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<title>GAP (CTblLib) - Chapter 5:
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<div class="<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80F14D1C86CF1E9C">5.8-3 ConstructGS3</a></span<div class="ContSect"><span class="java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 12
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<p>An important ingredient of the constructions 

<p>Section <ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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<p><a id="X7C32AA0784A79D53" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>5.1 <span class="Heading">Character Tables of Groups of Structure <span class="SimpleMath">M.G.A</span java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0

<p>For><div

p><a pan

<dtstrongspan=C>

<div class="func"><table class="func" dd><p>a possible ordinary character table=ContSS>br/<class=">;&bsp;/pan< java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<>pthefusionmapfrom class">/> into  table stored in <code ="ode></code./>

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<dl>
<dt><strong class=func><table class=func" width="100
<dd><p>a possible ordinary <"<span> <=">N<pan>,< ="impleMath</span> be as described at the beginning of the section, let <var <p>The functions described in this chapter deal with the construction of character tables from other character tables. So they fit to the functions in Section <a href="../../."><span 

</d
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
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<h5>

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java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0

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< span""/pan group anormal  class=SimpleMath<spansuch thatclass=func>ableclass""width100><<>p  map varclass"tbl2/>  the  span class"><> and>

<p>>Information the  is  if info levelof codeclass="java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 0

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<>codeclass=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<l>codefjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 104
<dt><strong classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
dd<< java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 353

<p id"82"=pLet span=SimpleMath><spanbe  groupdd><p>the ordinary or <span class="SimpleMath">p</span> classSimpleMath>H> ,<p>
< =>codeclass=code>/<strong/>
<dd><p>the java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

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<dd><p>the fusion map from <java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0

</java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
</dl>
<p>The returned tablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

<>52-java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<iv =""><table class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p> thearguments be described

<p>Information about the progress is reported java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 0

<><a =X83EB02D38624821C name=X83EB02D38624821C><java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 0

<h4>5.3 <span class="Heading">java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0

<p are

<p><">table class=f"width"100%>tr>td ="dleft"<code

<h5

<divclass="unc><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"width="100">tr>class=t">< class=func>&#227; PossibleCharacterTablesOfTypeGV4>(<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><p><code class="func">PossibleCharacterTablesOfTypeGV4</code> returns a list <p>Let <span class="SimpleMath">H</span> be a group with a normal subgroup <spanSimpleMath"G<span> such that <span class="SimpleMath">H/G</span> java.lang.StringIndexOutOfBoundsException: Index 157 out of bounds for length 0

<java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 764

<>Inthesecond version,let <var class="Arg">modtblG</var> be the <java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 0

<p>p> the second ffusion  the  var =A"></var> into  <code java.lang.StringIndexOutOfBoundsException: Range [101, 100) out of bounds for length 135


<p>T possiblejava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 136
<dt<strong class"">
<dd>> possible (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</dd>
<dl
<>pthe list

</>
</dl>
<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<p>The returned dd><p>the list of fusion maps from the tables in</dl

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<div<>a id="X7CCD5A2979883144"name"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<p><a id=<< id="79142119791673EF

<h4.4<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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<p><a id

<>5.4-1 span class=<Let<span ""H/> be    subgroupspan class="SimpleMath">N</span> of type <span

<div ="func"><able class=func" width=100"<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<</span> be 

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<p><div<div class ,let< =><var  java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 828

<p><code class="pLet spanclass"SimpleMath">H/>be agroupwitha central subgroup < class=<p>et < class=java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<<aid"8536F9027F097C79 name=""><a</<>In  ,let<">java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 828

<h5>5.4-2 <span

<div class="func"><table class="func" width="100%"><tr><td classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table p< id="X8536F9027F097C79" name="X8536F9027F097C79"></a></p>
<>Let<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The class fusions from <var class="Arg">ordtblV4G</var> p>Let <span class="SimpleMath">H</span> be a group with a central

<p>code class=<

<p><a id"X7B7282A37F293B36 name="X7B7282A37F293B36

<h4>h5.5-1 CharacterTableOfIndexTwoSubdirectProduct>

p>The  functionisthoughtforconstructing java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 0

<p<<a =X7B7282A37F293B36=X7B7282A37F293B36>/</p

<5.<h55.5- CharacterTableOfIndexTwoSubdirectProduct>

<div class"
<<p>Returns recordthedirect  of < class=SimpleMath">_>and <pan ="SimpleMath">G_2</span> contains a unique subgroup <span class="SimpleMath">G</span> of index two that contains the direct product of <span class="SimpleMath">H_1</span> andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<<dt><strong class=Mark><code class=code">table</code</code</strong></>

<p>In this situation,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 5


<dl
<<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>p><d<>java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 114

<dt<strong ="Mark"
<dt><strong class="p>If the  fusion from < class="<d>>the  fusion var="Arg"</> java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 0
<dd><p>the 

</dd>
<dt4B802DA949name"X8714D24B802DA949"<a/p
<p> thejava.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 5

</dd>
</l
<p<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0

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

<

<52 ConstructIndexTwoSubdirectProduct</5

<iv ="< "unc" java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 432
<pcode<code> constructs the characters of the ordinary character table java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 118

<p><a id="X8254CA95814DF613

<h5<Returns:tstrong class"<code class="">kernels</code></strong></dt>

<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Returns: a <ppCalled<dlist orders < class=="SimpleMath">_1/,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Called java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4


<dl>
<dt><dt><strong"
istofjava.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 145

</
<dt><strong class=<>
<dd><java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 0

</dd>
<dt><strong class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<dd><pthe  of <code

</dd>
<>strong class="ark"< class=code>subgroups</(ahref=..././//.#"><class=""Reference: Identifier  character tables</span></a>) values of the <strong class="pkg">GAP</strong> library java.lang.StringIndexOutOfBoundsException: Index 248 out of bounds for length 0
<><>the listof<code class="func"Identifiercode> (<a href="../../../doc/java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 0

</dd>
</dl>
<p>If thedivclassp>If  returned is emptythen < ="rg><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Called   

<p><a id<>Note that java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 0

<h4>5.6 <span class="Heading">Brauer<>/
</p<pAs fortheconstruction of Brauer character tables from known tables, the functions <code

<p>As for the constructionh5>5.6-1 IBrOfExtensionBySingularAutomorphism</h5>

<java.lang.StringIndexOutOfBoundsException: Range [27, 2) out of bounds for length 62

<BrOfExtensionBySingularAutomorphism>

<div p>The second argument <var class="Arg">p>Let <var class="Arg">modtbl</var> be ajava.lang.StringIndexOutOfBoundsException: Range [58, 4) out of bounds for length 335
<<>h45  =H"java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0

<p

<p>Note that java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 0

<p><java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0

< ="Heading"Character of Coprime  Extensionsspan<h4

<p><a id="X82CEF31D7815C53D" <h5>5.7-1 CharacterTableOfCommonC</>

<h5>.7-1 dt><strong class="Mark"><codecode>< =">java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 335

<div class="func"><table<>  java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 533
/>bethejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<dl<>
<t<trong M"code ="tblmnG/code/><dt
<dd><p>the character table <span class="SimpleMath">t</span>, say, of the

</ddpcode ="eyw>true</code> if the <code class="func">Irr</code>
<dt><strong class="<</
<d<>ode class="keyw">true</java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0

</dd>
<dt<strong class"Mark"<codetstrong java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<dd><p>java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 5

</p>No  theconjugacyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</dl>
<p>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

dinthe Character  Library/pan</4

<>java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 520

<p><a id="X7A948B397B5BC0AD" name

<h5>p>< class="unc"55.- <>

<divp="<>var class=""plan</var> is a list, each entry being a list containing positions of characters of <span class="SimpleMath">m.G</span> that form an
<<class"<//>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<">plan</var> is a list, java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0

<

<p>Examples where <code class="<55java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p><a id="div class="func"=func =100%>trclass"dleft"< p>Let <var class="Arg">tblmGa</var> be the ordinary character table of a group of structureArg></ be java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 551

h55.8-2 ConstructMGAInfo</5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 0
<><varclass"">tblmGa/var> be java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0

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

<p><ah5>Let "></ar>be the character table of a group of type <span class="SimpleMath">2^2.G</span> where an outer automorphism of order <span

<h5>5.8-3 ConstructGS3</h5>

<v="java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 396
<<divclass="func><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstructGS3Info</code>( <var class="Arg">tbl2</var>, <var class="Arg">tbl3</var><>< =">ConstructGS3/code constructs theirreducibles ofan ordinary character table<var div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstructProjInfo">/>  ="">kernel</ar </td> class="dright(&;function </><tr</<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConstructProjInfo</code>( <var class="Arg">tbl</var>, <var class="Arg">kernel</var> )</td><td class="tdright">( function )</td></tr></table
java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 879

<p><code class="func">ConstructGS3Info</code> returns a record with the components <code class="code">ind2

<p><a id="X7C1CE54A864936E0" name="X7C1CE54A864936E0"></a></p>

<

<div class=p><code"7p<a ""X7C1CE54A864936E0">/<
<p>Let <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0

<p><aid=X8726ADCC7BCEA96F name"8726ADCC7BCEA96F"></a></p>

<h5>5.8-5 ConstructProj</h5p <code class="func">java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ivclass="func"><table class="func" width="100%"><tr><td dd>p> valueof<class="irrinfo<var>that is needed for constructing the irreducibles of the <code class="code">tbl</code> component 
<< ="java.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds for length 0

<<=func">onstructProjInfo<code> takes an ordinary character table <var class="Arg">tbl</var> and a list <var class="Arg">kernel</var> of class positions of a cyclic kernel of order dividing <span class="SimpleMath">12</span>, and java.lang.StringIndexOutOfBoundsException: Index 247 out of bounds for length 0


<
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
> java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 235

<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<>strong
<dd><p recordthe entrydd><p>a character table that is permutation isomorphic with <var class="Arg">tbl</var>, and sorted such that 

</>
<dt><strong class="java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
<dd><<bothclassf"< ="" width=100%><  the   java.lang.StringIndexOutOfBoundsException: Range [103, 102) out of bounds for length 289

</dd>
</>
<p><a id

<h5>5.8-6 ConstructDirectProduct</h5>

<div class="class=""><tablepNote  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The direct product of the library character <p><a id="X7F26BEC67B744999" name="X7F26BEC67B744999"></p>

<p>h5>5.8-7 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

<p>If the java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<p><

<p><iv class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

<h5>5.8-7 ConstructCentralProduct

<class=>java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 369
<s="Arg"tbl/>  withjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 0

<p 

<p< = in    java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 165

<h5>5.8-8 </5< =7java.lang.StringIndexOutOfBoundsException: Range [33, 27) out of bounds for length 62

<divjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class"rg>>   java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 289

tionthejava.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 165

<> java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 282

<.9ConstructWreathSymmetric/5java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

<class="< "func width="00%>< class="<div class="func"><table class="func" width="100%"><tr><td class="tdleft
<p>The wreath-ConstructIsoclinic<>

<p>If the =""< java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 394

<p><a id="X7F6A9FE57C77D778p>If thejava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 391

<>5.8-10 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="8-11 ConstructPermuted
<>construction  to java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 517

<java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 391

<p>In both cases, if the optional arguments <var class="Arg">permclasses</var>, <var class="Arg">permchars</var> are given then the classes and characters of the result are sorted accordingly.</p>

<p><a id="X7D3F0AC2825C8CA1" name="X7D3F0AC2825C8CA1"></a></p>

<58-11 ConstructPermutedh5>

<div class="func"><</h5>
<p>The div class="func"><table class="func" width="100%"><tr><td class="tdleft>java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 208

<p>So <var class="Arg">tbl</var> and the library table with the name <var class="Arg">libnam</var> are permutation equivalent. With the more general function <code class="func">ConstructAdjusted</code> (<a href="chap5.html#X85FE8954830F85A0"><span class="RefLink">5.8-12</span></a>), one can derive character tables that are not necessarily java.lang.StringIndexOutOfBoundsException: Index 349 out of bounds for length 0

<p>The two permutations are optional. If they are missing then the lists of irreducible characters and the power maps of the two character 

<p><a id="X85FE8954830F85A0" name="X85FE8954830F85A0"></a></p>

<h5>5.8-12 ConstructAdjustedhr /<>.-12ConstructAdjusted</h5java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

<div class="func"><table class="func" width="100%"><tr><td class="tdleft<body>
<  values  library class=Arg><vararegiven bytheattribute values  bythe list<var class""pairs/var and–thoseattributes which do appear in<class="rg></>–by the attributejava.lang.StringIndexOutOfBoundsException: Range [269, 268) out of bounds for length 502

<p>This construction can be used to derive a character table from another library table (the one with the name <var class="Arg">libnam</var>) that is <em>not</em> permutation equivalent to this table. For example, it may happen that the character tables of a split and a nonsplit extension differ only by some power maps and element orders. In this case, one can encode one of the tables via <code class="func">ConstructAdjusted</code>, by prescribing just the power maps in the list <var class="Arg">pairs</var>.</p>

<p>If no replacement of components is needed then one should better use <code class="func">ConstructPermuted</code> (<a href="chap5.html#X7D3F0AC2825C8CA1"><span class="RefLink">5.8-11</span></a>), because the system can then exploit the fact that the two tables are permutation equivalent.</p>

<>

<h5>5

<java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 286
<p> <="rg>java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 208


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