class="code">chars</code></strong></dt>
<dd><p>an integer or a list of integers to select a sublist of the irreducible characters of <<uljava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

</dd>
<dt><strong class="Mark"><code class="code">charnames</code></strong></dt>
<dd><p<><a idX7A0CBD1884276882 =X7"<a>p>

</dd>
<dt><strong class="< "unc"<table " width"100"><>td ="codeclass=func"‣</(  =Arg>tbl<<var>,<var =p/java.lang.StringIndexOutOfBoundsException: Range [187, 186) out of bounds for length 261
<>java.lang.StringIndexOutOfBoundsException: Range [21, 17) out of bounds for length 109

<dd
<dt><strong class="Mark"><code class="code">classnames</code></strong></dt>
<><>aof strings tothe java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 131

</dd>
<dt><strong class="Mark"><code class="code">indicator</code></strong></dt>
<>p<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 301

</java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
<tstrong class="Mark"<class=""letter/</<dt<
<dd><p><p>Preceded a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</dd>
<dt><strong class="Mark"><code class="code">powermap</code></strong></dt>
<dd><p>an <pan class="< valuesnot java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 252

</dd>
<dt><strong class="Mark"><code class="code">Display</code></strong></dt>
<><the java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 215

</dd>
<dt><strong class="Mark"><code class="code">StringEntry
<dd><p>a function that 

</dd>
dt<strong class="Mark"><code class="code">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<dd><p>a unary function that is called once with argument <li><p>as the value of the attribute <code class="func">DisplayOptions<java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 0

</dd>
<dt><strong class="Mark"><code class="code">Legend</code></strong></dt>
<>p> java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 310

<dd
</dl>
<p><a id="X85E883A87A190AA4" name="java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 0

<h5>71.13-4 DisplayOptions</h5>

<div class="func"><table class="func" width="100%"><tr><td div "unc"<able <><c">/java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 77
<> no defaultto java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 123



<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" );;</span>
<span class="APprompt>gt;<span> <The java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 265
A5

     2  2  2  .  .  .
     3  1  .    ..
     5  1  .  .  1  1

       1a 2a 3a 5a 5b
    2P 1 a5 a
    3P 1a 2a 1a 5b 5a
    5P 1a 2a

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
X>>The  spanclass=SimpleMath><span if java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 110
X.3     3 -1  ./java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
X 4. 11-java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
X.5     5  1 -1  .  .

A = -E(5)-E(
  = (1-Sqrt(5))/2 = -b5
< =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class<>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( CharacterTable( "A5" )
A5

    60  4  3  5  5


 p'     A  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    2A 3 AB*

<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 126
< >gt;/>< "java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 123
<span class="GAPprompt">></>
<="java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 93
A5

          3a 5a
       2P 3a 5bdt>class"java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 12
       2
X.4    +   1 -1
<class=G"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
A5

          3a 5a
       2P 3a 5b
       2
X.4    +   1 -1
<span class"GAPprompt"gap&/pan><spanclass=GAPinput"Unbind.ser);/>
<

<p><a id="X79EC9603833AA2AB" name="X79EC9603833AA2AB"></a></pp>Inthecenter ofthe algebra  span class=<l>

<h5>71.13-5 PrintCharacterTable</h5>

<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<p>Let <var class="Arg">tbl</var> be a nearly

<p>The output of <code class="func">PrintCharacterTable</code> is <java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 62

<p>This is usedjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintCharacterTable( java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 21
tbl:= function()
local tbl, ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
tbl:=rec();
tbl.Irr:=
[ [ 1, 1 ], [ 1, -1 ] ];
tbl.IsFinite:=
true;
tbl.NrConjugacyClasses:=
2;
tbl.Size:=
2;
tbl.OrdersClassRepresentatives:=
[ ,2 ]java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
tbl.java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 23
[ 2, 2 ];
tbl.UnderlyingCharacteristic:=
;
tbl.span class="GAPprompgap> span "APinput">DisplayCharacterTable"5  )<span>
[ [ 1, 0 ], [ 1, 1 ] ];
tbl.CharacterParameters:=
[ [ 1, 0 ], [ 1, 1 ] ];
tbl.Identifier:=
"C2";
InfoText:java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
"java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 59
tbljava.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 23
[ , [ 1, 1 ] span"gapg;span> <=Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 116
ConvertToLibraryCharacterTableNC(tbl);
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 11
end;
tbl:= tbl();
<pre><div>

<p><a id="X79BC08C6846718D9" name="X79BC08C6846718D9"></a></p>

<h4>71.14 <span class="Heading">Computing the Irreducible Characters of a Groupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Several algorithms are =java.lang.StringIndexOutOfBoundsException: Range [35, 27) out of bounds for length 62

<p>These functions are installed in methods for <code class="func/

<p><a id="X7ED39DB680BFEA96" name="X7ED39DB680BFEA96"></a></p>

<h5>71.14-1 IrrDixonSchneider</h5>

<div class="
<p>computes the irreducible characters of the finite group <var java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 26

<><idjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

<h5>71.14-2 IrrConlon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IrrConlon</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></2;
<p>For a finite solvable group <var class="Arg">G</var>, <code class="func">IrrConlon</code> returns a list of monomial irreducible characters of <var class="Arg">G</var>, among those all irreducibles that have the supersolvable residuum of <var 

<p>The characters are computedtbl.SizesCentralizers:

<p><a id="X7BF15729839203FCn java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 240

<h5>71.14-3 IrrBaumClausen</h5>

<class="func">&8227 IrrBaumClausen</> varclass"rg></ar>)<td<InfoText:java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
>=funcIrrBaumClausencode theabsolutelyirreducible charactersofthefactorgroup  the finite java.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 23

<p>The characters are computed using the algorithm by Baum and Clausen (see <a href= 


<>>/
spanclass="GAPprompt">gap></span> <span class="GAPinput">g:= SL
<span class=<h4> theIrreducible a <span<h4>
[ Character
  java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 39
[ 32 () ,) (3)2   ,
  Character( CharacterTable(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 1, E(3), E(3java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  Character( CharacterTable( SL(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (( SL23 ,
    [ 2, E(3)^2, E(3), -2, -E(3), -E(3)^2, 0 ] ),
  (CharacterTable SL23 ),
    [ 2, E(3), E(3)^2, -2, -E(3)^2, -E(3), 0 ] ),
  Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 
<span class="GAPprompt">gap></span> <span class="GAPinput">irr2:= IrrConlonh5>71.142 <h5>
[=funcclassfunc"=100>td "> java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 217
  Character( CharacterTable( SL(2,3) ),
    [ 1,
  java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 39
    [ 1, E(3)^2, E(3), 1, E(3), e="X7BF15729839203FC"></a></p>
  Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">irr3:= IrrBaumClausen( g );</span>
[ Character( <div class="func"><tableclass="func" width"100%><tr><td class="dleft"><code class="func">‣; IrrBaumClausen<</ode( var =Arg"G/ar>)/d>tdclass=tdright"(&bsp;attribute&bsp;)/d>/r>/table<div>
  Character(CharacterTable(SL2,3) ),
    [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
  Character( CharacterTable( SL(2,3) ),
     ,(3^,3,1 (3 3^   ,
  Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
<span class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
true
</pre></div>

<p><a id="X7F29C5447B5DC102" name="X7F29C5447B5DC102"></a></p>

<h5>71.14-4 IrreducibleRepresentations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IrreducibleRepresentations</code>( <var class="Arg">G</var>[, <var class="Arg">F</var>] )</td><td class="tdright">(  (CharacterTable( (2,))  , 11 0])
<p>Called with a finite group <var class="Arg">G</var> and a field <var class="Arg">F</var>, <code class="func">IrreducibleRepresentations</code> returns a list of representatives of the irreducible matrix representations of <var class="java.lang.StringIndexOutOfBoundsException: Index 240 out of bounds for length 39

<p>If <var class="Arg">G</var> is the only argument then <code class="func">IrreducibleRepresentations</code> returns a list of representatives of the absolutely irreducible complex representations of <var class="Arg">G</var>, up to equivalence.</p>

<p>At the moment, methods are available for the following cases: If <var class="Arg">G</var> is abelian by supersolvable the method of <a href="chapBib.html#biBBC94<[ 2 E(3) E()^,-,-E()2 E(),0  ,

<p>Otherwise, if <var class="Arg">F</var> and <var class="Arg">G</var> are both finite, the regular module of <var class="Arg">G</var> is split by MeatAxe methods which can make this an expensive operation.</p>

<p>Finally, if <var class="Arg">F</var> is not given (i.e. [Character CharacterTable(SL2,)) [,1 ,1 ,1   ,

<p>For other cases no methods are implemented yet.</p>

<>representations obtained are <em>not</em> guaranteed to be <q>nice</q> (for example preserving a unitary form) in any way.</p>

<p>See also <code class="func">IrreducibleModules</code> (<a href="chap71 3^ 3,1 ) ()2   ,


<div class="example"><pre>
<span [Character((SL23 ,[1 ,1,1,,1,1] )
<span class="GAPprompt">gap></span> <span class="GAPinput">repr:= IrreducibleRepresentations( g );</span>
[ Pcgs([ (2,4     1,E3)E3^,1 ()2 (3,1  )
    [ ] 1] [    ,
  Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) -[1 32), 3,E)2,1])
    ] ] [[  ,   ]]]
  Pcgs([ (2,4,3), (1,3<span ="APprompt""gap><span><span =GAPinput"chi irr2[4];;  HasTestMonomialchi)<span
    [ [ [ E(3)^2 ] ], [ [ 1 ] ], [ [ 1 ]     ><div
  Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
    [[ 0,0 1]  ,0 
      [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
      [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( repr, IsGroupHomomorphism );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( repr );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );</span>
[ (1,2,3), (2,3,4) ]
<span class="GAPprompt">gap></span> <span
[ [    ,[[1]]]
<span class="GAPprompt">gap></span> <span class="GAPinput"> List( gens, x -> x^repr[4] );</span>
[ [ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ],
  [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
</pre></div>

<p><a id="X8493ED7A86FFCB8A" name="X8493ED7A86FFCB8A"></a></p>

<h5>71.14-5 IrreducibleRepresentationsDixon</h5>

<div class="func"><table class="func" width="100%"><tr><td class
<p>Called with one argument, a group <<p>For other cases no methods are implemented yet.</p>

<p>This method can be quite expensive for larger groups, for example it might involve calculation of the subgroup lattice of <var class="Arg">G</var><p>he representations obtained are <em>not</em> guaranteed to be <q>nice</q> (for example preserving a unitary form) in any way.</p>

<p>A character <var class="Arg">chi</var> of <var class="Arg">G</var> can be given as the second argument, in this case only a representation affording <var class="Arg">chi</var> is returned.</p>

<p>The second argument can also be a list of characters of <var class="Arg">G</var>, in this case only representations for characters in this list are computed.</p>

<p> there is no subgroup in which itsreduction has a linear constituent with multiplicity one.</p>

> option<="unitary</var> is given,<strong ""GAP/   extra ,   a unitary representation (and will issue an error if it cannot do so).</p>


< class=example>prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="  ( (43) (3(,) 12(3,4 ]-gt;
Alt( [ 1 .. 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">char:= First( Irr( a5 ), x -> x[1] = 4 );</span>
Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 4, 0, 1, -1     [ [0 0,1 ,[1 ,0] [0 ,0 ],
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:=IrreducibleRepresentationsDixon( a5, char: unitary );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( a5.1*a5.2 ) = Order( Image( hom, a5.1 )*Image( hom, a5.2 ) );</span>
true
<span class="GAPprompt">gap></span> <<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( repr, IsGroupHomomorphism );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Listtrue
 , 0 1,-1,- ]
</pre></div>

<p><a id="X7E51AACD79CE0BC8" name="X7E51AACD79CE0BC8"(12,,(,,)]

<[[[1  ,[ [ 1 ]]]

<p>This section describes functions that return certain modules of a given group. (Extensions by modules can be formed by the command <code class="func">Extensions</code> (<a href="chap46.html#X8236AD927A5A0E5A"><span class="RefLink">46.8-4</span></a>  ,0 - ]  ,0  ,0 -1   ]

<p><a id="X87E82F8085745523" name="X87E82F8085745523"></a></p/re><div>

<h5>71.15-1 IrreducibleModules</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IrreducibleModules
< of <var class=Arg><var>. secondentrya  of  irreducible of< =A"<var overthe field <var class=""F<var dimension var class="><var, givenas  modules(see<code class=func></code>(ahref".tml#"><spanclass="">9.1/<a>)<java.lang.StringIndexOutOfBoundsException: Index 419 out of bounds for length 419

<p><a id="X7D0BD5337D1C069B" name="X7D0BD5337D1C069B"></a></p>

<>71.52AbsolutelyIrreducibleModules

< java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 290
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AbsoluteIrreducibleModules</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">dim</var> )</td><td class="tdright">( operation )</td></
< =< ""=100"<>java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 287
<p><code class="func">AbsolutelyIrreducibleModules</code> returns a list of length 2. The first entry is a generating system of the group <var class="Arg">G</var>. The second entry is a list of all those absolutely irreducible modules of <

<p>The other two names are just synonyms.</p>

<p><a id="X7EB88B2E87AF5556" name="X7EB88B2E87AF5556"></a></p>

<h5>71.15-3 RegularModule</h5>

<div class="func">Character(CharacterTable [1.5 ] ,[ ,0,1 - 1 ]java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
<p>returns a list of length 2. The first entry is a generating systemspan ">><span><class="">(Image(hom,.)Image(hom, .  );/>

<p><a id="X86CDA4007A5EF704" name="X86CDA4007A5EF704"></a></p>

<h4>71.16<4 ,1,1 1 ]

<p>The <strong class="pkg">GAP</strong> library implementation of the Dixon-Schneider algorithm first computes the linear characters, using the commutator factor group. If irreducible characters are missing afterwards, they are computed using the techniques described in <<p<a id="X7E51AACD79CE0BC8" name="X7E51AACD79CE0BC8"></a></p>

 <class="java.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 1062

<p>The computation of irreducible characters from the group needs to identify the classes of group elements very often, so it can be helpful to store a class list of all group elements. Since this is obviously limited by the group order, it is controlled by the global function <code class="func">IsDxLargeGroup</code> (<a href="chap71.html#X8089009E7EA85BC8"><span class="RefLink">71.17-8</span></a>).</p

<p>The routines compute in a prime field of size <span class="SimpleMath">p</span>, such that the exponent of the group divides <span class="SimpleMath">(p-1)</span> and such that <span class="SimpleMath">2 sqrt{|G|} < p</span>. Currently prime fields of size smaller than <span class="SimpleMath">65536</span> are handled java.lang.StringIndexOutOfBoundsException: Index 329 out of bounds for length 0

<p>The routine stores a Dixon record (see <code class="func">DixonRecordjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7C083207868066C1" name="X7C083207868066C1"></a></p>

<h4>71.17 <span class="Heading">Advanced Methods for Dixon-Schneider

<p

<p>When the info level of <code class="func">InfoCharacterTable</code> (<a href="chap71.html#java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 0

<p><a id="X7C398F2680C8616B" name="X7C398F2680C8616B"></a></p>

<h5>71.17-1 DixonRecord</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DixonRecord</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <code class="func">DixonRecord</code> of a group contains information used by the routines to compute the irreducible characters and related information via <p><code class="func">AbsolutelyIrreducibleModules</code> returns a list of length 2. The first entry is a generating system of the group <var class="Arg">G</var>. The>/ < java.lang.StringIndexOutOfBoundsException: Range [317, 316) out of bounds for length 543

<p><a 

<h5>71.17-2 DixonInit</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DixonInit</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function does all the initializations for the Dixon-Schneider algorithm. This includes calculation of conjugacy

<p><a id="X868476037907918F" name="X868476037907918F"></a></p>

<h5>71.17-3 DixontinI</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DixontinI</code>( <var class="Arg">D</var> )</td><td class="tdrightjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This function ends a Dixon-Schneider calculation. It sorts the characters according to the degree and unbinds components in the Dixon record that are not of use any longer. It returns

<p><a id="X87ABE0B081DAD476" name="X87ABE0B081DAD476"></a></p>

<h5>71.17-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DixonSplit</code>( <var 
<p>This function performs one splitting step in the Dixon-Schneider algorithm. It selects a class, computes the (partial) class sum matrix, uses it to split character spaces and stores all the irreducible characters obtained that way.</p>

 here.</p>

<p><code class="func">DixonSplit</code> returns the number of the class that was used for splitting if a split was java.lang.StringIndexOutOfBoundsException: Range [0, 124) out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 62

<>.-5BestSplittingMatrix>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><<p>When the info  of< class="unc">InfoCharacterTable<code (a "chap71.html#C6F3D947E5188D1"><span class="RefLink">71.4-2</span></a>) if positive, information about the progress of splitting is printed. (The default value is zero.)</p>
<p>returns the number of the class sum matrix C8616B"<a<p>

p option code =code"maxclasslen/ (efaulting  code class="uncinfinity/ < href".tml#">< class=">8.21/>/a>))is  by code class="func">BestSplittingMatrix</code>: Only classes whose length is limited by the value of this option will be considered for splitting. If no usable class remains, <code class="keyw">fail</code> is returned.</p>

<p><a id="X7C85B56C80BFA2E3" name<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<h5>71.17-6 DxIncludeIrreducibles</h5>

< ""<tableclass=func width"100"<tr>tdclass=tdleft">code ="unc"‣DxIncludeIrreducibles</code>( <var class="Arg">D</var>, <var class="Arg">new</var>[, <var class="Arg">newmod</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function takes a list of irreducible characters <var class="Arg">new</var>, each given as a list of values (corresponding to the class arrangementp<a idX7E33C03E7BDDC9B0"<li><p>as the value of the attrib <code "func"isplayOptions</code> (<a href="chap71.html#X85E883A87A190AA4"><span class="RefLink">71.13-4</span></a>) if this value is stored in the table,</p>

< closes thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X87A5B5C77F7F348E" name="<>This <i<p>   of   variable <code =code"CharacterTableDisplayDefaults</ode<

</

<div class="func"><table class="func" width="100%</5>>
<p>java.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 0

<p><a id="X8089009E7EA85BC8" name="X8089009E7EA85BC8"></a></p>

<

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsDxLargeGroup</code>( <var class="java.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 0
<

<<"C1153637 ="X7C1153637E7D2133<a</>

<h4>71.18 <span class="Heading">Components of a Dixon Record</span></h4>

<p>The <q>Dixon record</q> <var <dd<p>aninteger a list of integers to select a sublist of the classes of <java.lang.StringIndexOutOfBoundsException: Range [0, 82) out of bounds for length 0


<dl
strong="code=code"group/<strong><java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
<dd><p>the group <span class="SimpleMath">G</span> of which the character table is to be computed,</p>

</dd>
<dt><strong class="Mark"><code class="code">conjugacyClasses</code></strong></dt>
<d>pclasses of<panclass=SimpleMath>G<span>(llcharactersstoredinDixonrecordcorrespondthisjava.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 38

<>
<dt><strong class="Mark"><code class="code">irreducibles</code></strong></dt>
<dd><p>the already known irreducible characters (given as lists of their values on the conjugacy classes),</p>

</dd>
<dt><strong class="Mark"><code class="code">characterTable</code></strong></dt>
<dd><p>the <code class="java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0

</dd>
<dt><strong 
<dd><p>a function that returns the number of the class of <span class="SimpleMath">G</span> that contains the element

</dd>
</dl>
B5E37848786BCX782B5E37848786BC<<>

<h4</dd

<p>First<dd>pafunction that takes the result of the <code class="code">StringEntryData</code> call as its only argument, after the character table has been displayed; the return value <dd><p>a function that takes the result of the <code class="code">StringEntryData</code> call as its only argument, after the character table has been displayed; the return value is



<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoCharacterTablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>

forprintoutof someinternal . now  our group,hich  isomorphic  <


<div"xample"<>
<spanclass=GAPprompt"gap&</span>><span class="<dt><strong class="Mark"><code class="code">group</code></st
PSL(4, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span>
6065280
<span class="GAPprompt">gap></span> <span class="GAPinput">d:= DixonInit( g );;</span>
#I  29 classes
  65521
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
/>/

<p>After java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DixonSplit( d );</span>
#I  Matrix 2,Representative of Order 3,java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 21
#I  Dimensions: [ [ 1, 6 ], [ 2, 3 ], [ 4, 1 ], [ 12, 1 ] ]
2
</pre></div>

<In   of the listed dimensionsare aresultof  splitting process. The<

<p>We obtain several irreducible characters by tensor products and notify them to the Dixon record.</p>


<div class="example"><pre>
<span class="GAPprompt">gapriate info level higher.</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">ro:= ReducedCharacters( c, d.irreducibles, asp );;</span>
<span class="GAPprompt">gap></span> <span class<div ="example"><pre
3
<span class="GAPprompt">gap></span> <span class="GAPinput">DxIncludeIrreducibles( d, rojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>

pFinally we   characters  from all  subgroups obtain themissing byapplyingthe -  them./p>

  35java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<div class="6065280
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 101
<span class="GAPprompt">gap></span> <span class="GAPinput"I   29classes
<span class="GAPprompt">gap></span> <span class="GAPinput">Length<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
0
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= LLL( c, ro.remainders );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( l.irreducibles );</span>
13
</pre></div>

<java.lang.StringIndexOutOfBoundsException: Range [106, 2) out of bounds for length 292


<div class="xample"><pre>
<span class=GAPprompt">gap>/span> <span class="GAPinput">l.irreducibles[1];</span>
Character( CharacterTable( PSL(4, 3) ),
 [ 640, -8, -8, -8, 1, 1, 1, 0, 0, 0,</pre><div
  0, 0, 0, 0, E(13)^7+E(13)^8+E(13)^11, E(13)^4+E(13)^10+E(13)^12,
  E(13)^2+E(13)^5+E(13)^6, E(13)+E(13)^3+E(13)^9
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=List(l.irreducibles,ValuesOfClassFunction);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DxIncludeIrreducibles( d, l );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( d.irreducibles );class="example"><pre>
29
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 11
29
</pre></div>

<p>It turns out we have found all irreducible characters. As the last step, we obtain the irreducible characters and tell them to the group. This makes them available also to the characterNrConjugacyClasses:=


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">irrs:= DixontinI( d );;</span>
#  :1 ,[2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SetIrr(g,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pan ="APprompt">gap></span> <span class="GAPinput">Length(Irr(c));</span>
29
<span class="bl.ClassParameters:=
</pre></div>

<p><a id="X7C38C5067941D496" name="X7C38C5067941D496"></a></p>

<h4>71.20 <span class="Heading">Constructing Character Tables from Others</span></h4>

<pThe  operations  take one )<span>

<p><em>Note</em> that whenever fusions between input and output tables occur</re></div>

<p>(The interactive construction of character tables using character theoretic methods

<p><a id="X7BE1572D7BBC6AC8" name="X7BE1572D7BBC6AC8"></a></p>

<h5>71.20-1 CharacterTableDirectProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 0
<p>is the table of the direct product of the character tables <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var>.</p>

<p>The matrix of irreducibles of this table is the Kronecker product (see <code class="func">KroneckerProduct</code> (<a href="chap24.html#X8634C79E7DB22934"><span class="RefLinklling one   E(13)^2+E(13)^5+E(13)^6, E(13)+E(13)^3+E(13)^9, 0 ] )

<>Products of ordinary and Brauer character tables are supported.</p>

<p>In general, the result will not know an underlying group, so missing power maps (for prime divisors of the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The embeddings of the input tables into the direct product are stored, they can be fetched with <code class="func">GetFusionMap</code> (<a href="chap73.html#X8464DD23879431D9"><span class="RefLink">73.3-3</span></a>); if <var class="Arg">tbl1</var> is equal to <var class="Arg">tbl2</var> then the two embeddings are distinguished by their <code class="code">specification</code> 29

<java.lang.StringIndexOutOfBoundsException: Range [47, 2) out of bounds for length 177

<p>The attribute <code class="func">FactorsOfDirectProduct</code> (<a href="chap71.html#X7C97CF727FBDFCAB<571.4-2  IrrConlon</h5>

<p>The <code class="code">*</code> operator for two character tables (see <a href<ivclass="func"><table class="func" width="100%"><tr><td class="tdleft<pan class="GAPprompt"gap&;<span>< class="APinput>=   ;<java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c2:= CharacterTable( "Cyclic", 2 );;</span>
<panclass="GAPprompt">gap></span> <span class="GAPinput">s3:= CharacterTable( "Symmetric", <panclass="&</> <panclass=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 107
<<pre<div
C2xSym(3)

2221    java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
     3  1.1   . 1

       1a 2a 3a 2b 2c 6a
    2P11 3java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    3P 1a 2a 1a 2b 2c 2b

X.1     1 -1  1  1 -1  >ap&<<p><em>Note</em> that whenever fusioninputand output tablesoccur inthese operations, they are stored on the concerned tables, and the <code class="func">NamesOfFusionSources</code> (<a href="chap73.html#X7F6569D5786A9D49"><span class="RefLink">73.3-5</span></a>) values are updated.</p>
1 2   -
X.3     1  1  1  1  1  1
X.4     1 -1  1 -1  1 -1
X.5     2  . -1 -2  .  1
X.6     1  1  1 -1 -1 -1
</pre></div>

<p><a id="X7C97CF727FBDFCAB"( 23 ,

<h5>71.20-2 FactorsOfDirectProduct</h5>

<div class="func">
<Forordinarycharactertablethat   constructedvia<code class="CharacterTableDirectProduct> (a ".#X7BE1572D7BBC6AC8"spanclass=java.lang.StringIndexOutOfBoundsException: Index 177 out of bounds for length 91

<>Notethatthere is no default method for <em>computing</em> the value of <code class="func">FactorsOfDirectProduct</code>.</p>

<p><a id="X7C3A4E5283B240BE" name="X7C3A4E5283B240BE"></a></p>

<h5>71.20-3 CharacterTableFactorGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227<>nalogously,   from   productonto   tables arestored java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds for length 177
<p>is the character table of the factor group of the ordinary character table <var class="Argjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 39

<p>The <code class="code">/</code> operator for a character table and


<"><re
<span class="GAPprompt">gap></span> <span class="GAPinput">s4:= CharacterTable( "Symmetric", 4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalSubgroupsh5>71.- IrreducibleRepresentations</h5>
[ [ 1 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1 .. 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= CharacterTableFactorGroup( s4, [ 3 ]
div class="unc><tableclass="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleRepresentations</code>( <var class="Arg">G</var>[, <var class="Arg">F</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( f );</span>
Sym       1 2a 3a  2c 6java.lang.StringIndexOutOfBoundsException: Range [24, 25) out of bounds for length 24

     2  1  1  .
     3  1  .  1

       1a 2a 3a
2a3
    3P 1a 2a 1a

X.1     1 -1  1
X.2     2  . -1
X.3     1  1  1
<<div

<p><"></a></p>

<h5>71.20-4 CharacterTableIsoclinic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CharacterTableIsoclinic</code>( <var class="Arg">tbl</var>[, <var class="Arg">arec</var>] )</td><td class="tdright">( operation )</td></tr></table<panclass="GAPprompt">gap></span> <span class="GAPinput">repr:= IrreducibleRepresentations class="unc">table class" =100><tr<class""> java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 232
< class"unc"<ableclass=func"width"100%>tr<td class="tdleft"><code class="func">&#'color: green'>8227; CharacterTableIsoclinic</code>( <var class="Arg">tbl</var>[, 
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CharacterTableIsoclinic</code>( <var class="Argjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div     [ [  E3^ ]], [ [ 1 ]] [[1]]]
<p>Let <var class="Arg">tbl</var> be the (ordinary or modular) character table of a group <span class="SimpleMath">H</span> with the structure <span class="SimpleMath">p.G.p</span> for some prime <span class="SimpleMath">p</span>, that is, <span class="SimpleMath">H/Z</span> has a normal subgroup <span class="java.lang.StringIndexOutOfBoundsException: Index 321 out of bounds for length 0

<p>Then <code class="func">CharacterTableIsoclinic</code> returns the table of an isoclinic group in the sense of the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 6, Section 7]</a>.</p>

<p>If <span class="SimpleMath">p = 2</span> then also the case <span class="SimpleMath">H = 4.G.2</span> is supported, that is, <span class="SimpleMath">Z<p> <ode ="ode"// foracharactertableand ofclasspositions(<ahref=chap71htmljava.lang.StringIndexOutOfBoundsException: Range [140, 139) out of bounds for length 249

<p>The optional arguments are needed if <var class="Arg">tbl</var> does not determine the class positions of <span class="SimpleMath">N</span> or <span class="SimpleMath">Z</span> uniquely, and in the case <span class="SimpleMath">p > 2</span> if one wants to specify a <q>variant number</q> for the result.</p>


<ul>
<li><p>In general, the values can be specified via a record <var class="Arg">arec</var>. If <span class="SimpleMath">N</span> is not uniquely determined then the positions of the classes forming <span class="SimpleMath">N</span> must be entered as the value of the component <code class="code">normalSubgroup</code>. If <span class="SimpleMath">Z</span> is not unique inside <span class="SimpleMath">N</span> then the class position of a generator of <java.lang.StringIndexOutOfBoundsException: Index 456 out of bounds for length 15

</java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<li><p>If <span class="SimpleMath">p = 2</span> then one may specify the positions of the classes forming <span classX.1       1  1 1

</li>
</ul>
<p>Note that also if <var class="Arg">tbl</var> is a Brauer table then <code class="code">normalSubgroup</code> and <code class="code">centralElement</code>, resp.  <var class="java.lang.StringIndexOutOfBoundsException: Index 179 out of bounds for length 21

<p>If <span class="SimpleMath">p</span> is odd then the <strong class="pkg">Atlas</strong> construction describes <span class="SimpleMath">p</span> isoclinic variants that arise from <span class="SimpleMath">p.G.p</span>. (These groups need not be pairwise nonisomorphic.) Entering an integer <span class="SimpleMath">k ∈ { 1, 2, ..., p-1 }</span> as the X.     2  .-2  . .


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">d8:= CharacterTable( "Dihedral", 8 );</span>
CharacterTable( "Dihedral(8)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( d8 );</span>
[ [ 1 ], [ 1, 3 ], [ 1 .. 3 ], [ 1, 3, 4 ], [ 1, 3 .. 5 ], [ 1 .. 5 ]
 java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<span class="GAPprompt">gap></span> <span class="GAPinput">isod8:= CharacterTableIsoclinic( d8, nsg[3] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(  class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutations lt;g ;/>
Isoclinic(Dihedral(8))

     2  3  2  3  2  2

       14a 2a4 4c
    2P 1a 2a 1a 2a 2a

X.1     1  1  1  1  1
X.     1  1
X.3     1 -1  1  1 -1
X.4     1 -1  1 -1  1
X.5     2  . -2  .  .
<span class="GAPprompt">gap></span> <span class="GAPinput<="<java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 70
<span class="GAPprompt">gap></span> <span class="GAPinput">t2:= CharacterTable( SmallGroup( 27, 4 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( t1 );</span>
[ [ 1 ], [ 1, 4, 8 ], [ 1, 2, 4, 5, 8 ], [ 1, 3, 4, 7, 8 ],
  [ 1, 4, 6, 8, 11 ], [ 1, 4, 8, 9, 10 ], [ 1 .. 11 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iso1:= CharacterTableIsoclinic( t1, rec( k:= 1,</span>
<span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">iso2:= CharacterTableIsoclinic( t1, rec( k:= 2
<span class="GAPprompt">></span> <span class="GAPinput">              normalSubgroup:= nsg[3] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( iso1, t1 ) <> fail;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( iso1, t2 ) <> fail;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( iso2, t2 ) <> fail;</span>
true
</pre></div>

<p>For java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0



<dt><strong classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>character< ="tbl/,/p>

</dd>
<>strong="Mark"<code class=code></ode<strong</dt
<dd><p>the prime <span class="SimpleMath">p</span>,</p>

</dd>
<dt><strong class="Mark"><code class="code">k</code></strong>class"APinput>=( t3 iso ;<>
<dd><p>the variant number <span class="SimpleMath">k</span>,</p>

</dd>
<dt><strong class="Mark"><code class="code">outerClasses</code></strong></dt>
<dd><p>the list of length <span class="SimpleMath">p-1</span> that contains at position <span class="SimpleMath">i</span> the sorted list of class positions of the <span class="SimpleMath">i</span>-th coset of the normal subgroup <span class="SimpleMath">N</span></p>

</dd>
<dt><strong class="Mark"><code class="code">centralElement</code></strong></dt>
<dd><p>the class position of a generator of the central subgroup <span class="SimpleMath">Z</span>.</p>

</dd>
</dl>
pisnodefaultmethod em<em>    <code class==f"SourceOfIsoclinicTable</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 26
[ CharacterTable( "Dihedral(8)" ), [ 1 .. 3 ], [ 3 ], 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SourceOfIsoclinicTable( iso1 );</span>
rec( centralElement := 4, k := 1,
<  length2 Thefirstentryisa generating system of <var class="Arg">G</var>. The second entry is a list of all irreducible modules of <var class="Arg">G</var> over the field <var class="Arg">F</var> in dimension <var class="Arg">dim</var>, given as MeatAxe modules (see <code class="func">GModuleByMats</code> (<a href="chap69CharacterTable("est(4),1 .5]) )]
  table := CharacterTable( <pc group of size 27 with 3 generators> ) )
</pre></div>

<p>If the arguments of <code class="func">CharacterTableIsoclinic</code> are a Brauer table <var class="Arg">modtbl</


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GL(2,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( g );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso:= CharacterTableIsoclinic( t );;</span>
< classGAPprompt"gap>/span>><span class=GAPinput">t3:= t mod 3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso3:= CharacterTableIsoclinic( t3, iso );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( iso3,</span>
<span class="GAPprompt">></span> <span class="GAPinput">       CharacterTableIsoclinic( t3 ) ) <> failjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</pre></div>

<

"X806E55A58397B11B<<p>

h571205CharacterTableOfNormalSubgroup><spanclass=GAPinput"ndex2= Filtered( rest x -gt; x &t;)=240);</pan>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CharacterTableOfNormalSubgroup</code>( <var class="Arg">ordtbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">ordtbl</var> be the ordinary character table of a group <span class="SimpleMath">G</span>, say, and <var class="Arg">classes</var> be a list of class positions for this table. If the classes given by <var class="Arg">classes</var> form a normal subgroup <span class="SimpleMath">N</span>, say,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( "Symmetric", 4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [307, 308) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( t );</span>
[ [ 1 ], [ 1, 3 ], [ 1, 3, 4 ], [ 1 .. 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">rest:= List( nsg, c -> CharacterTableOfNormalSubgroup( t, c ) );</span>
[ CharacterTable( "Rest(Sym(4),[ 1 ])" ), fail, fail,
  CharacterTable( "Rest(Sym(4),[ 1 .. 5 ])" ) ]
</pre></div>

< variants of <span class="..22span>./p>


<div class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
<span class="GAPprompt">gapdiv "xample"<
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= span class="GAPprompt">gap></span> <span class="GAPinput">c3:= CharacterTable
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 28
<span class="GAPprompt">gap></span> <span class="GAPinput">diag:= First( Elements( Centre( dp ) ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">                 x -> Order( x ) = 2 and</span>
<span class="GAPprompt">></span> <span class="GAPinput">                      not x in Image( Embedding( dp, 1 ) ) and</span>
<span class="GAPprompt">>p> ="E ="<>
<span class="GAPprompt"<>.-<
<span class="GAPprompt">></span> <span class="GAPinput">                     Subgroup( dp, [ diag ] ) ));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( fact );;</span>
 ="&;/java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 79
480
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( t );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rest:= List( nsg, c -> CharacterTableOfNormalSubgroup( t, c ) );;</span>3       /-    
<span X.4     1 1/    -1-A-/ 
<>Length
2
<span class="GAPprompt">gap></span> <span class="GAPinput">tg:= CharacterTable( java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">SortedList(List(
< class=G"&;<span><=GAPinput"      (xtg))<span
[ true, false ]
<<div

< +3 =b3

<p><a id="X79B75C8582426BC5" name="X79B75C8582426BC5"></a></p>

<h5>71.20-6 CharacterTableWreathSymmetric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227
<returnsthecharactertableof wreathproduct  a  <=java.lang.StringIndexOutOfBoundsException: Range [90, 87) out of bounds for length 261

<pTheresult valuesfor<java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 470


<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 105
<span class="GAPprompt">gap></span> <span class="GAPinput">wr:= CharacterTableWreathSymmetric( c3, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( wr );</span>
C3wrS2

         java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39


       1a  3a  3b  3c 3d  3e 2a  <pan=<<aid"=X87"<>
    2P 1a  3b  3a  3e 3d  3c 1a  3c  3e
    3P 1a  1a  1a  1a 1a  1a 2a  2a  2a

X.1     1   1   1   1  1   1 -1  -1  -1
22   A/ B. java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
< class"GAPprompt"gapgt<>span class="GAPinput">lasspos:=Position(ClassParameters( wr ), classpara );;</span>
X.4     1 -/A  -A  -A  1 -/A -1  /A   A
.-1     java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
     1- / -A  1-A -   A/java.lang.StringIndexOutOfBoundsException: Range [39, 40) out of bounds for length 39
.71111  1  1   
./  java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 39
X.9     1  -A -/A -/A  1  -A  1  -A -/A

A = -E(3)^2
  = (1+Sqrt(-3))/2 = 1+b3
B2*)
  = -1+Sqrt(-3) = 2b3
<span class="dl
[ [ 1, 1 ], [  ], [  ] ]
</pre></div>

<p><a id="X83E71B1F7FA70134" name="X83E71B1F7FA70134"></a></p>

<h5>71.20-7 CharacterValueWreathSymmetric</h5>

=%trdclass=dleft>=func>8227CharacterValueWreathSymmetric> classA"tblvar> <arclass"></> var=Arg>eta/ar> varclass="rg><var )<td>><class"tdright>&;unction;<table<div
< ordinary charactertable of a group< class=SimpleMath>/> aimof thisfunctioniscompute a single value   characterofthe product of <span class="SimpleMath><span>  symmetricjava.lang.StringIndexOutOfBoundsException: Range [290, 289) out of bounds for length 329

<p>The conjugacy classes and the irreducible characters of this wreath product are java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>We describe the conjugacy class for which we want to compute java.lang.StringIndexOutOfBoundsException: Range [10, 67) out of bounds for length 33


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:= <p>is  thejava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 117
<span class="GAPprompt">gap></span> <span class="GAPinput">classpara:= [ [], [ 2, 1, 1 ] ];;</span>
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 101
<span class="GAPprompt">gap></span> <span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">c2:= CharacterTable( "Cyclic", 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterValueWreathSymmetric( c2, n, betas, classpara );</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">wr:= CharacterTableWreathSymmetric( c2, n );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">classpos:= Position( ClassParameters( wr ), classpara );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">charpos:= Position( CharacterParameters( wr ), charpara );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( wr )[ charpos, classpos ];</span>
0
</pre></div>

This  if java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 450

<p><a id="X816FCD5A805F9FE8" name="X816FCD5A805F9FE8"></a></p>

<theisa   classesof the result are sorted by  this permutation<java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124

<p><a id="X7D9C4A7F8086F671" name="X7D9C4A7F8086F671"></a></p>

<h5>71.21-1 CharacterTableWithSortedCharacters</h5>

<div class=<57121- java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 37
<#I Matrix 2,Representativeof Order 3,entralizer:5832

<p>If no second argument is given then a permutation is used that yields irreducible characters of increasing degree for the result. For the succession of characters in the result//>

p resulthas those attributes   of< class=Arg">/>that stored  code class""</code <"hap71.X7DBEF4BF87F10CD6><pan

<p><a id="X87E3CF317D8E4EC7" name="X87E3CF317D8E4EC7"></a></p>

java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 33

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SortedCharacters</code>( <var class="Arg">tbl</var>, <var class="Arg">chars</var>[, <<p>The second form does the same successively for all kernels <span class="SimpleMath">k_i</span> in the list <span class="SimpleMath"><var class="Arg">normalseries</var> = [ k_1, k_2, ... bea < class"Pprompt"&t<span java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
<p>is a list containing the characters <var class="Arg">chars</var>, ordered as specified by the other ./>

p are threepossibilities   : can be   to <llrearrangements   charactersarestable, ..,the  positions classes characters  are not distinguished byany relevantproperty is not changed./

>java.lang.StringIndexOutOfBoundsException: Range [26, 22) out of bounds for length 94

<p>The trivial character, if contained in <var class="Arg">chars</var>, will always be sorted to the first position.</p>

<p><a id="X7E3DE0A47E62BE6B" name="X7E3DE0A47E62BE6B"></a></p>

<h5>71213 />

<div class="func"><table class="func" width="100<pan class="APpromptgapg;/  "APinput">ength<aid"=X8099java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
< obtainedpermutationthe  < class=A"/.If  <class"rg><var>is <pan "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 94

p> thesecond argument  a  then the  theresultare   of .p>

<p>The result has all those attributes and properties of <var

<p><a id="X82DCAAA882416E24" name="X82DCAAA882416E24"></a></p>

<>.- /java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SortedCharacterTable</code>( <var div ="xample>pre>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SortedCharacterTable</code>( <var java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 108
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SortedCharacterTable</code>( <var class="Arg">tbl</var>, <var class="Arg">facttbl</var>, <var CharacterTable( "Sym(4)" )
<p>is a character table obtained on permutation of the classes and the irreducibles characters of <var class="Arg">tbl</var>.</p>

<p>The first form sorts the classes at positions contained in the list <var class="Arg">kernel</var> to the beginning, and sorts all characters in <code class="code">Irr( <var class="Arg">tbl</var> )</code> such that the first characters are those that contain <var java.lang.StringIndexOutOfBoundsException: Index 269 out of bounds for length 2

<p>The second form does the same successively for all kernels <span class="SimpleMath">k_i</span> in the list <<p>The second form does the same successively for all kernels <span class="SimpleMath">k_i</span> in the list <span

<.     3--1  .1

<p>(<em>Note</em> that the transformation is only unique up to table automorphisms of <span class="SimpleMath">F</span>, and this.431-  -

<p>All rearrangements of classes and characters are stable, i.e., the relative positions of classes and characters that are not distinguished by any relevant property is not changed.</p>

<p>The result has all those attributes and properties of <var class="Arg">tbl</var> that are stored in <code class="func">SupportedCharacterTableInfo</code> (<a href="chap71.html#X7DBEF4BF87F10CD6"><span class="RefLink">71.3-4</span></a>) and do 

<p>The <code class="func">ClassPermutation</code> (<a href="chap71.html#X8099FEDC7DE03AEE"><spanCharacterTable( "Sym(4

<p><code class="func">SortedCharacterTable</code> uses <code class="func">CharacterTableWithSortedClasses</code> (<a href="chap71.html#X7E3DE0A47E62BE6B"><span class="RefLink">71.21-3</span></a>) and <code class="func">CharacterTableWithSortedCharacters</code> (<a href="chap71.html#X7D9C4A7F8086F671"><span class="RefLink">71.21-1<[24,, 8,,4]

<p><a id="X8099FEDC7DE03AEE" name="X8099FEDC7DE03AEE"></a></p>

7121-ClassPermutation/h5>

class">table class="unc width"%"<>tdclass="tdleft"><code class="func">&#8227; ClassPermutation</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a permutation <span class="SimpleMath">π</span> of classes of the character table <var class="Arg">tbl</var>. If it is stored then class fusions into <var span class=GAPprompt>gap>/span><span class=GAPinput">nsg:= ClassPositionsOfNormalSubgroups( srt3 );</span>

<p>This attribute value is bound only if <var class="Arg">tbl</var> was obtained from another table by permuting the classesspan class="APprompt">gap></span> <span class="GAPinput">Display( srt3 );</span>

<p>It is necessary because the original table and the sorted table have the same identifier (and the same group if known), and hence the same fusions are valid for the two tables.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Symmetric", 4 );</span>
CharacterTable( "Sym(4)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( tbl );</span>
Sym(4)

     2  3  2  3  .  2
     3  1  .  .  1  .

       1a 2a 2b 1     java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    2P 1a 1a 1a 3a 2b
    3P 1a 2a 2b 1a 4a

X.1     1 -1  1  1 -1
X.2     3 -1 -1  .  1
X.3     2  .  2 -1  .
X.4     3  1 -1  . -1
X.5     1  1  1  1  1
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 36
CharacterTable( "Sym(4)" )
<class="GAPprompt">gap></span> span class="GAPinput">List( Irr( srt1 ), Degree );</span>
[ 1, 1, 2, 3, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">srt2:= CharacterTableWithSortedClasses( tbl );</span>
CharacterTable( "Sym(4)" )
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 0
[24 4,8,3  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SizesCentralizers( srt2 );</span>
[ 24, 8, 4, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( tbl );</span>
 1,,3 ,[,3,4]  1. 5]]
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
CharacterTable( "Sym(4)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">nsg:= ClassPositionsOfNormalSubgroups( srt3 );</span>
 1 ] [ 1  ,[1.  ]  1. 5 ]java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 85
Sym(4)

     2  3  3  .  2  2
     3  1  .  1  .  .

       1a 2a 3a 2b 4a
    211 a
    3P 1a 2a 1a 2b 4a

X.1     1  1  1  1  1
X.2     1  1  1 -1 -1
X.  1  .
X.4     3 -1  . -1  1
X.5     3 -1  .  1 -1
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPermutation( srt3 );</span>
(2,4,3)
</pre></div>

<p><a id="X7B0A669484470D09" name="X7B0A669484470D09"></a></p>

<h4>71.22 <span class="Heading">Automorphisms and Equivalence of Character Tables</span></h4>

<p><a id="X84353BB884AF0365" name="X84353BB884AF0365"></a></p>

<h5>71.22-1 MatrixAutomorphisms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MatrixAutomorphisms</code>( <var class="Arg">mat</var>[, <var java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 104
<p>For a matrix <var class="Arg">mat</var>, <code class="func">MatrixAutomorphisms</code> returns the group of those permutations of the columns of <var class="Arg">pre</div

<p>If the arguments <var class<p><a id"7D721E3D7AA319F5" nameX7D721E3D7AA319F5"</<p

<p>Each entry in <var class="Arg">maps</var> must be a list of same length as the rows of <var class="Arg">mat</var>. For example, if <var class<h5>71.22- TransformingPermutations</h5>

<p><a id="X8082DD827C673138" name="X8082DD827C673138"></a></p>

h57122-TableAutomorphisms<h5

class> ="func" width"100%">trtd =tdleft">code class="unc&8227 TableAutomorphisms/code( < class="Arg">tbl</var>, <var class="Arg">characters</var>[, <var class="Arg">info</> </td>&operation&;<td>/tr<table<div>
<p><code class="func">TableAutomorphisms</code> returns the java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0

<p>If <var class="Arg">characters</var> is closed under Galois conjugacy –this is always fulfilled for the list of all irreducible characters of ordinary

<p>The attribute <code class="func">AutomorphismsOfTable</code> (<a href="chap71.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>) can be used to compute and store the table automorphisms for the case that <var class="Arg">characters</var> equals the <code class="func">Irr</code> (<a href="chap71.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) value of <var class="Arg">tbl</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbld8:<div class=="unc><ableclass="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableDirectProduct</code>( <var class="Arg">tbl1</var>, <var class="Arg">tbl2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<span class="GAPprompt">gap></span> <span class="GAPinput">irrd8:= Irr( tbld8 );</span>
[ ("Dihedral8") [1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  Character( CharacterTable( "java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 0
  Character( CharacterTable( "Dihedral(8)" ), [ 1, -1, 1,
   CharacterTable(Dihedral8))" , [1 -1, 1 1 1 ] ),
  Character( CharacterTable( "Dihedral(8)" ), [ 2, 0, -2, 0, 0 ] ) ]
< "APprompt>gt</pan>< "">:=OrdersClassRepresentatives( tbld8 ;/>
[ 1, 4, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MatrixAutomorphisms( irrd8 );</span>
Group([ (4<p>   java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 177
<span class="GAPprompt">gap></span> <span class<p>Theattribute <code class"func">actorsOfDirectProduct</ode>(ahref"hap71.#X7C97CF727FBDFCAB"><span class="RefLink"7120-2/toof<
Group([ (4,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">TableAutomorphisms( tbld8, irrd8 );</span>
Group([ (4,5) ])
</pre></div>

<p><a id="X7D721E3D7AA319F5" name="X7D721E3D7AA319F5<div class="example">pre>

<h5><panclass=G>&;/>< =GAPinput>: ("" 3);/

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TransformingPermutations</code>( <var class="Arg">mat1</var>, <var class="Arg">mat2</var> )</td><td class="tdright">( operation )</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Let <var class="Arg">mat1</var> and <var class="Arg">mat2</var> be matrices. <code class="func">TransformingPermutations</code> tries to construct a permutation <span class="SimpleMath">π</span> that transforms the set of rows of the matrix <var class="Arg">mat1</var> to the setjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>If such a permutation exists, a record with the components <code class="code">columns</code>, <code class="code">rows</code>, and <code class="code">group3 1a 2a 1 2b  2

<p><var class="Arg">r</var><code class=X.1      -  1  1 -11

<p><a id="X849731AA7EC9FA73" name="X849731AA7EC9FA73"></a></p>

<h5>71.22-4 TransformingPermutationsCharacterTables</h5>

<div class="func"><<iv class=">table class=" =100%>tr<class">code =""&8227; </code>( <var class=Arg">tbl</var> )</td><td class="tdright> attribute )</td></tr></table></div>
<p>Let <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var> be character tables. <code class= class="RefLink">71.20-1</spanclass="efLink>.201/>/>,the  of code class"func"FactorsOfDirectProduct</code> is the list of arguments in the <code class="func">CharacterTableDirectProduct</code> (<a href="chap71.html#X7BE1572D7BBC6AC8"><span class="RefLink">71.20-1</span></a>) call.</p>

<p>If such a permutation <span class="SimpleMath">π</span> exists then a record with the components <code class="code">columns</code> (<span class="SimpleMath">π</span>), <code class="code">rows</code> (the permutation of <code class="code">Irr( <var class="Arg">tbl1</var> )</code> corresponding to <span class="SimpleMath">π</span>), and <code class="code">group</code> (the permutation group of table automorphisms of <var class="Arg">tbl2</var>, see <code class="func">AutomorphismsOfTable</code> (<a href="chap71.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)) is returned. If no such permutation exists, <code class="keyw">fail</code> is returned.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tblq8:= CharacterTable( "Quaternionic", 8 );;</span>
<span <571.20- CharacterTableFactorGroup>
[ Character( CharacterTable( "Q8" ), [ 1, 1, 1, 1, 1 ] )java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  Character( CharacterTable( "Q8" ), [ 1, 1, 1, -1, -1 ] ),
  Character( CharacterTable( "Q8" ), [ 1, -1, 1, 1, -1 ] ),
  Character( CharacterTable( "Q8" ), [ 1, -1, 1, -1, 1 ] ),
  Character( CharacterTable( "Q8" ), [ 2, 0, -2, 0, 0 ] ) ]
<span class="GAPprompt">gap></span> <span classcode">/</code> operator for a character table and a list of class positions (see <a href="chap71.html#X7CADCBC9824CB624"><span class="RefLink">71.7</span></a>) delegates to <code class="func">CharacterTableFactorGroup</code>.</p>
[ 1, 4, 2, 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutations( irrd8, irrq8 );</span>
reccolumns : (,group: Group([45),(4 ) rows :=())
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( tbld8, tblq8 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">tbld6:= CharacterTable( "Dihedral", 6 );;</span>
class""gapgt;span spanclass="GAPinput"tbls3: CharacterTable "ymmetric",3);/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TransformingPermutationsCharacterTables( tbld6, tbls3 );</span>
rec( columns := (2,3), group := Group(()), rows := (1,3,2) )
</pre></div>

<p><a id="X8117D940835B0B47" name="X8117D940835B0B47"></a></p>

<h5>71.22-5 FamiliesOfRows</java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 15

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code java.lang.StringIndexOutOfBoundsException: Range [0, 85) out of bounds for length 15
<distributes rowsof thematrixvarclass=Arg></ar>into families follows.Tworows  <arclass"rg></var>belong to the same family there  a of  thatmaps one  to the other . Each in the list var =Arg"maps/var   to  family length 1<p

<p><code class="func"


<dl>
<dt><strong class="Mark"><code class="code">famreps</code></strong></dt>
<dd><p>the list of representatives for each family,</p>

</dd>
<dt><<div class=func"<table class="" width=="100%><tr><td "tdleft"<
 representative,</p>

</dd>
<strong"">code">/>/>/t>
<dd><p>the list that div "func"><ableclass"func"width=100">tr><d class=tdleft"<class"unc"&#tyle='color: green'>8227; ;CharacterTableIsoclinic/(< "rg>/> var java.lang.StringIndexOutOfBoundsException: Range [175, 174) out of bounds for length 267

</< ="/varbe (rdinaryormodular charactertable a  span"impleMath></>with  structure span =S"pGp/ forsome prime<panclass="SimpleMath">p</span>, that is, <span class="SimpleMath">H/Z</span> has a normal subgroup <span class="SimpleMath">N</span> of index <span class="SimpleMath">p</span> and a central subgroup <span class="SimpleMath">Z</span> of order <span class="SimpleMath">p</span> contained in <span class="SimpleMath">N</span>.</p>
/>
<p><a id="X81272CEE79F13E7B" name="X81272CEE79F13E7B"></a></p>

<h4>71.23 <span class="Heading">Storing Normal Subgroup Information</span></h4>

<p><a id="X7E66174C7C7A8C0C" name="X7E66174C7C7A8C0C"></a></p>

<h5>71.23-1 NormalSubgroupClassesInfo</h5>

div class=func"< class==func"width"%"<<d="tdleft"<code =func>&8227;NormalSubgroupClassesInfocode> <varclass=Arg><var )<td><td class="dright">(nbspattribute&bsp)&lt;/d</tr</table<div>>
<> var=">tbl</var> be the ordinary character table of the group <span class="SimpleMath">G</span>. Many computations for group characters of <span class="SimpleMath">G</span> involve computations in normal subgroups or factor groups of <span class="SimpleMath">G</span>.</p>

<In  java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 601

<p>In order to do these computations only once, one stores in the group a record with components to store normal subgroups, the corresponding lists of conjugacy classes, and (if necessary) the factor groups, namely</p>


<dl>
<dt><strong class="Mark"><code class="code">nsg</code></strong></dt>
<java.lang.StringIndexOutOfBoundsException: Range [61, 58) out of bounds for length 92

</dd>
<dt><strong class="Mark"><code class="code">nsgclasses</code></strong></dt>
<dd><p>at position <span class="SimpleMath">i</spank</span>,</p>

</d
<dt><strong class="Mark"><code class="code">nsgfactors</code></strong></dt>
<p>t  <spanclass"impleMath>i</span>  bound, the factor group modulo the span ="SimpleMath>i</>th entry the<codeclass=code"nsg</ode>component.<p>

</dd>
</dl>
<p><code class="func">NormalSubgroupClasses</code> (<a href="chap71.html#X87E7391F7F92377C"><span class="RefLink">71.23-3</span></a>), <code class="func<d><>the   java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 103

<p>So if you need information about a normal subgroup for that you know the conjugacy classes, you should get it<>Thereis no  method <computing/>the value < ="unc"SourceOfIsoclinicTable<pjava.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119

<p

<p><a id="X7C2A87E085111090" name="X7C2A87E085111090"></a></p>

<h5>71.23-2 ClassPositionsOfNormalSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ClassPositionsOfNormalSubgroup</code>( <var class="Arg">tbl</var>, <var class="Arg">N</var> )</td><td class="tdright">( java.lang.StringIndexOutOfBoundsException: Index 230 out of bounds for length 76
<p>is the list of positions of conjugacy classes of the character table <var class="Arg">tbl</var> that are contained in the normal subgroup <var class="Arg">N</var> of the underlying group of <var class="Arg">tbl</var>.</p>

<p><a id="X87E7391F7F92377C" name="X87E7391F7F92377C"></a></p>

<h5<span class="GAPprompt">gap></span> <span class="GAPinput">t:= CharacterTable( g );;</span>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NormalSubgroupClasses</code>( <var class="Arg">tbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the normal subgroup of the underlying group <span class="SimpleMath">G</span> of the ordinary character table <var class="Arg">tbl</var> that consists of those conjugacy classes of <var class="Arg">tbl</var> whose positions are in the list <var class="Arg">classes</var>.</p>

<p>If <code class="code">NormalSubgroupClassesInfo( <var class="Arg">tbl</var> ).nsg</code> does not yet contain the required normal subgroup, and if <code class="code">NormalSubgroupClassesInfo( <var class="Arg">tbl</var> ).normalSubgroups</code> is bound then the result will be identical to the group in <code class="code">NormalSubgroupClassesInfo( <var class="Arg">tbl</var> ).normalSubgroups</code>.</p>

<p><a id="X79D451F0808EB252" name="X79D451F0808EB252"></a></p>

<h5>71.23-4 FactorGroupNormalSubgroupClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FactorGroupNormalSubgroupClasses</code>( <var class="Arg">tbl</var>, <var class="Arg">classes</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the factor group of the underlying group <span class="SimpleMath">G</span> of the ordinary character table <var class="Arg">tbl</var> modulo the normal subgroup of <span class="SimpleMath">G</span> that consists of those conjugacy classes of <var class="Arg">tbl</var> whose positions are in the list <var class="Arg">classes</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( 4 );</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( g, "S4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );</span>
CharacterTable( S4 )
<span class="GAPprompt">gap></span> <span class="GAPinput">irr:= Irr( g );</span>
[ Character( CharacterTable( S4 ), [ 1, -1, 1, 1, -1 ] ),
  Character( CharacterTable( S4 ), [ 3, -1, -1, 0, 1 ] ),
  Character( CharacterTable( S4 ), [ 2, 0, 2, -1, 0 ] ),
  Character( CharacterTable( S4 ), [ 3, 1, -1, 0, -1 ] ),
  Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">kernel:= KernelOfCharacter( irr[3] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsSet(kernel);</span>
[ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName(kernel,"V4");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasNormalSubgroupClassesInfo( tbl );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalSubgroupClassesInfo( tbl );</span>
rec( nsg := [ V4 ], nsgclasses := [ [ 1, 3 ] ], nsgfactors := [  ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassPositionsOfNormalSubgroup( tbl, kernel );</span>
[ 1, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := FactorGroupNormalSubgroupClasses( tbl, [ 1, 3 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalSubgroupClassesInfo( tbl ).nsgfactors[1] = G;</span>
true
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap70.html">[Previous Chapter]</a>    <a href="chap72.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

Messung V0.5 in Prozent
C=98 H=100 G=98

¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.224Bemerkung:  ¤

*Bot Zugriff






Entwurf

Suchen

PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.






                                                                                                                                                                                                                                                                                                                                                                                                     


Neuigkeiten

     Aktuelles
     Motto des Tages

Open Source Software

     Quellcodebibliothek
     Eigene Quellcodes
     Fremde Quellcodes
     Suchen

Jenseits des Üblichen ....

Besucherstatistik

Besucherstatistik

Statistik
#Sources=1127926
#Domains=2039723