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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7F8AB7CB7A46002F">71.2 <span class="Heading">History of Character Theory Stuff in GAP</span></a>
</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 /><span class="nocss"> </span><a hrefdiv ">a hrefc.#B7"71 span ="Character Tables</span></a>
<span
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X85DB8AE7786A2DB5">71.3-3 CharacterTableRegular</a></span>
< class=ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7DBEF4BF87F10CD6">71.3-4 SupportedCharacterTableInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8195BC057B1DFAD5">71.3-5 ConvertToCharacterTable</a></span>
</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>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X82FF82C87CF82ADF">71.4-1 IsNearlyCharacterTable</a></span>
<span class="ontSS>br /><span class="nocss"> </span><a href="chap71.html#X7C6F3D947E5188D1">71.4-2 InfoCharacterTable</a></span>
<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 Character Tables</span></a>
</span>
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<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</spanspan class="ontSS><br/<span class="nocss" nbsp;/span><a href=chap71.html#X8195BC057B1DFAD5">71.3-5 ConvertToCharacterTable</a></span>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> <span
<spanclass="ContSS"<br< classnocss"&&bsp</pan><a href="chap71.html#X849A38F887F6EC86">71.6-2 ConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X84DC12AA804C8085">71.6-3 IdentificationOfConjugacyClasses</a><span class="ContSS"<br/<span class"nocss">nbsp; <span<ahref=="hap71.#X7C6F3D947E5188D1""71.-InfoCharacterTable</a></span>
<span class=""<br /><span class="nocss"> </span><a href="chap71.html#X8788C6C7829C1ADE">71.6-4 CharacterTableWithStoredGroup</a></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 class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X849A38F887F6EC86">71.6-2 ConjugacyClass
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  < </><spanclass>nbsp <pan<a"html"71.6-CharacterTableWithStoredGroup<a><java.lang.StringIndexOutOfBoundsException: Index 153 out of bounds for length 153
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X873B3CC57E9A5492">71.8-2 <span class="Heading">Irr</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8549899A7DE206BA">71.8-3 <span class="Heading">LinearCharacters</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8011EEB684848039">71.8-4 <span class="Heading">OrdinaryCharacterTable</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href"chap71.tml##81EFD9FE804AC6EE">71.5 < class"">Operations to Character Tables</><a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss">&<span>
<span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X86F455DA7A9C30EE">71.9-1 OrdersClassRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7CF7907F790A5DE6">71.9-2 SizesCentralizers</a></span>
<#X7D9D2A45879A6A62"7193 />spanjava.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 7
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7F58A82F7D88000A">71.9-5 <span class="Heading">UnderlyingCharacteristic</span></a>
</spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<span class="ContSS"><br /><span class="nocss"> </spanclass="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X86F455DA7A9C30EE">71.9-1 OrdersClassRepresentatives</a></span>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8333E8038308947E">71.9-7 <span class="Heading">Class Parameters and Character Parameters</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7F58A82F7D88000A">71.9-5 <span class="Heading">UnderlyingCharacteristic</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7932C35180C80953">71.9-9 InfoText</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 139
<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="java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 7
<span class="ContSS><<r ><panclass=="nocss"> &bsp;<span><<href=chap71.html#F863B15804E0835"71.9-13 ClassRoots</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X79CEBC3C7E0E63DF">71.10 <span class="Heading">Normal Subgroups Represented by Lists of Class Positions</span><span class="ontSS"><br /><span class="nocss"> </span><a ".#">71.910 <a<span>
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<r /<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 156
<span class="ContSS"><br /><span class="nocss"> </span><a href classContSS>br/spanjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 156
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7A6B1F8A84A495DC">71.<spanclass"ontSS><br ><pan class="ocss" nbsp;</span><a href="chap71.html#X8491DA0981D6F264">71.10-2 ClassPositionsOfAgemo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7D53F60785AB22B1">71.10-4 ClassPositionsOfDirectProductDecompositions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X79EE7BE17BD343D5">71.10-5 ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X79EE7BE17BD343D5">71.10-5 ClassPositionsOfDerivedSubgroup
<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""711011ClassPositionsOfSupersolvableResiduum</></>
<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>
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<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>
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<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 href="chap71.html#X7FF4CE4A7A272F88">71.11-3 BlocksInfo</a></span>
<="chap71.html#X84701640811D2345>71.114DecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X83EC921380AF9B3B">71.11-5 LaTeXStringDecompositionMatrix</a></span>
<<div
<div class="ContSect"><span class="tocline"><spanspan ""<br>< "ocss>nbsp&;span><a href="chap71.html#X83EC921380AF9B3B">71.11-5 LaTeXStringDecompositionMatrix</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8441983C845F2176">71.12-1 Index</a></span>
<spanclass=C"br/>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 147
<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</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7E2EA9FE7D3062D3">71.12-7 ClassMultiplicationCoefficient</a></span>
<span class"ContSS">br/<class=nocss>nbsp;nbsp</span><a href"htmlX7A19F56C7FD5EFC7">71.12-8 ClassStructureCharTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X809E67E57D4933B3">71.12-9 MatClassMultCoeffsCharTable</a></span>
<div/div
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7C1941F17BE9FC21">71.13 <span class="Heading">Printing Character <pan class=""><br />class=""  <ahref="chap71.#X7A19F56C7FD5EFC7">71.12-8 ClassStructureCharTable</a></span>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7D45224B86D802E5">71.13-1 ViewObj</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X836554207C678D41">71.13-2 PrintObj</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7B41F36478C47364">71.13-3 Display</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X85E883A87A190AA4">71.13-4 DisplayOptions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X79EC9603833AA2AB">71.13-5 PrintCharacterTable</a></span>
</div></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>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7ED39DB680BFEA96">71.14-1 IrrDixonSchneider</a></span>
<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"><span class="nocss"> </span><a href="chap71.html#X7E51AACD79CE0BC8">71.15 <span class="Heading">Representations Given by Modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><?xml version="1.0" encodingUTF-">
<h:www.orgjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 61
<span=C"< >span class="nocss"> </span><a href="chap71.html#X7EB88B2E87AF5556">71.15-3 RegularModule</a></span>
<div/div>
<div class="ContSect"><span class="tocline"><span class="nocss"> <eta name="generator"content"APDoc2HTML java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
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<java.lang.StringIndexOutOfBoundsException: Range [55, 4) out of bounds for length 205
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7C398F2680C8616B">71.17-1 DixonRecord</a></span>
<span class="ContSS"><br /0"name"X7B7A9EE881E01C10"</>/>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X868476037907918F">71.17-3 DixontinI</a></span>
<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/spanjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 6
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7C85B56C80BFA2E3">71.17-6 DxIncludeIrreducibles</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X87A5B5C77F7F348E">71.17-7 SplitCharacters</a></span>
< class"ontSS"> /<span class=nocss">">nbsp;&bsp;</span><a href="chap71.#E7EA85BC8>71.178 IsDxLargeGroupjava.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 139
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7C1153637E7D2133">71.18 <span class="Heading">Components of a Dixon Record</span></a>
</span>
</div>
<div class="ContSect"><span "ContSS">br>n">java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 151
<
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"><java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X836554207C678D41">71133Display</a>/pan>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7C97CF727FBDFCAB">71.20-2 FactorsOfDirectProduct</a></span>
<pan "">br >< class">nbsp <span><href==ch#7C3A4E5283B240BE">.-CharacterTableFactorGroup></span
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X85BE46F784B83938/iv>/>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X806E55A58397B11B">71.20-5 CharacterTableOfNormalSubgroup</a></span>
classContSS"></>span ="">nbsp&bsp;<span>href=chap71.X79B75C8582426BC5">71.20-6 CharacterTableWreathSymmetric</a></span>
span class="ContSSbr/>spanclass=njava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 154
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&span class="ContSS"><br /><span class="nocss"> </span><aX7ED39DB680BFEA9671.-IrrDixonSchneider<a<span>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7D9C4A7F8086F671">71.21-1 CharacterTableWithSortedCharacters</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X87E3CF317D8E4EC7">71.21-2 SortedCharacters</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7E3DE0A47E62BE6B">71.21-3 CharacterTableWithSortedClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X82DCAAA882416E24">71.21-4 SortedCharacterTable</a></span>
span=ContSS">br />span class="nocss"> </span><a href="chap71.html#X8099FEDC7DE03AEE">71.21-5 ClassPermutation</a></span>
</div>div class"ContSSBlock>
<div class="ContSect"><span class="tocline"><span class="spanclass="ontSS> />span class"ocss>nbsp;n;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 143
<java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss<panclass"ContSS">br/>span class=="ocss> /span><a href="chap71.html#X7EB88B2E87AF5556">71.15-3 RegularModule</a></span>
<="ContSS">br><span class="ocss">nbsp;nbsp;<span<ahref=chap71.#X8082DD827C673138>71.22-2 TableAutomorphisms<a><span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7D721E3D7AA319F5">71.22-3 TransformingPermutations</a></span>
<span class="ContSS"><</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="divclass="ontSect"<class="tocline"><span class="nocss>nbsp;<span><href=chap71html##7117 span class""Advanced Methods -Calculations</span></a>
</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 span ="< /<spanclass==">nbsp;&bsp</< href=".X868476037907918F"17- ixontinIa<span>
<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 "ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X79D451F0808EB252">71.23-4 FactorGroupNormalSubgroupClasses</a></span>
</div></div>
<div>
< java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 57
< operationsforemtablesfinite/m.<<p
<p>Operations for <em>characters</em> (or, more general, <em>class functions</em>) are described in Chapter <a href="chap72.html#X7C91D0D17850E564"><spandiv<div>
<p>For a description of the <strong class="pkg">GAP</strong> Library of Character Tables, see the separate manual for the <strong class=java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 7
<p>Several examples in this chapter require the <strong class="pkg">GAP</strong> Character Table Library to be available. If it is not yet loaded then we load it now.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib" );</span>
true
</pre></div>
p< =X7B9FCBBC7B95F91B" name=X7B9FCBBC7B95F91B">/>/>
<h4>71.1 <span class="java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 7
<p>It seems to be necessary to state some basic facts –and maybe warnings– at the beginning of the character theory package. This holds for people who are familiar with character theory because /panjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<p>Intuitively, <em>characters</em> (or more generally, <em> class="ContSS"><br /><span class="nocss"> </spanhtml#A38F887F6EC86>716- ConjugacyClasses/a><span>
<p>At this stage one could ask where there is a problem, since all these algebraic structures are supported by <strong class="pkg">GAP</strong>. But in practice, these structures are of minor importance, compared to individual characters and the <class=ContSS"<br/<class"ocss" <span>href"html#java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 153
<p>For computations/span>
<>AsAsaconsequence, 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/span>
<p>So typical calculations with characters involve loops over /java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<p class="pcenter">[ χ, ψ ] = ( ∑_{g ∈ G} χ(g) ψ(g^{-1}) ) / |G|</p>
<p>can be written as</p>
<class="example"><pre>
Sum(/span>
ComplexConjugate( psi[]))/ ( ;
</prediv><//iv>
<p>where <code class="code"<span>
<p theory that one has translated a problem concerning groups into a problem concerning only characters, the necessary calculations are mostly simple. For example, one can often prove that a group is a Galois group over the rationals using calculationsspan class="ontSS">br/<span class=nocss" nbsp;;/span><ahref=chap71.html#X7D9D2A45879A6A62">71.9-3 SizesConjugacyClasses</a></span>
<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; in the example above, we<span>
<p>Note that the character table of a group <span class="SimpleMath">G</span> in <strong class="pkg">GAP</strong> must <em>not</em> be mixed up<span>
<p>Two further important instances of information that depends on the ordering of conjugacy classes are <em>power maps</em> and <em>fusion maps</em>. Both are represented as lists of integers in <strong class="pkg">GAP</strong>. The <java.lang.StringIndexOutOfBoundsException: Range [1, 238) out of bounds for length 7
<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 they are regarded as class functions etc.java.lang.StringIndexOutOfBoundsException: Index 183 out of bounds for length 136
<p>One can construct class function objects that store their underlying character span class"ontSS"br/<span class=nocss" &bsp;<< =java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 135
<p><a id="X7F8AB7CB7A46002F" name="X7F8AB7CB7A46002F"></a></p>
<h4class=ContSect"<span class"tocline"<span class="ocss"&;</span< href="chap71.#79java.lang.StringIndexOutOfBoundsException: Range [117, 115) out of bounds for length 212
<p><strong class="pkg">GAP</strong> provides functions for dealing with group characters since the version <strong class="pkg">GAP</strong> 3.1,span>
<p>For integrating the existing character theory algorithms, <strong class="pkg">GAP</strong<class=ContSS"<br/>spanclass="nocss>  /span><href="hap71.html#DA0981D6F264">.- ClassPositionsOfAgemo</a></span>
<p>The development of character theory code for <strong class="pkg">GAP</strong> has been supported by several DFG grants, in particularspan class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#spanclass="ContSS"</<span class="nocss" nbsp;/span><a =chap71.html#X79EE7BE17BD343D5">71.10-5 ClassPositionsOfDerivedSubgroup</a></span>
<p>The major contributions can be listed as follows.</p>
<ul>
<i<p>The arithmetic for the cyclotomics data type, following <a href="chapBib.html#biBZum89">[Zum89]</a>, was first implemented by Marco van Meegen; <span class="ContSS"<br />span class="nocss" nbsp<span>><a href=chap71.html#X877FDE8A84A9F52C71.-10 ClassPositionsOfSolvableRadical<a<span>
</li>
<li><p>The span class"ontSS"><br /><span class="nocss"> </span><a href="chap71.html#X7FCF905D7FD7CC40">71.10-13 java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 12
</li>
<li><p>The lattice related functions, java.lang.StringIndexOutOfBoundsException: Range [1, 42) out of bounds for length 7
</li class"ContSSbr /<pan class=">;nbsp;</span><ahref="chap71.tml#X7E80E35985275F35">.112SameBlocka>/pan>
<i<p>unctions computing possible class ,possible power maps, and table automorphisms were written by Thomas Breuer (see <a href="chapBib.html#biBBre91">[Bre91]</a>).</p>
</li>
<li><p>Functions for computing possible permutation characters were written by Thomas Breuer class"java.lang.StringIndexOutOfBoundsException: Range [28, 19) out of bounds for length 155
</li>
<li><p>Functions for computing character tables from groups were written by Alexander Hulpke (Dixon-Schneider/div>/div>
</li>
<li><p>Functions for dealing with Clifford matrices were written by Ute Schiffer (see <a href="chapBib.html#biBSchiffer94">[Sch94]</a>).<//span
</li>
<li><p>Functions for monomiality questions were written spanclass=C><r /<nocss"& </><a href="chap71.html#X8123650E817926FC">71.12-2 IsInternallyConsistent</a></span>
</li>
</ul>
<p>Since then, the code has been maintained and extended further by Alexander Hulpke (code related to his implementation of the Dixon-java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 7
<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>713 < =java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 68
<p>There are in general span class="ContSS"><br / class="nocss"  <span>><a href="hap71.html#X7C3A4E5283B240BE">71.20-3 CharacterTableFactorGroup</a></span>
<ol>
<li><p>compute the table from a group,</p>
</li>
<li><p>read a file that contains the table data,</pspan=>/<spannocss>nbsp;nbsp;/< =chap71.html#X79B75C8582426BC5">71.20-6 CharacterTableWreathSymmetric</a></span>
</li>
<li><p>construct the table using generic formulae,</p>
</li>
<li><p>derive it from java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 12
</li>
<li><p>combine partial information about conjugacy classes, power maps of the group in question, and about (character tables
</li>
</ol>
<p>In 1., the computation of the irreducible charactersspanclass=>< /<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 156
<>The codeclass"">/> =cjava.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 178
<p><a id="X7FCA7A7A822BDA33" name="X7FCA7A7A822BDA33"></a></p>
<h5>71.3-1 <span class="Heading">CharacterTable</span></h5>
<divspan>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; CharacterTable</code>( <var class="span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8082DD827C673138">span =">< /<class"nocssnbsp;n;span><a href=.#7".- TransformingPermutations<span
<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>] )</td><td class="tdright">( operation )</td></tr></table></div>
<>Called with <var=Gvar>java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 603
<p>Called with a string <var class="Arg">name</var> and perhaps optional parameters <var class="Arg">param</var>, <code class="func">CharacterTable</code> tries to access a character table from the <strong class="pkg">GAP</strong> Character Table Library. See the manual of the <strong class="pkg">GAP</strong> package <strong class="pkg">CTblLib</strong> for an overview of admissible arguments. An error is signalled if this <strong class="pkg">GAP</strong> package is not loaded in this case.</p>
<pdivdiv
<p>Thereason <ode "func"CharacterTable</code> does not compute the irreducible characters is that there are situations where one only needs the <q>table head</q>, that is
<p>For details about different algorithms to compute the irreducible characters, see <a href="chap71.html#X79BC08C6846718D9"><span class="RefLink">71.14</span></a>.</p>
<p>If the group <var class="Arg">G</var> is given as an argument, <code class="func">CharacterTable</code> accesses the conjugacy classes of <var class="Arg">G</var> and therefore causes that these classes are computed if java.lang.StringIndexOutOfBoundsException: Range [0, 226) out of bounds for length 0
<p><a id="X8476B25A79D7A7FC" name="X8476B25A79D7A7FC"></a></
<h5>71.3-2 <span class="Heading">BrauerTable</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; BrauerTable</code>( <var class="Arg">ordtbl
<div =func"<class" ="%">><java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 245
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; BrauerTableOp</code>( <var class="Arg">ordtbl</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<p or a group var Arg><var>< class"unc><>returns itsvar =">var-modular charactertableif<java.lang.StringIndexOutOfBoundsException: Range [220, 219) out of bounds for length 318
<p>The <var class="Arg">p</var>-modular table can be computed in the following cases.</p>
<>
<li><p>The java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
</li>
<li><p>the Sylow <var class="Arg">p</var>-subgroup of <var class="Arg">G</var> is cyclic,
</li>
<li><p>the table <var class="Arg">ordtbl</var> stores information how it was constructed from other tables (as a direct product or as an isoclinic variant, for example), and the Brauer tables of java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p><var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
</li>
</ul>
<>The method for aand aprime delegates <code class="BrauerTable</java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 452
<p>So if one wants to install a new method for computing Brauer tables then it is sufficient to install it for <code class="func">BrauerTableOp</code>.</p>
<p>The <code class="keyw">mod</code> operator for a character table and a prime (see <a href="chap71.html#X7CADCBC9824CB624"><span class="RefLink">71.7</span></a>) delegates to <code class="func">BrauerTable</java.lang.StringIndexOutOfBoundsException: Index 209 out of bounds for length 63
<p><a id="X85DB8AE7786A2DB5" name="X85DB8AE7786A2DB5"></a></p>
<h5>71.3-3 CharacterTableRegular</h5>
<iv "><java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 256
<p>For an ordinary character table <var class="Arg">tbl</var> and a prime integer <var classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The returned table head can be used to create <var class="Arg">p</var>-modular Brauer characters, by restricting ordinary characters, for example when one is interested in approximations of the (unknown) irreducible Brauer characters.</p>
<div class="example"><pre>
spanclass"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 93
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( g );; HasIrr( tbl );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2:= CharacterTable( tbl, 2 );</span>
BrauerTable( Sym( [ 1 .. 4 ] ), 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = CharacterTable( tbl, 2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = BrauerTable( tbl, 2 );</span>
true
<span class="GAPprompt">gap></span<>aid"X7F8AB7CB7A46002F" name="X7F8AB7CB7A46002F"></a></p>
true
<spanh471.<span class"Heading>History of Stuff inGAP<span>/h4>
CharacterTable(>class"kg>AP<strong java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 1068
">ap></span> <span class="GAPinput">CharacterTableRegular( libtbl, 2 );</span>
BrauerTable( "M", 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">BrauerTable( libtbl, 2 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "Symmetric", 4 );</span>
"4")
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ , BrauerTable( Sym( [ 1 .. 4 ] ), 2 ) ]
</pre></div>
<p><a id="X7DBEF4BF87F10CD6" name="X7DBEF4BF87F10CD6"></a></p>
<h5>.4 </>
<div class="func"><table class="func" width="100racter were writtenby java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 114
<p>codeclass"unc<code>java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 534
<p>When (ordinary or Brauer) character table objects are created from records, using <code class="func">ConvertToCharacterTable</code> (<a href="chap71.html#X8195BC057B1DFAD5"><span class="RefLink">71.3-5</span></a>), <code class="func">SupportedCharacterTableInfo</code> specifies those record components that shall be used as attribute values; other record components are <em>not</em> be regarded as attribute values in the conversion process.</p>
<p>New attributes and properties can be notified to <code class="func">SupportedCharacterTableInfo</code> by java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 0
<p><a id="X8195BC057B1DFAD5" name="X8195BC057B1DFAD5"></a></p>
<.5/java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 39
<div class="func"lipFunctions monomiality questions were writtenby java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 98
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227<li>
<>Letvar"Arg"><>be record <codeclass">ConvertToCharacterTablecode>converts<varclass=Arg"><var intoa object(see<a href=".X866"<span java.lang.StringIndexOutOfBoundsException: Range [220, 219) out of bounds for length 658
<p>If inconsistencies in <var class="Arg">record</var> are detected, <code class="keyw">fail</code> is returned. <var class="Arg">record</var> must have the component <code class="code">UnderlyingCharacteristic</code> bound (cf. <code class="func">UnderlyingCharacteristic</code> (<a href="chap71.html#X7F58A82F7D88000A"><span class="RefLink">71.9-5</span></a>)), since this decides about whether the returned character table lies in <code class="func">IsOrdinaryTable</code> (<a href="chap71.html#X82FF82C87CF82ADF"><span class="RefLink">71.4-1</span></a>) or in <code class="func">IsBrauerTable</code> (<a href="chap71.html#X82FF82C87CF82ADF"><span class="RefLink">71.4-1</span></a>).</p>
<p><code class=">
<p>An example of a conversion from a record to a character java.lang.StringIndexOutOfBoundsException: Index 1 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="X82FF82C87CF82ADF
<h5>71.4-1 IsNearlyCharacterTable</h5>
=
<div="">class="unc 100>=java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 225
<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>( <var class="Arg"obj/ <td>< ""nbspcategory&</></><<
<iv class=""<table class"func"width=100%"<tr>tdclass="dleft>code class"func">8227; IsCharacterTableInProgress</code>( var ="Arg"obj/ar )/td< =""&;ategory)</d/></table/
<p>Every <q>div class="func>table class="func" width="100%"><tr><td <span class="ontSS"<br>="">&;n;><ahref=c.htmlX85E883A87A190AA471.34 DisplayOptionsa<spanjava.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 139
<java.lang.StringIndexOutOfBoundsException: Range [88, 2) out of bounds for length 413
<p>Furthermore, <code class="func">IsOrdinaryTable</code> and <code class="func">IsBrauerTable</code> denote character tables that provide enough information to compute all power maps and irreducible characters (and in the case of Brauer tables to get the ordinary table), for example because the underlying group (see <code class="func">UnderlyingGroup</code> (<a p>Called with a group <var class="Arg">G</var>, <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 argument a group <var class="Arg">G</var> or an ordinary character table <var class="Arg">ordtbl</var>, and
p character objects code ="IsCharacterTable<code> are always immutable (see <a href="chap12.html#X7F0C119682196D65"><span class="RefLink">12.6</span></a>). This means mainly that the ordering of conjugacy classes used java.lang.StringIndexOutOfBoundsException: Range [1, 248) out of bounds for length 188
<p>The <strongdivclass="ContSSBlock"
>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SymmetricGroup( <><div>
<span"GAPprompt"gap&;/ <class=GAPinput>: CharacterTable g ;modtbl: tblmod 2;/pan
CharacterTable( Sym( [ 1 .. 4 ] ) )
BrauerTable( Sym( [ 1 .. 4 ] ), 2 )
<h5>71..3s="ontSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7C083207868066C1">71.17 <span class="Heading">Advanced Methods for Dixon-Schneider Calculations</span>/span>
true
true
pt<span> <span class=GAPinput">IsBrauerTable( modtbl ); IsBrauerTable( tbl );</span>
true
false
<span class="GAPprompt">gap>Calledwith<pan "<r/>span classocss"">nbsp;nbsp<span<a ="hap71.html#X7BFD4C1A821731FB">71.17-5 BestSplittingMatrix</a></span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCharacterTable( g ); java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 0
alse
false
<li>
<p><a id="X7C6F3D947E5188D1"
<h5>71.</h5>
<div class=java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 5
</p>
p< id"FA867637EBB30F9 name="7"<a/p>
<NearlyCharacterTablesFamily/>
<ivclass=="func"<class="unc width=="100%<tr<td class="tdleft"<code class=func"‣ NearlyCharacterTablesFamily</code></td><td class="tdright">( family )</td></tr></table></div>
="3.<span>/a>>.</p>
<p< id"829C4B6E83998F40"name"X829C4B6E83998F40"></a></p>
java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 75
<p>The following few conventions should be noted.</p>
<
<li<java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 7
</li>
<li><p>The <em>trivial character</em> =div =func>class=""width=%>trtd=""> =unc&8227;CharacterTableRegular<code>var "rg>var, var=Arg"p/ar<pan => >java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 156
</li>
<li><pMost java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 12
</li>
<>p<span class="gapgt;</><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 93
</li>
<p><a < java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 191
<h4>71divjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 25
<p>For a span class="GAPprompt>gap&;</>spanclass="GAPinput>tblmod2= tbl 2)</<span ="ContSS"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 155
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<> and a ofa isomorphic spanclass="/pan (orexample character table from the <strong class="pkg">GAP</strong> table library), one can tell <strong java.lang.StringIndexOutOfBoundsException: Index 256 out of bounds for length 4
<71<classH"java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 57
<p><a id="X7FF4826A82B667AF" name=" class="GAPprompt">gap></span> <spanGAPinputB
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
<anordinary java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 166
<p>Brauer tables do not store the underlying group, they access it via the ordinary table (see <code class="func">/><divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
</>/>
<h5>71.6-2 ConjugacyClasses</h5>
<div ="< p><code class="func">Supporte</code isa that java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 534
<p>For a character table <var class="Arg">tbl</var> with known underlying group <spanp> ordinaryorBrauerjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
<p>This ordering need <em>not</em> p>As a consequence, also other information involving characters is given implicitly. For example, we can talk about the kernel of
<>The class of the identity element must be the first class (see <a Let <var class="Arg">record</var> be a record. <code class="func">ConvertToCharacterTableclass="Arg>record</arinto
<p>Ifjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<
<p><a id="X84DC12AA804C8085" name="X84DC12AA804C8085"></a></p>
<h5>71.6-3 IdentificationOfConjugacyClasses</h5>
<div class="func"><table class="func" width="100%"><tr
<>java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 562
<div class="exampleh5>71can construct class function objects that store their underlying character table and other attribute values (see Chapter <a href="chap72.html#X7C91D0D17850E564"><span class="RefLink">72</span
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsCharacterTable</code>( <var classdiv class="func"><table class="func" width="100%java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class="APprompt">gap></span> <span class="GAPinput">repres
< -> ConjugacyClass( g, x) ;;/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetConjugacyClasses( g, ccl );</span>
<">gap&t;;</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 136
<span class="GAPprompt">gap></span> <span class="GAPinput">HasConjugacyClasses( tbl ); HasUnderlyingGroup( tbl );</span>
true
true
"GAPinput<=java.lang.StringIndexOutOfBoundsException: Range [81, 79) out of bounds for length 722
true
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 0
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentificationOfConjugacyClasses( tbl );</span>
[ 5 1,2,3, 4]
</pre></div>
<p><a
<h5>71.6-4 CharacterTableWithStoredGroup</h5>
<</code>( <var classspan class=>ap;</span <class=GAPinput>:CharacterTable ;modtbl:mod2;<pan
<p>Let <var class="Arg">G</var> be a group and <var class="Arg">tbl(Sym( 1.. 4] ) 2 )
<p this identification is unique up to java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p>Otherwise, i.e., if java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>If a record is present as<li<p>Functions for monomialityquestions were written by Thomas Breuer and Erzsébet Horváth.</p>
<span pCurrently <strong class="strong>does provide functionality for Brauer tables, but there is an interface to the <strong class="pkg">MOC</strong> system (see <a href="chapBib.html#biBHJLP92">[HJLP]</a>), and the
<p><a id="X790019E87CFDDB98" name="X790019E87CFDDB98"></a></p>
<h5>71.
< java.lang.StringIndexOutOfBoundsException: Range [11, 8) out of bounds for length 117
<p>If the arguments <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>If >43NearlyCharacterTablesFamily
<p>The usual approach div class="func"><table" width=""><tr<td java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The following invariants are used.</p>
<
<li><p>element orders (see <code class="func">li><p>combine partial information about conjugacy classes, power maps in question, java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 0
/>
<li><p>class lengths (seejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</>
<li><p>power maps (see <code class="func">PowerMap</code> (<a href="chap73.html#X781FAA497E3B4D1A"<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<
<li><p>symmetries of the table (see <code class="func">AutomorphismsOfTable</code> (<a/java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
</li>
</ol>
cord java.lang.StringIndexOutOfBoundsException: Range [98, 97) out of bounds for length 214
<dl>
<dt><strong class="Mark"><code class="code">natchar</code> </strong></dt>
<dd><p>if <span class="SimpleMath">G</span> is a permutation group or matrix group
</dd>
<dt><strong class="Mark"><code class="code">java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 5
ddpalistdescribing partial ;the <<span=S<dd><p>a list describing a partial bijection; the <span class="SimpleMath id==E0EBF84B07313 name"793E0EBF84B07313"<a>/p
/
</dl>
<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [82, 5) out of bounds for length 95
Alt( [ 1 .. 5
<span">gapgt;/span <span class"GAPinput">tbl:= CharacterTable( "A5");</java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
CharacterTable "A5")
<span class="GAPprompt">gap></span> <<iv class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">8227; BrauerTable</code>( <var java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 0
<<id=X7FF4826A82B667AF"ame"F</<java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
false
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibleConjugacyClasses
[ ]
<span class="GAPprompt">gap></span>
CharacterTable( Alt(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span"">><><span=GAPinputIrr
true
<span classli
true
true
<span class="GAPprompt
[
<span class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
<<p>Th method a<a class==Arg><var with known group<span class="SimpleMath">G</span>, the <code class="func">ConjugacyClasses</code> value of <var class="Arg">tbl</var> is a list ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 17, ,[ 9,10]]
</pre></div>
<<id"=X7CADCBC9824CB624"/>p
>
<p>The returned table headcan java.lang.StringIndexOutOfBoundsException: Range [0, 38) out of bounds for length 0
<dl( [1.. java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
<dt><java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<>pthe product two
<
<dt< class=Mark"code "code> class"rg">tbl<pordinaryjava.lang.StringIndexOutOfBoundsException: Range [34, 28) out of bounds for length 562
<dd><p>the table of the factor group modulo the normal subgroup spanned by<div class"example<prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
</dd>
<dt<strong class="ark"<code class=code">var class="Arg"><var mod <var class=Arg>p<</ar></ode><<rue
<dd><p>the <var class="Arg">p</var>-modular Brauer character table corresponding to the class="GAPprompt">gap></span> <span class="GAPinput">ccl:= List( repres, x -> ConjugacyClass;/<
</dd>
<dt><strong class="Mark"><code class=<span "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 94
<dd><p>the position of the class with name <var class="Arg">name</var> in <var class="Arg">tbl</var> (see <code class="func">ClassNames</code> (<a href="chap71.html#X804CFD597C795801"><span class="RefLink">71.9-6</span></a>)).</p>
</dd>
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 5
<p><a id="X7F9D58208241D35E" name="X7F9D58208241D35E"></a></p>
<>attributesand propertiescanbenotified < class"unc"SupportedCharacterTableInfobycreating them with <code class="code">DeclareAttributeSuppCT</code> and <code class="code">DeclarePropertySuppCT</code> instead of <code class="func">DeclareAttribute</code> (<a href="chap13.html#X7A00FC8A7A677A56"><span class="RefLink">13.5-4</span></a>) and <codejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>Severalemattributes groupsem>arevalidalso forcharactertables.</
<p>These are first those that have the same meaning for both the group and its character table, p> a record is as the third argument var class==Arg></var>, its meaning is the same as the optional argument <var class="Arg">arec</var> for <code class="func"><div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConvertToCharacterTableNC</code>( <var class="Arg">record</var> )</td><td class="tdright">( funct
<p>Second, there java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
<p><a id="X81FEFF768134481A" name="X81FEFF768134481A"></a></p>
<h5>71.8-1 <p>If the arguments arguments"Arg>G/ var="cclvar java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 1089
<="Arg">G</var>[, <var class="Arg">p</var> )/>td class=tdright"( ttribute&bsp;)<><tr>/able><divjava.lang.StringIndexOutOfBoundsException: Index 252 out of bounds for length 252
<div class="func"><table class="funcpIf<arclass=Arg"tbl</ is thefirst then it<
<p>In the <
<p>(The default method for the call with only argument a group is to call the operation with second argument <code class="java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 0
<p><>lementjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 167
<div =pre>
<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
[ [ 1, 2 ], [ 2, 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class="APprompt"gapgt;/>< ="GAPinput"CharacterDegrees( SymmetricGroup(4) 2)<span
[ [ 1, 1 ], [ 2, 1 ] ]
<span class="GAPprompt">gap>/>
[ [ 1, 1 ], li<> table( codeclass="unc><code(a chap71.#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)).</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( p>If the optional argument <var class="Arg">arec</var> is present then it must be a recordjava.lang.StringIndexOutOfBoundsException: Range [141, 140) out of bounds for length 214
[ [ 1, 1 ], [<>
</pre></div>
<p><a id="X873B3CC57E9A5492" name="X873B3CC57E9A5492"></a></p>
<h5>71.8-2 <span class="Heading">Irr</span></h5>
<div class="func"><table/>
< =>class=func" width="100">tr>< ="<java.lang.StringIndexOutOfBoundsException: Range [85, 79) out of bounds for length 213
<p>Called with a group <var class="Arg">G</var>, <code class="func">Irr</code> returns the irreducible characters of the ordinary 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 the <var class="Arg">p</var>-modular Brauer table of <var class="Arg">G</var>. java.lang.StringIndexOutOfBoundsException: Index 415 out of bounds for length 253
<p>For a character table <var class="Arg">tbl</var> with underlying group, <code class="func">Irr</code> may delegate to the group. For a([1. 5] java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
<p>(If <var class="Argjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 5
<pNote thatthe ordering ofcolumns[ ]
<>The
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( SymmetricGroupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ Character(java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) ),
[ 3, -1, -1, 0, 1 ] ),
Character(java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
, Character( CharacterTable( Sym( [ 1 .. 4 ] ) ),
[ 3, 1, -1, 0, -1 ] ),
Character(CharacterTable Sym 1 . 4 ])) [1 1 ,1 1])
< BSym( 1.4]) )
[ Characters class"APprompt"&;/span<pan=GAPinput>( tbl ); IsCharacterTable modtbl )<span
Character( BrauerTable( Symtrue
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( CharacterTable( "A5" ) );</span>
[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ),
p following infix java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ),
Character( CharacterTable( "A5" ),
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ),
Character( CharacterTable( <p<a id"X7C6F3D947E5188D1dd of character tables (see <code class="func">CharacterTableDirectProduct</code> (<a href="chap71.html#X7BE1572D7BBC6AC8"><span class="RefLink">71.20-1</span></a>)),</p>
<n> <span class"GAPinput"> CharacterTable<>sthe java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 146
[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ),
Character( BrauerTable<><a ="" name"">/>p
[ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ),
Character(BrauerTable("5
[2 -,E()2E()<pvar- java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 270
Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ]
</pre></div>
<p><a id="X8549899A7DE206BA" name
<h5>71.8-3 <span class="Heading">LinearCharacters</span></h5>
<="func"width="%<dd>p>the position of class varclass""></var> in<var class="tbl/>(< class=uncCjava.lang.StringIndexOutOfBoundsException: Range [136, 135) out of bounds for length 230
<div class="func"><table class="func" widthul
<p><code class/dl
<Forjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<p>The ordering
<div class="example"><pre>
<panclass=APprompt>>ap>< < =GAPinput"( SymmetricGroup 4 ;/>
[
Character(CharacterTable(Sym 1 ..4])) 1,1, 1,1-
</><div
<p><a id="X8011EEB684848039" name="X8011EEB684848039"></a></p>
<h5>71.8-4 <span class="Heading">OrdinaryCharacterTable</span></h5>
<div class="func"><table class="func" width="100%"><tr><
<div ="unc"<table class="func" width="100%"><tr><td class="tdleft"><code class="func">le='color: green'>8227; OrdinaryCharacterTable</code>( <var class="Arg">modtbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>Since arefrom tables, theattributevalue <ar=Arg"modtbl/var> is already stored (cf. <a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span( SymmetricGroup( 4 ) );</span
CharacterTable( Sym( [ 1 .. 4 ] ) )
<pan class=GAPprompt"gapgt</span class=GAPinput">=CharacterTable [ 1, ] 3,2 ],[ 4, 1 ] [ ,1 ]
BrauerTable( "A5", 2 )
<( modtbl ) = tbl;</span>
true
/>><divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<p><a id="X81EFD9FE804AC6EE" name="X81EFD9FE804AC6EE"></a></p>
<java.lang.StringIndexOutOfBoundsException: Range [16, 3) out of bounds for length 92
< "unc><table class=""width="00%<>< =tdleft">code ="unc"‣AbelianInvariants</> =Arg"tbl/ar> )</><tdclasstdrightjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<""< ""width"100"<>td ""> =>#227;</code var"Arg">tbl/ar)/< classt"(nbsp<iv"nc"table class="func"width="100"rtd class="left>code class="func""#8227; </> class"rg>tbl/var )/>< class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsAbelian</code>( <var class="ordering need <em>not</em> coincide with the ordering of conjugacy classes as stored in the underlying group of the table (see <a href="chap71.html#X816FCD5A805F9FE8"><span class="RefLink">71.21</span></a>). One reason for this is that java.lang.StringIndexOutOfBoundsException: Index 248 out of bounds for length 0
< class"func<table class=func" width="100">tr>td class="tdleft"><code class="func">e='color: green'>8227; IsAlmostSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<pNote that the 5716- IdentificationOfConjugacyClassesjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
<div>The of the the code=func"Irr/ a group need <em>not</em> coincide with the ordering of its <code class="func">IrreducibleRepresentations</code> (<a href="chap71.html#X7F29C5447B5DC102"><span classFor an ordinary character table <var class="Arg">tbl</var> with known underlying group <span class="SimpleMath">G</span>, <code java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<divclass=func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFinite</code>( <var class="Arg">tbl<[ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1
<ivclass=="unc"<table java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 219
java.lang.StringIndexOutOfBoundsException: Range [145, 4) 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> )</td><td class="tdright">( property )</td></tr></table></div>
<pan="APprompt"></ [3 1,-, 0, ],
<">table class="unc" width="100%"<tr><class="dleft"><code class="func">‣ IsSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func><table class="unc [haracter( rauerTable((Sym([1 .. 4 ,2 , 1, ],
<iv class=func"< class="unc" width="100%><tr<d ="dleft">code ="unc">227;IsSporadicSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright">
<java.lang.StringIndexOutOfBoundsException: Range [64, 4) out of bounds for length 224
<C( java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 57
<Character "")
0%"><tr><tdclass="dleft"<code ="unc#; Size/( var=A"</var <td><td class="tdright">( attribute )</td></tr></table></div>
<p>These operations for groups are applicableCharacter(CharacterTable("" )java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
<div class="example"><pre>
< Character CharacterTable( "A5" ), [ 5, 1, -1, 0, 0 ] ) ]
<<pan="APprompt"gap&< =GAPinput">rr( CharacterTable(" ) mod 2 );>
<span class="GAPprompt">></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">></span> <span class="GAPinput"> CharacterTable( <span class="GAPprompt">></span> <span class="GAPinput"> CharacterTable( SL/code(<var class="rg> Character(("5 ,
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, AbelianInvariants );</span>
[ [ 3 ], [ 2 ], [ ], [ ] ]
<pan class="">gap&;/><spanclass"GAPinput">ist tables,CommutatorLength;/span>
[ 1, 1, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, Exponent );</span>
[3, 12, 30 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsAbelian );</span>
[ true false, false,false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsAlmostSimple );</span>
[ java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<pan=GAPprompt>&t<span><span class""List tables, java.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 94
[ true, java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<class=GAPprompt"gap></span> <<>code ""LinearCharacterscode>the linear (e, span class="><)characters ="func"Irr><a".#">span-span<a) thegroup<varclass=Arg></var> orthe character table <var class=Arg"tbl/var>,respectively. thesecond form code class="func">LinearCharacters</code> returns the <var class="Arg">p</var>-modular linear characters of the group <var class="Arg">G</var>.</p>
[ true, true, true, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsMonomial );</span>
[ true, true, false, false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, IsNilpotent );</span>
[ true, false, false, false ]
<</span> <spanclass"GAPinput"List(tables,IsPerfect;<span>
[ false, false, true, true ]
<span class="GAPprompt">gap></span> <span class<divclass"example"><pre>
[ false, false, true, true ]
<span class="GAPprompt">gap></span Character ( Sym( . 4 , 1, 1 1,, 1 ])java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
[>If< ="tbl</>java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 10
<="GAPinput">List( java.lang.StringIndexOutOfBoundsException: Range [0, 74) out of bounds for length 0
,,java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 28
<spanclass="APprompt>ap&t;</span="unc">OrdersClassRepresentatives/code>(<a href="chap71.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>)),</p>
[ true, false, false, false ]
span=">&;/>span class="GAPinput">List( tables, NrConjugacyClasses );</span>
3, 5,5,9 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( tables, Size );</span>
[/li>
<class="GAPprompt"gapgt<span character areconstructed from ordinary tables, the attribute value for <var class="Arg">modtbl</var> is already stored (cf. <a href="chap71.html#X789FAC077AEF088A"><span class="RefLink">71.4</span></a>).</p>
rec(name:= "(5)" parameter : 5, series : "",shortname := "C5"
Ch(([ .4 java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<span class="GAPprompt">gap></span> <span class="GAPinput">p>If the optional argument <var class="Arg">arec</var> is present then it must be a record .java.lang.StringIndexOutOfBoundsException: Range [0, 174) out of bounds for length 22
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 12
rec(name : "3,)=S(6,),parameter := [ 3, 3 ], series := "C",
shortname := "S6(3)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "O7(3)" ) );</span>
rec( name := "B(3,3) = O(7,3)", parameter := [ 3, 3 ], series := < ="<table class"unc width"%>>< class="tdleft"<code class=func">#8227 AbelianInvariants</code(<ar <
shortname := "O7(3)" )
<span class="GAPprompt">gap></span> <spandivclass="unc><table class="width"00"<><td =">code class="func">‣ Exponent</code>( <var class<d><p>a describing a partial ; the <class="impleMath"i/< class=">able=f""width="00">tr>td class=t">java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 218
rec( namediv=func class""=%>>g )/java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
<div""tableclass=="java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 228
<an class=APprompt"gapgt</span><span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( CharacterTable( "L3(4)" ) );</span>
rec( name := "A(2,4) = L(3,4)", parameter := [ 3, 4 ], series := "L",
shortnameCharacterTable(("" java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
</pre></div>
<>< id=7995A2AD83BC58A0java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<h4>71.9 <span class="<div class="func"><table class"unc width="100%"><tr><td class="tdleft"><code class="func">8227; IsNilpotentspan ="APprompt>></span><span ="APinput">new:= CharacterTableWithStoredGroup( g, tbl );</span>
<p>The following three <em>attributes forjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p>The other attributes introduced in this section apply only to character tables, notspanclass">gt= 100"""<code func">&8227<code> var class=Arg">tbl/ <tdtd =tdright"&;property&bsp;/td>/>/table>/>
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 225
<h571.-1 OrdersClassRepresentatives</>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OrdersClassRepresentatives</code>( <var class="Arg">tbl</<tdright&;attributenbsp/> | |