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Quellcode-Bibliothek chap71.html
Interaktion und
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< div class= "chlinkprevnexttop"> < a href= "chap0.div class="ContSect"><span class="tocline"><span class="nocss "> </span><a href="chap71. html#X7C083207868066C1 ">71.17 <span class="Heading ">Advanced Methods for Dixon-Schneider Calculations</span></a>
< p id= "mathjaxlink" class= "pcenter">< a href= "chap71_mj.html">[MathJax on]</ a></ p>
0 =X7B7A9EE881E01C10>/ a>/>
< div class= "ChapSects">< a href= "chap71.html#X7B7A9EE881E01C10"> 71 < span class= "Heading">Ch aracter Tables</span></a>
<div class="ContSect"><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>
<span>
</div>
<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="nocssspan =C>br>< =>nbsp&><href"chap71htmlX8089009"7117-8 </a></span>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8476B25A79D7A7FC">71.3-2 <span class="Heading">BrauerTable</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X85DB8AE7786A2DB5">71.3-3 CharacterTableRegular</a></span>
<class=ContSS< /<span class="ocss">nbsp; </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 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="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7C6F3D947E5188D1">71.4-2 java.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">&java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 4
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><<>aid"X7B9FCBBC7B95F91B="X7B9FCBBC7B95F91B<a<p
</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>
</>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7FF4826A82B667AF">71.6-1 UnderlyingGroup</a></span>
<span><a href="chap71.X84938"71.- <//>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X84DC12AA804C8085">71.6-3 IdentificationOfConjugacyClasses</a></span>
<pan "ContSS> ><pan =nocss>;/>a =chap71.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>
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
</div>
<div class="ContSectp>As also java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 402
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X873B3CC57E9A5492">71.8-2 <span class="Heading">Irr</span></a>
</span>
<span class="ContSSjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
<span div java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 26
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<div class="ContSect"><span class="tocline"><span class=
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<div class="ContSSBlock">
<span class="ContSS"><br
<span class="ContSS"><br /><span >It is one of the advantages of characterafter java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 608
<span C">< ><=>nbsp;;span< "java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 145
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7C2753DE8094F4BA">71.9-4 AutomorphismsOfTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7F58A82F7D88000A">71.9-5 <span class="Heading">UnderlyingCharacteristic</span></a>
/java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<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>
<java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<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#X79C40EE97890202F">71.9-8 Identifier</a></span>
<span class="ContSS"><br /><span class="nocss"> &
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7919E2897BE8234A">71.9-10 InverseClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X87FF547981456932">71.9-11 RealClasses</a></span>
<class=C"< / class=nocss>nbsp;n/span><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=>spanclass">nbsp;<span>ahref"html#CEBC3C7E0E63DF">71.10 <span class="Heading">Normal Subgroups Represented by Lists of Class Positions</span></a>
</java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 0
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<span class="ContSS"><br /><span class="nocss"> </span><a java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 0
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<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="
<span class="ContSS"><<>p java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 384
span>br>"nocss>nbsp;bsp;/<ahref=html">711010 />/>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X8392DD5B813250A4">71.10-11 ClassPositionsOfSupersolvableResiduum</a></span>
<span class="ContSS"><br /><span class="nocss"> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< =Cjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 155
</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 class=""><><class=nocss" nbsp<span< "h#>7111-- </a</>
<span class="ContSS"><br /><span class="nocss"> </span<i>p>forpossiblefusions java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 184
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 0
<span="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X83EC921380AF9B3B">71.11-5 LaTeXStringDecompositionMatrix</a></span>
<><div
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>
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<span class="ContSS"><br /><span class="nocss"> </span><a hrefjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7A0CBD1884276882">71java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span <li><p>combinepartial informationaboutconjugacyclasses, power java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="
<span class="ContSS"><br < 1,the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">&p>The operation <code class="func">CharacterTable</code
<span class="ContSS"><br /><span java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
</div></div>
<div class="ContSect">div class=
</span>
<class""table ="width"100%"<tr>><td =tdleft"><code class="func">‣ IsCharacterTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<span class="ContSS"><br /><span class="nocss"> rclassfunc><=""100"tr> class""< =<classArg></> )>tdclass="dright>(nbsp; </><tr</><div>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7B41F36478C47364">71.13-3 Display</a></span>
span class"> /<span class="ocss>&bsp&bsp;/span<a href="hap71#X85E883A87A190AA4">71.3--4 </></>
<span ="ContSS>br/>span class="nocss"> </span><a href="chap71.html#X79EC9603833AA2AB">71.13-5 PrintCharacterTable</a></span>
<
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a p>We want to distinguish ordinary and Brauer tables because a Brauer table may delegate tasks to the ordinary table of the same group, for example the computation of power maps. A Brauer table is constructed from java.lang.StringIndexOutOfBoundsException: Index 215 out of bounds for length 0
</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=n>nbsp;nbsp;/pan>a href=chap71.tml#7BF15729839203FC71.3IrrBaumClausen<a</>
<span class="ContSS"><br /><span class="
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</div<p>The tableobjects in<class"func"IsCharacterTable/java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 475
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7E51AACD79CE0BC8">71.15 <span class="Heading">Representationsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
< >
<span class="ContSS"><br /><span class="nocss"> </span><a
<spanclass"ontSS">re
<span class="ContSS"><br /><span class="nocss">
<div/>
<div class="ContSect">span class="tocline"><span class="nocss"> </span><a href="chap71.html#X86CDA4007A5EF704">71.16 <span class="Headingspan class=GAPprompt>gt;span>span ""tbl=() :tbl 2;</pan>
</span>
</div>
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<span
<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="div class="func"><table class="func" width=
<span classclass=func<<g;/span =">java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 116
<span classdiv class=func"table java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 5
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<span class="ContSS"><br /><span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<panclass=ContSS>br >span class"java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 4
<span class="ContSS"><br /><java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 0
</div></div>
<div class="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 4
</span>
</div>
<divli>p>The group <class=Afalse
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.html#X7C38C5067941D496">71.20<><p>Sylowvar class"rg><var-subgroupjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&li><p><var class="Arg">ordtbl</var> is a table from=pkg"GAP<strong>character java.lang.StringIndexOutOfBoundsException: Range [106, 105) out of bounds for length 210
<><aid=X7"XFA867637EBB30F9"/><java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 62
<span <h5>71.4-3 java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 5
<span class="ContSS"><br /><span class="nocss"> p>default iv">table f" "%>>td => =>#8227java.lang.StringIndexOutOfBoundsException: Range [130, 128) out of bounds for length 204
<panclass=C>br > class=nocss"&;<span=c.html#">71205=efLink"3.>><
<span class="ContSS"><br /><span class="nocss"<p>a =" =java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<span class="ContSS"><br /><span class="nocss"> </span<h4>71.5 span class="Heading">Conventions for Character Tables</span></<class"keyw/>fora java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 219
</div></div>
<div class="ContSect"><span class="tocline"><span <><a idul>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7D9C4A7F8086F671">71.21-1 CharacterTableWithSortedCharacters</a><
<pan class=ContSS classfunc>"100">> classtdleft>code class"java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 256
<pan classContSS">br/>pan 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 class="ContSS"><br /><span class="nocss">&java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 5
</div></java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
<div class="ContSect"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><div
<<i<>For<"java.lang.StringIndexOutOfBoundsException: Range [52, 23) out of bounds for length 93
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7D721E3D7AA319F5S( java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 5
<span java.lang.StringIndexOutOfBoundsException: Range [11, 11) out of bounds for length 5
</div></div>
<ivclass="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap71.htmlBrauerTable( Sym( 1 ..4 ],
</span>
<div class=java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap71.html#X7E66174C7C7A8C0C">71.23-1 NormalSubgroupClassesInfo
<class"<r ><pan class=nocss" &;<span><a href"chap71.#7C2A87E085111090">>71232 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"> &<p>However, given a group <span class="SimpleMath">G</span character table groupto< =SimpleMath">G</> ( a tablejava.lang.StringIndexOutOfBoundsException: Range [177, 176) out of bounds for length 543
</div></div>
</div>
<h3>71 span ="eading>Character Tables</span></h3>
<p>This chapter describes operations for <em>characterjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ass functions</
<p>For a description of the <java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<p>Several examplesspan class""gap&;<>< classGAPinput">omputedBrauerTables p>For an ordinary character table <var class="Arg">ordtbl</var> of a finite group, the group can be stored as value of <code class="func">UnderlyingGroup</code>.</p>
<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [50, 5) out of bounds for length 94
true
</re<div
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
<h4>71.1 <span class="Heading">Some Remarks
<p>It seems class=func>tablejava.lang.StringIndexOutOfBoundsException: Range [51, 49) out of bounds for length 534
<p>Intuitively, <em>characters</em> (or more generally, <em>class functions</em>) of java.lang.StringIndexOutOfBoundsException: Range [0, 86) out of bounds for length 0
<p>At this stage one could ask where there is a
<p>For computations with characters of a finite group <span class="java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [43, 2) out of bounds for length 402
<p>So typical calculations with charactersh5.- ConvertToCharacterTable<h5>
<p class="pcenter">[ χ, ψ ] = ( ∑_{g ∈ G} χ(g) ψ(g^{-1
<p>can bewritten <p>
<div class=Thejava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 149
java.lang.StringIndexOutOfBoundsException: Range [143, 144) out of bounds for length 63
* ComplexConjugate( psi[i] ) ) / Size( t );
</pre></div>
<p>where <code class="code">t</code> is the character tablepIfinconsistencies var classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>It is one of the advantages of character theory java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 0
<p><strong class="pkg">GAP</strong> uses character <em>tables</em> to store
<p>Note that the character table of a group <span class="SimpleMath">G</span> injava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Two further important instances of information java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0
<
java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 366
<p<a =X7" name="X7F8AB7CB7A46002F"></a></p>
<h4>71.<ivclass"unc>table class=""width="00"><tdclass=tdleft>codeclass=func">8227 /ode>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p><strong class="pkg">GAP</strong> provides functions for dealing with group span class="GAPprompt">gap></span> <span class="GAPinput">ccl:= List( repres, x );<spanjava.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123
<pFor integrating span class="GAPprompt&;;span <<pan class="GAPinput">tbl:= CharacterTable g ); table stores already the java.lang.StringIndexOutOfBoundsException: Range [0, 129) out of bounds for length 124
<p>The development of character theory code for <strong <p>We want to distinguish ordinary and Brauer tables because
<p>The major contributions can be listed as <span class="GAPprompt">gap></span> <span class= codeclassfunc>IsBrauerTable</code>denote charactertablesthat enough information to all java.lang.StringIndexOutOfBoundsException: Range [179, 178) out of bounds for length 722
<ul>
<li><p>The arithmetic
</li>
<li><p>The basic routines for characters and java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 12
</li>
<li><p>The
</li>
<li><p>Functions for computing possible class fusions, possible power maps, and table automorphisms were written by div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableWithStoredGroupclass="">apg;span span ="bl CharacterTable(g ;modtbltbl java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 116
</li>
<li><p>Functions for>If java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 995
</li>
<li><p>java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 4
</li>
<li<>Functionsfor Clifford matrices
</li>
> java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 98
</li>
</ul>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p <strongclass"kg">GAP</strong> notspecialfunctionalitycomputingBrauercharacterjava.lang.StringIndexOutOfBoundsException: Range [134, 133) out of bounds for length 361
<p><a id="X8701174D86B586AF" name="X8701174D86B586AF"></a></p>
<h4>71.3 <span class="Heading">Creating Character Tables</span></h4>
<e ingeneral java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 117
<ol>
<li><p>compute the table from a e"java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 62
</li>
<li><p>read a file that contains java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
</li>
<li><p>construct the table using genericinjava.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 145
</li>
<li><p>derive it from known character tables, or</p>
<li
< of the group java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 164
</li>
</ol>
<p>In 1., the computation of the irreducible <li>
<p>The operation <code class="func">CharacterTable</code> (<a href="chap71.html#X7FCA7A7A822BDA33"><spanuljava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<<java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for length 62
<h5>71.3-1 <span class="Heading">CharacterTable</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classli>pMost functions a tableas and charactersexpect
<java.lang.StringIndexOutOfBoundsException: Range [115, 4) out of bounds for length 253
<div class="func"><table class="func" width="100%"><tr><td class=<
<p>Called with ajava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
<p>Called with a string <var class="Arg">name</var> and perhaps optional parameters <var class="Argjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Probably the most interesting information about the character table is <dd><> describing apartialbijection; <class"impleMathp< = =">/<>
<pThe reasonwhy codeclass==func>CharacterTable</code </d>
<p>For details about different algorithms to compute the irreducible characters, see <a href=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>If the group <var class="Arg">G</var> is given as an argument, <code span class="GAPprompt">gap></span> <span class="GAPinput">g:= AlternatingGroup
<p><a id="X8476B25A79D7A7FC" name="X8476B25A79D7A7FC"></apCharactertablesjava.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 98
<h5>71.3-2 <span class("A5 )
<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 250
<div class="func"><table class="func" width="100%"><pTasksmay be agroup to java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 151
<div class="func
<div class="func"><table>a ""ame"7>/><>
<p>Called with an ordinary character table <var class="Arg">ordtbl</var> or a group <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The <var class="Arg">p</var>-modular table java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 228
<ul>
<java.lang.StringIndexOutOfBoundsException: Range [30, 3) out of bounds for length 215
</li>
<li><p>the Sylow <var class="Arg">p</var>-subgroup of < class=GAPpromptgap;<span class"">Irr( new
</li>
<li
</li>
<li><p><var class="Arg">
</li>
</ul>
< defaultmethod for < charactertable < java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 514
<
<pThecodeclass"keyw"mod/ operatorfora java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 219
<p><a id="X85DB8AE7786A2DB5" name="X85DB8AE7786A2DB5"></a></p [17 18] [9 java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<h5>71.3-3 CharacterTableRegular</h5>
<div class="func"><table class="func<>a ="X7CADCBC9824CB624"name"><a</>
<p>For an ordinary character table <var class="Arg">tbl</var> and a <h4>71.7 <span class="Heading">Operators for Character Tables</span></h4
<p>Thehead bejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
Sym [1 ..4])
<span class="GAPprompt">gap></
falseh5>71.6-3 IdentificationOfConjugacyClasses</h5>
<span class="GAPprompt">gap></span> <spandd<> direct oftwo
rauerTableSym( [1. 4 , 2)
<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2>strongclass=Mark"< class=">varclass=Atbl<pFor an character table <var class="Arg">tbl</var> with known underlying group <span class="SimpleMath">G</span>, <code class="func">
true
<span span class=GAPprompt"gapgt</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 125
<span class="GAPprompt">gap></span> <span class="GAPinput">tblmod2 = BrauerTable( g, 2 );</span>
rue
<span class="GAPprompt">gap></span> <span class="GAPinput">libtbl:= CharacterTable( "M" );</span>
CharacterTable( "M" )
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTableRegular( libtbl, 2 );</span>
BrauerTable "M,2)
< class="GAPprompt">gap></span> <span class="GAPinput">BrauerTable( libtbl, 2 );</span>
fail
<pan="java.lang.StringIndexOutOfBoundsException: Range [153, 19) out of bounds for length 230
CharacterTable( "Sym(4)" )
<span class="GAPprompt">gap></span> <span class="GAPinput">span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingGroup( tbl ) = g;</span>
[, Sym [. 4 ) 2 )]
</pre></div>
<p><a id="X7DBEF4BF87F10CD6" name="java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 17
<h5>71.3-4 SupportedCharacterTableInfo</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 0 out of bounds for length 0
<p><code class="func"> class" width="100%>tr>>td class="tdleft"><code class="func">‣ CharacterTableWithStoredGroup</code>( <var class="Arg">G</var>, <var class="Arg">tbl</var>[, <var class="Arg">info</var>] )</td><td class="tdright">( functionp"/var a java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 645
<p>
<p>ew attributes properties can tocodeclass="></code> java.lang.StringIndexOutOfBoundsException: Range [118, 117) out of bounds for length 497
<p><a id="X8195BC057B1DFAD5" name="X8195BC057B1DFAD5"></a></p>
<h5>71.3-5 ConvertToCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func"><>Ifthis to <ar ="rg><var> (see <code class="func">AutomorphismsOfTable</code> (<a href="chap71.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)) then <var class="Arg">tbl</var> java.lang.StringIndexOutOfBoundsException: Range [0, 255) out of bounds for length 0
ion )</td></tr></table></div>
<p>Let <var class="Arg">record</var> be a record. <code class="func">ConvertToCharacterTablep>If a list is entered as third argument <var class="Arg">info</var> it is used as value of <code class="func">IdentificationOfConjugacyClasses</code> (<a java.lang.StringIndexOutOfBoundsException: Index 157 out of bounds for length 0
<p>If inconsistencies in <var class="Arg">record</var> are detected, <code class="keywdiv =func><class=funcjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<p><code class="func">ConvertToCharacterTableNC</code> does the same except that all checks of <var class="Arg">record</var> are div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterDegrees</code>( <var classpvar]<td<class"> ttributen)</td<</<>
<p>An example of a conversion from a record to a character table object can be found in Section <code class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><a id="X789FAC077AEF088A" name="java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
<h4>71.4 <span class="Heading">Character Table Categories</span></h4>
<p><a id="li><p>class lengths (see <codefunc">/ <hrefdiv=example"<
<h5>/li>
<<var class="Arg">obj</var> )</td><td java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 5
<div class="div class="example"><pre>
<div java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
< class="tdleft>codeclass=="unc>#8227;IsBrauerTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 5
<p>Every <q>character table like object</q> in> that java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 4
<p>We want to
<p> [ . )java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<p>The character table objects in <code class="func">IsCharacterTable</code> are always immutable (see <a href="chap12.html#X7F0C119682196D65"><span class="java.lang.StringIndexOutOfBoundsException: Index 161 out of bounds for length 67
<p>The <strong class="pkg">GAP</strong> objects java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 17
<span "APprompt>gap><span><class=G>= CharacterTable (([1.4 ] ,[,1,1 , 1 java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
CharacterTable( Sym( [ 1 .. 4 ] spanclass>>/pan><<span class="GAPprompt">gap></span> <span class="GAPinput
rauerTable([1 .4],2java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 35
<pan=G"gap&t;/> < class="IsCharacterTable ( modtbl )<>
true
<span class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
true
false
<span class="GAPprompt">gap><
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCharacterTable( g ); IsCharacterTable( Irr( g ) );</span>
false
false
</pre></div>
> =dd><p>the direct product java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 197
<h5>71.4-2 InfoCharacterTable</h5>
<div java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 199
p> the infoclass (ee<a href="chap7h#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>) for computations with character tables.</p>
<idX7FA867637EBB30F9 =X7FA867637EBB30F9</>/>
<h5>71.4-3 NearlyCharacterTablesFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"[2,1 5)2+5^pjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 270
<p>Every character table like object lies in this family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>).
<><aid="java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<h4>71.5 <span class="Heading">Conventions for Character Tables</span></h4>
<p>The following few div class="func"><table class" =100>s< class=">java.lang.StringIndexOutOfBoundsException: Range [136, 135) out of bounds for length 230
<>
<li><p>The class of the <em>identity element</java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 5
</li>
<li><p>The <em>trivial character</em>p> a java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
</li>
<li><p>Most functions that take a character table as anpSeveralemfor
</ijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<li><p>For a stored class fusion, the image table is denoted by its <code class="func">Identifier</<>< class=">;/><pan java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 109
</li>
</ul>
<p><a id="X793E0EBF84B07313" name="X793E0EBF84B07313"></a></p>
<h4>71.6 <span class="Heading">The Interface between Character Tables and Groups</span></h4>
<p>For a character table with underlying group (see <code class="funcp first form, <code class="func">CharacterDegrees</code> returns
<>java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 211
<p>However, given a group <span class="SimpleMath">G</span> and a character table of a group isomorphic to <span class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Tasks may be delegated from a group to its character table or vice versa only if java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 0
<p><a spanclass="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( class="GAPprompt">gap></span> <span class="GAPinput">OrdinaryCharacterTable>
<h5>71.6-<class"GAPprompt>&;/span> <span =>tbl: CharacterTable(" 32 , , 51
<div class="<pan class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( span class="GAPprompt">gap></span> <span class="GAPinput">OrdinaryCharacterTable
<p>For an ordinary character table <var class="Arg">ordtbl</var> of a finite group, the group can be stored</pre>//<div>
<p>Brauer tables ><a id="X873B3CC57E9A5492>a><>
<p
<h5>71.6-2 h5>71.8-5 <span class="Heading">Group Operations Applicable to java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 0
<class=u<class=func width"%><< ">"func"8227 java.lang.StringIndexOutOfBoundsException: Range [226, 117) out of bounds for length 226
<p>For a character table <var class="Arg">tbl</var> p>Called with a group <var class="Arg">G</var>, <code class var ""G<,code func"/ode>returns irreducibleof java.lang.StringIndexOutOfBoundsException: Range [130, 129) out of bounds for length 623
<p>This java.lang.StringIndexOutOfBoundsException: Range [96, 16) out of bounds for length 546
<p>The java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
<> <ar class="">>wasfromspanS">< thentheconjugacyhavebeen java.lang.StringIndexOutOfBoundsException: Range [132, 131) out of bounds for length 679
<p>Brauer tables do not store the (<span class="SimpleMath">p</span>-regular) conjugacy classes, they access them via the ordinary table (see <code class="func">OrdinaryCharacterTable</code> (<a href<ivclass="">class= width100%>>java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 223
<p><a id="X84DC12AA804C8085" name="X84DC12AA804C8085"></a></p>
<5.63IdentificationOfConjugacyClassesjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
<div classdiv java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 287
<p>anjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 562
<div class="example"><pre>
<( 4 );;</span>
<span class="GAPprompt =java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 217
<pan class="">gap&t<span>< class=GAPinputccl= List ,x -gt;ConjugacyClass< u<class=""width"%"<tr><class==tdleft"code class="unc>#8227 </> < ="Arg">tbl<<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotent</code>( <var class="Arg">tbl</va&java.lang.StringIndexOutOfBoundsException: Index 220 out of bounds for length 220
<span class="GAPprompt">gap></span> <span class="GAPinput">SetConjugacyClasses( g, ccl );</span>
< class"APprompt"gap; ,1-,,-]java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="div class="func>"unc"<<td tjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 217
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingGroup( tbl ) = g;</spandiv"""unc"C( 1. ) [1 java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 61
java.lang.StringIndexOutOfBoundsException: Range [179, 4) out of bounds for length 4
<span class="GAPprompt">gap></span> <span class="GAPinput">HasIdentificationOfConjugacyClasses( tbl );</<panclass==G>></> span "APinput">div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSupersolvable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table><java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 57
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IdentificationOfConjugacyClasses 3 - ()^2-E)3 E()E)4 ] ),
[ 5, 1, 2, 3, 4 ]
</pre></div>
<p><a id="X8788C6C7829C1ADE" name="X8788C6C7829C1ADE"></a></<span class="GAPprompt">></span> <span class="GAPinput"> >gt/>>( A5" )/span>
<h5>71.6-4 CharacterTableWithStoredGroup</h5>
">‣ CharacterTableWithStoredGroup<//> varclass=A"<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 36
<p>Let <var class="Arg">G</var> be a group and <var class="Arg">tbl</var> a character table of (a group isomorphic to) <var java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 36
<p>If thistrue, java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 29
<p>Otherwise, i.e., if<iv class=="func"><table class"func" "100<>="dleft<class=="java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 94
<p>If a record is present as the third argument <var class="Arg">info</var>, its meaning is the same as the optional argument <var class="Arg">arec</var> for <code class="func">CompatibleConjugacyClasses</code> (<a href="chap71<pan ">java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 94
<p>If a list is entered as third argument <var class="Arg">info</var> it is used as value of <code class="func">IdentificationOfConjugacyClasses</code> (<a href="chap71.html#X84DC12AA804C8085"><span class="RefLink">71.6-3</span></a>), relative to the <code class="func">ConjugacyClasses</code> (<a href="chap71.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X790019E87CFDDB98" name="X790019E87CFDDB98"></a></p>
<h5>71.6-5 CompatibleConjugacyClasses</h5>
<div class "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 109
<p>If the arguments <var class="Arg">G</var>>(CharacterTable Sym [ 1 . 4 ])),[, ,1 11],
pIf <ar class="Arg">/var ]) ]
<p>The usual approach is that one first calls <span class="GAPprompt">gap></span> <span class
<p>The[true true false, false ]
<ol>
java.lang.StringIndexOutOfBoundsException: Range [78, 72) out of bounds for length 167
</li>
<li><p>class lengths (see <code class="func">SizesConjugacyClasses</code> (<a href="chap71.html#X7D9D2A45879A6A62"><span class="RefLink">71.9-3</span><div class"func"< class"" width< class"GAPprompt"gap&t<span <java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 104
<li>
<li><p>power maps (see <code class="func">PowerMap</code> (<aspan class="GAPprompt"&;/<>Since Brauer tablesare java.lang.StringIndexOutOfBoundsException: Range [53, 48) out of bounds for length 231
</li>
<li><p>symmetries of the table (see <code class="func">AutomorphismsOfTable</code> (<a href="chap71.html#X7C2753DE8094F4BA"><<span class=G"gap></span> <( ="5,= =Z,java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 69
<li
</ol>
< whose components describe additional information for theclassidentification TheBrauerTable("A5", 2 )
<dl>
<dt><strong class="Mark"><code class="code">natchar</code> </strong></dt>
<>p <classSimpleMath"</span>is apermutation group or matrix groupthen the value this component is regarded as the list of values of the natural character (see <code class="func">NaturalCharacter</code> (<a href="chap72.html#X82C01DDB82D751A9"><span class="RefLink">72.7-2</span></a>)) of <var class="Arg">G(name := "(3) 6," java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 69
</dd>
<dt><strong class="Mark"><code class="code">bijection</code> </strong></div""< java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 226
<>palistdescribingpartialbijection;span class=S><< class"func>< =unc" ==00>tr<=dleft"< java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 218
</dd>
</dl>
<div class="example"><pre>
<"= 5;>
Alt( [ 1 .. 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5 classu"= =100"<<class< ""&8227; /(<class=Arg><var><td< "">(n;< class=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 135
A5 )
< =java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 128
false
false
< ">&<> =X7995"name"7995A2AD83BC58A0">a>/>
[ ]
<classG"gapgt;/span spanclassGjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 115
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 35
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( new java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<class"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 124
true
true
< class=GAPprompt"gap>width100"tr< d"<class"&java.lang.StringIndexOutOfBoundsException: Range [111, 109) out of bounds for length 217
[ 1, 2, 3, 4, 5 ]
<span class="GAPprompt">gap></span> <span classfunc>table"
<pan=>gt;/< java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 122
[ [h5>719-1/5
</pre></div>
<p><a id="div ">table class=func"width"%"<>< =""code =func">#8227 <java.lang.StringIndexOutOfBoundsException: Range [144, 143) out of bounds for length 246
<>.7< "> is listjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 194
<p>The following infix operators are defined for java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 0
<dl>
<dt><strong class="Mark"><code class="code"><var class=< "ap&;<span <spanclass"java.lang.StringIndexOutOfBoundsException: Range [65, 58) out of bounds for length 389
<dd
</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">tbl</var> / <var class="Arg">list<div=java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 120
<dd><p>the table of the factor group modulo the normal subgroup spanned by the classes in the list <var class="Arg">list</var> (see <code class="func">CharacterTableFactorGroup</code> pispan="">java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 103
</dd>
<>strong""< =code"varclassArg"tblvarmodvar""p/><code<strong<java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 120
<dd><p>the <var class="Arg">p</var>-modular Brauer character table corresponding to>class=gap/<"List java.lang.StringIndexOutOfBoundsException: Range [76, 74) out of bounds for length 100
</dd>
<dtdivclass">tableclass"java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 231
<dd><p>the position of the classspanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<< "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<>
<p><a id="X7F9D58208241D35E" namespan=>&</< ="java.lang.StringIndexOutOfBoundsException: Range [85, 83) out of bounds for length 98
<="Heading"AttributesandPropertiesfor and CharacterTablesiv
<<namejava.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 62
<p>These are first java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 28
<p>Second, there are attributes whosetrue false,false false java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
<> = =>/<p>
<h5>71.8-1 <span class="Heading">CharacterDegreesspan ="GAPprompt>gt;<span> span class=GAPinput">(tables,Size class=rg>/ java.lang.StringIndexOutOfBoundsException: Range [110, 109) out of bounds for length 342
<div class="func"><table class="func" width="100%"> 3 24 ]
<class=">table " =100%>tr>td ""< java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 108
<p>In the first form, <code class="func">rec( name := "Z(5)", parameter := 5, series := "Z", shortname := "C5"
<p>(The default method for the call with only argument a group is to call the operation with 1,,2
<p>For solvable groups, the default method is based on <a href, ,
<"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( SymmetricGroup
]
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( SymmetricGroup( 4 )
[ [ 1div=" class="GAPprompt">gap< java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 132
<span class=( < "unc><=f"width"00"<>=java.lang.StringIndexOutOfBoundsException: Range [74, 72) out of bounds for length 234
[ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterDegrees( CharacterTable( "A5" ) mod 2 );</span>
[ [ 1, 1 ], [ 2, 2 ], [ 4, 1 ] ]
</pre></div>
<><=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 135
<h5>71.8-2 <span class(name=A(,) 3,4" :[ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Irr</code>( <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Called with a group <var class="Arg">G</var>, <code class="func">Irr</code> returns the irreducible
<">java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 42
<p>(If <var class="Arg">G</var> is <var class="Arg">p</var>-solvable (see <code class="-">java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 76
<p>Note that the ordering div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterNames</code>( <var class="Arg">tbl</var> )</td><td class="><code class="func">ClassNames</code> and <code class="func">CharacterNames</code> return lists of strings, one for each conjugacy class or irreducible character, respectively when<var =Arg>/ar>> java.lang.StringIndexOutOfBoundsException: Index 284 out of bounds for length 0
<p>pre/>
<div classp> default code class==f"haracterNames<code> java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 0
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, -1, 1, 1, -1
] ), Character( CharacterTable( Sym( [ 1 .. 4 ] ) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 3, -1, -1, 0, 1 ] ),
Character [ .4 ) 20,2,-,0])
, Character( CharacterTable( Sym( [ 1 .. 4 ] ) ),
[ 3, 1, -1, 0, -1 ] ),
Character( CharacterTable( Sym( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1, 1 ] )
]
<span class="GAPprompt"span class=GAPprompt>gap><span < class"APinput>( ble( " ;<>
[java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 61
Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 2, -1 ] ) ]
<span class="GAPprompt">gap>pre>div
[ Character( CharacterTable( "A5" ), [ 1, 1, 1, 1, 1 ] ),
Character( CharacterTable( "A5" ),
[ 3, -1, 0, -E(5u>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
Character( unc>class=func" "100"<tr>td class="tdleft"<codeclass=func">&8227; </><="></var> )<td><java.lang.StringIndexOutOfBoundsException: Range [168, 167) out of bounds for length 231
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ),
Character( CharacterTable(div="<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 229
<span class="GAPprompt">gap></span> <span class="GAPinput">Irr( CharacterTable( "A5" ) mod 2 );</span>
[Character BrauerTable("5, 2) [ ,, ,1 ,
Character( div class="func"><table="""%>tr<class"java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 224
[ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ),
Character( BrauerTable( "A5", 2 ),
[ 2, -p<code ="">ClassNames/code and<odeclass"">CharacterNames/ode return listsofstrings one for conjugacyclass or irreducible,respectively ofspanclass"GAPprompt"gap&<pan>>"">(( A5" ) );</span>
Character
</pre></div>
<p><a id="X8549899A7DE206BA" name="X8549899A7DE206BA"></a></p>
<h5>71.8-3 <span class="Heading">LinearCharacters
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LinearCharacters</code>( <var class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p<ode class="</> returns the linear (i.e., degree <span class="SimpleMath">1</span>) characters in the <code class="func">Irr</code> (<a href="chap71.html#X873B3CC57E9A5492"><span class="RefLink">71.8-2</span></a>) list of the group <var class="Arg">G</var> or the character table <var class="Arg">tbl</var>, respectively. In the second 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>
<>Fora java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 316
<p>The ordering of linear characters in <var class="Arg">tbl</var> need not coincide with the ordering of linear[ 1,"a","3","5","
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LinearCharacters( SymmetricGroup( 4 ) );</span>
[ Character( CharacterTable( Sym( [ 1 .. 4
Character ([ 1 . 4]<p<id"X8333E8038308947E" name="X8333E8038308947E"></a></p>
] ) ]
</pre></div[ java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<p
<h5>71.8-4 <span class="Headingdiv "">table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassParameters</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<=<classfunc "><class=tdleft"<code"unc"#8227 </code(var=Arg"G/var> )/td<td class="tdright">( attribute )</td></tr></table></div>
< class=""tableclass="width="100%>tr><td class="tdleft"> class=func"&8227; </> <ar ="rg"modtbl/> </>< class=> attribute <td<tr</table</>
<>code=func"OrdinaryCharacterTable</code> returns ordinary table of the group <var class=Arg"></>or charactertable<class="Arg"modtbl/var>,respectively./java.lang.StringIndexOutOfBoundsException: Index 205 out of bounds for length 205
<p>Since Brauer character tables are constructed from ordinary tables, the attribute value for <var class="Arg">modtbl</var> is already stored (cf
<div class="example"><pre>
<spanclass"">gapgt
CharacterTable( Sym( [ 1 .. 4 ] ) )
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "<iv ="example">prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 22
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 0
true
</pre></div>
<>< =X7java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<h5>71.8-5 <span class="Heading">Group Operations Applicable to Character Tables</span></h5>
class<="< ="">table class="unc" width="00">tr>td class="tdleft"><code class="func">‣ ClassRoots</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<tdright">( attribute )</td></tr></table></java.lang.StringIndexOutOfBoundsException: Index 224 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Exponent</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 26
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsAlmostSimple</code>( <var class="Arg">tbl</var> )</td><td class="tdright"GAPpromptjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 4
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCyclic</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table>< , ][
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsElementaryAbelian</code>( <var
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsFinite</code>( <var class="div=func"<classf"=100"tr> java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 215
< class=func">table ="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonomial</code>( <arclass="><var>)<td>td => ropertyn)/>/<table>/java.lang.StringIndexOutOfBoundsException: Index 219 out of bounds for length 219
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsNilpotent</code>( <var java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 100
<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Range [0, 72) out of bounds for length 0
<<span class="APprompt">gap></span> <span class="GAPinput">Print(
<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 62
<pre>/div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSporadicSimple</code>( <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSupersolvable</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 113 | | |
