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<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=func8227;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= ( | | |