Quelle ctblm.htm

Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/ctbllib/htm/ctblm.htm

java.lang.StringIndexOutOfBoundsException: Range [65, 58) out of bounds for length 71
        "http://www.w3.org/TR/html4/loose.dtd">
<html>
<meta name="GENERATOR" content="TtH 3.59">
<style type="text/css"> div.p { margin-top: 7pt;}</style>
<style type="gap> Exten(
td div.comp { margin-top: -0.6ex; margin-bottom: -1ex;}
td div.comb { margin-top: -0.6ex; margin-bottom: -.6ex;}
td div.hrcomp { line-height: 0.9;
td div.norm {line-height:normal;}
span.roman {font-family: serif; font-style  ^1+)java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 64
     #  after3^1+12:6.Suz.2//[ 1,20]:found 12 classesnow have 165
java.lang.StringIndexOutOfBoundsException: Range [15, 5) out of bounds for length 74

             <title>Some stepsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h1 align="center">CharacterTable( "3<re
  <body bgcolor="FFFFFF">

<div class="p">gap#62 : java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 0

<h3 align="center"> T<font size="-2">java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 37

<div class="p"></div>

<h3 align="center">December 12th, 2024 </[java.lang.StringIndexOutOfBoundsException: Range [24, 20) out of bounds for length 54

<div class=" java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 47

<div class="p"></div>
We show the details of

<div class="p"></div>

<div class="p"></div>

<h1>Contents </h1><a href="#java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
>1&<prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
>2  Some restrictions of the natural character of <font size="+0">M</java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
>3  The permutation java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 28
>4  The conjugacy> S,OrdersClassRepresentatives S",
>4.1  Our strategy to describe the conjugacy classes of <font size="+0">M</font></a><br />    <a href      fusions java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
>4.&bsp;Utility</< /& compute the root classes for both candidates
>4.3  Classes of elements of even order</a><br />    <java.lang.StringIndexOutOfBoundsException: Range [0, 80) out of bounds for length 9
> first  a  <>head2/t  theinformation  <tt</t.
>4.5  Classes of elements of order divisible by 5</a
>4.6  Classes of java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
>4.7  Classes of     gap> Length( head.fusi ( headf );
>48  Classes ofelements of  13</>br>n;nbsp&;n< href    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
>4.9  Classes of elements of order divisiblegap#62 ([ 1.  ,i>.[]  head2.usionsi] ;
>4.10  Classes of elements of order divisible by 41</a><br />    <a href="#tth_sEc4.11"
>4.11&; Classesof elements of  divisible  59<>< >&;nbsp&; ap62 .[]m=ead2usions[];
>4.12  Classes of elements of order divisible by 71</a><br />    <a href="#tth_sEc4.13"
>4.13  Classes of elements of order divisible by 7</a><br /><a href="#tth_sEc5"
>5  The power maps of <font
>6  The degree 196 883 java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 35
>p; The irreduciblecharacters <fontsize"+><font>/a></< =#th_sEc8"
>8  Appendix: The character table of 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span    &tyle='color: green'>62         &#2 Positions( SizesConjugacyClassesf) 2 ;
>9  Appendix: <iv =""</div>

<div class="p"></div>

<h2><a name="tth_sEc1">
1/> n;Overviewh2java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

<div class="p"></div>
The aim of [<a href="#Mverify" name#  after 3^1+2:..2/[ 1,19 ]: 12,
of the Monster group <font size="+0">M</font>.
Here we collect,
in the form of an explicit and reproducible
<font face="helvetica">GAP</font> [<a href="#GAP" name="CITEGAP">GAP24</a>] session java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 0
the relevant computations that are needed in that paper.

<div class="p"></div>
We proceed as java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0

<div class="p"></div>
Section <a href="#natural">2</a> verifies thejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
ordinary irreducible java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0
groups.font "0><=>
as stated in [<a href="#Mverify" name="CITEMverify">BMW24</aThe  2.<ontsize="+">B</font    conjugacyjava.lang.StringIndexOutOfBoundsException: Range [80, 76) out of bounds for length 88

<div class="p">of order 5,
Section <a href="#suborbits">3</a> verifies the decompositions of
thetransitive constituents of the permutation character
of the action of C<sub><font size="+0">M</font></sub>(a)  ≅ 2.<font size="+0">B</font> on the conjugacy class a<sup><font java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 0
where a is a <tt>2A</tt> involution in <font size="+0">M</font>.

<div class="p"></div>
Sections <a     true
character table head of <font size="+0">M</font>, that is,
the  conjugacyclasslengths element orders andpower.

<div class="p"></div>
Section <a href="#sect:natcharM">    gap#62;headfusions[5ith  5&124< =">HN</>&124  55sup>7/> #1242< class=r">/>>2</sub#; .
degree 196 883 characterof <fontsize="</font>
and decides the isomorphism type of the <tt>3B</tt> normalizer in <font size="+java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>
With java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 6
of the java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
3<sup>1+12</sup><sub>+</sub>.2.<span class="roman
computing the irreducible characters of <font size="+0">div ="p">!--><div>
this corresponds to [<a href="#Mverify" name="CITEMverify">BMW24</a>,Section 5]
and is done in Section <a href="#sect:irreduciblesM">7</a>.

<div class="p"></div>
The final sections <a href="#sect:The question which of the two above candidate ta to
<a href="#sect:norm3B">9</a>, <of <font size="+0">M</font> will Section&bsp< =gap62= (java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 58
  java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 58

<div class="p"></div>
Wediv =">!-->/iv>
and the interface to the  A<font sizetrue
thus we load these <font face="helvetica"

<div class="p">

<pre>
    gap> LoadPackage( "ctbllib", false );
    true
        =p"></div>
    true

</pre>

<div class="p">java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 6
The <font face="helvetica">MAGMA</font> system [<a href="#Magma" name="CITEMagma"><div ="p"</div>
Ifthe followingcommand <tt>false/>
then these steps will not work.

<div class="p"></div>

<pre>
    gap> CTblLib.IsMagmaAvailable();
    true

</pre>

<div class="p"></div>
We set the line length to 72, like in other standard testfiles.

java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    &62; izeScreen<classp<!

</pre>

<div class="p"></div>

<h2><a name="tth_sEc2">
2</a>  Some restrictions of the natural characterpre>
</a>

<div class="p"    gap#62 s:= CharacterTable 2B );
We existence of  ordinary irreduciblecharacter&967;
of degree 196 883 of the Monster group <font size="+0">M</font>,
and that <font size="+0">M</font> has only two conjugacy classes of involutions.

<div class="p"></div>
First we compute the restriction of χ to 2.<font size="+0"gap#;pi  ;

<div class="p"></div>
Theonlyfaithfuldegree 196&;883 character of 2.font size=+"B< class="<java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 28
that two are rational
1a + 4371a + 96255a + 96256a, as claimed in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 1].
Thisfollows the dataabout2.font "0>B<font>

<div class="p"></div>

<pre>
    gap> table2B:= CharacterTable( "2.B" );;
    gap> cand:= Filtered( Irr( table2B ), x -> x[1] <= 196883 );;
    gap> List(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 1, 4371, 96255, 96256 ]
    gap> inv:= Positions( OrdersClassRepresentatives( table2B ), 2 );
    [ 2, 3, 4, 5, 7 ]
    gap> PrintArray( List( cand, This  tt</> >>re
    [ [       1,       1,       1,       1,       1,       1 ],
      [    4371,   4371    493 275,     275,      19 ]java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
        4863,    2047    2047,    255 ,
      [   96256,  -d centralizer orders5 |<pan class="oman"HN[1,45 java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

</pre>

<div class="p"></div>
Note that 96256a must occur as a java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 28
because is only  candidate,
and it can occur only once because otherwise only 4371a + 2 ·96256a
or 4371 ·1a + 2 ·96256a would be possible decompositions,
which have java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
Thus the values of χ<sub>2.<font size="+0">B</font></sub> on the classespre>

<div class="p"></div>
If 96255a would <b    #;     i-#;pi]  Size  )/ SizesConjugacyClasses(s[]);
on the s3and7 woulddiffer 512 times 4371,
but 65659 ·1a + 8 ·4371a +     1365154560000000, 94500000000]
Thus 96255a must occur exactly once.

<div class="p"></div>
sum 96255 and 96256  fourdifferent
n, hence  4371amust occur.

<with <ont =""</font,seeSectionnbsp< href"sect:"10/>,
We see that the values of χ on the classes of involutions are
4371 and 275, respectively.

<div we   character  from

<pre>
    gap> Sum( cand ){ inv };
    [ 4371, 4371, 4371, 275,

</pre>

<div class="p"</div>

<pre>
    gap> table3Fi24prime:= CharacterTable( "3.Fi24'" );;
    gap> cand:= we know that <font size="+0M/    gap&2  java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 42
    gap&(5(1+6)2.2.
    [ 4, 7 ]
    gap> mat:= List( cand>5A/>5  ="java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 82
   # (  mat;
    [ [      1,      1,      1 ],
      [   8671,    351,    -33 ],
      [  7477   1157,    133  ,
      [    783,     79,     15 ],
      [    783,     79,     15 ],
        4584,   1352      java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
      [  64584,   1352,     72 ] ]

</pre>

<div class="p"></div>
Since 1,4,37800,40320 ]
charactersahetwo classesare rational
783ab, and 64584ab.

<div class="p"></div>

<pre>
    gap&#    I 5^+)2J2.  3classes  havejava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
    [ 1, 8671, 57477, 783*E(3), 783*E(3)^2, 64584*E(3), 64584*E(3)^2 ]

</pre>

<div class="p"></div>
We see that the value on the first class of involutions must be 4371,
since all values of the possible constituents are positive
and too large for the other iv p</div>

<div class="p"></div>
Since the values of all possible constituents on the second class of
   equal the on firstclass,
and equal only for 1a,
we conclude that the value on the second class of involutions is 275.

<div class="p"></div>
We see from the    CharacterTable( "(10xHN).2")
and on the first class of involutions
that constituents of degree 57477<a

<div class="p"></div>

<pre>
    gap> Float( ,45]
     ">
    gap> List( mat, v -> Float( gap> ExtendTableHeadByRootClasses,s 45 ;
    .6776, 991139,,991139 47.7692,.7692 java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
    gap> Float( ( 196883 - 2 *Thepermutation character(1sub.
    40.6209
    gap> Float( ( 196883 - 57477 ) / ( 4371 - 1157 ) );
    43oforder 11 withcentralizer order 11 |M<sub>12</sub>&#yle='color: green'>124; in <font size="+0"
    gap> Float( ( 196883 <div class""</div>
    39.8294
    gap> Float( ( 196883 - 3*57477 ) / ( 4371 - 3*1157 ) );
    27.1689

</pre>

<div class="p"></div>
First suppose that 64584ab is not a constituent.
The above ratiosdiv="p"</div>
must occur.

<div class="p"></div>
However, then the
hence the value on the second class of involutions cannot be 275,
a contradiction.

<div class="p"></div>
means that both64584ab 57477 occur

<div class="p"></div>

<pre>
    &62; + 6] 7;
    [ 186645, 3861, 277 ]

</pre>

<div class="p"></div>
The second involution class forces one constituent of degree 8671
(java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 6
and then a character of degree 1567 remains to be decomposed.
The only solution for the degrees of its constituents is 1 + 1566.
We thedecomposition
1a + 8671a + 57477a + 783ab + 64584ab,
as claimed in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 2].

<div class="p"></div>

<pre>
    gap> Sum( mat );
    [ 196883, 4371, 275 ]

/>

<div class="p"></div>
The characters of the degrees 1, 8671, and
from 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> to 3.<<iv
In order decompose restriction
we have to determine which extensions from 3.<span class="roman">Fi</span><sub>24</sub><sup>&#tyle='color: green'>8242;</sup> occur.
The following irreduciblepre>
constituents &5:=java.lang.StringIndexOutOfBoundsException: Range [44, 42) out of bounds for length 51

<div class="p"></div>

<pre>
    gap> table3Fi24:= CharacterTable( "3.Fi24" );;
    :Filtered(Irr table3Fi24)x-62;] l; 196883 ;
    gap> inv:= Positions( OrdersClassRepresentatives( table3Fi24 ), 2 );
    [ 3, 5, 172, 173 ]
    &     -#; java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 6
    gap> PrintArray( mat );
    [ [       1,       1,       1,       1,       1 ],
      [       1,       1,       1,      -1,      -1 ],
       8671,    351,     -33,   1495     -1 ,
      [    8671,
      [   57477,    1157,     133,    5865,     233 ],
      [   57477,   1157,     133,   5865    -233java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      [    1566,,    158,     30,      0,      0 ],
      [  129168,    2704,     144,       0,       0 ] ]

</pre>

<div class="p">
We get the decomposition
1a++129168 java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 112

<div class="p"></div

<pre>
    gap> Sum( mat{ [ 1, 4, 5, 7, 8 ] } );
    [ 196883,,4371,275 4371, 275  ]

</pre>

<div class="p"></div>

<<name>
3</a>  The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></sup 2<size+></java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
</a>

=p
According <a href"GMS89" name"ITEGMS89">GMS89</a>,Tables VII, IX],
the restriction of the permutation character 1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></sup> to 2.<font size="+0">B</font>
decomposes into nine transitive permutation characters 1<sub>U</sub><sup>2.<font size="+0">B</font></sup>,
with the point stabilizers U listed in Table <a href="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>

<div class="p
<a name="tth_tAb1">
</a> <center>Table 1: Suborbit java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 5
</a>

<center    &62;s(2)

<table>
<>  </td><td alignjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 122
<tr><td><tt>1A</tt> </td><td>2.<font size="+0">B</font>            </td><td align="right">1 </td></tr>
<tr><td><tt>2A</tt> </td><td>2<sup>2</sup>.<sup>2</sup><span class="roman">E</span><sub>6</sub>(2) </td><td align="right"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<sup/>span =""><span>sub the arguments < href""name"></>],
<tr><td><tt>3A</tt> </td><td><span class="roman">Fi</span><sub>23</sub>        </td><td align="right">2031941058560000 </td></tr>
<tr><td><tt>3C</tt> </td><td><span class="roman">Th</span>             </td><td align="right">91569524834304000 </td></tr>
<tr><td>tt>A</> />td>sup+<sup><pan=r></>   <td>td ="rightjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<tr><td><tt>4B</tt> </td><td>2.<span class="roman">F</span><sub>4</sub>(2)       </td><td align="right">1254793905192960000 </td></tr>
<tr><td><tt>5A</tt> </td><td><span class="roman">    gap#62;java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 6
>>A /<>. />>

</center>

<div class="p"></div>
Here roup 2.<</div>
the action is that on the <font size="+0">M</font>-conjugacy class of a,
and c  &8712 <sup< size=+"M<font</

<div class="p"></div>
n sectionwe  the characters1<ub</ub<sup>.font size=+0"B/ont>/>,
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 62
Note that a  &
(and thus the character is an inflated character of <font size="+0">B</font>)
if and only if a and c commute;
this happens exactly for the first three orbits.

<""<-->/iv
All subgroups U except 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> and 2<sup>Sylowsubgroupsthatimplies
are  A<font size="-2">TLAS</font> groups whose character tables havedivclass""</div>brclear="ll /<table =""width="00%"tr>tdjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
The subgroup 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> is the java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 43
2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub> of <font size="+0">B</font> under the natural epimorphism java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 28
and the computation/verification of the character table of 2<sup>1+22</sup>.<span class="roman">Co<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
asdescribedjava.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 74
It will turn Note that either the two classes of elements of order p in 2.<font java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 68

<div class="p"></div>[118]
The characterswill bestored the variables, s, pi, pos );
<tt>pi1</tt>, <tt>pi2</tt>, ..., <tt>pi9</tt>.

<div class#   order 17  from .:175
For U = 2.<font size="+0">B</font>,
we1<subU<sub><up>2< size"0"Bfont<sup =1sub< "0><font<sub>

< =p<

pre
    gap> pi1: java.lang.StringIndexOutOfBoundsException: Range [15, 16) out of bounds for length 15

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

<div class="p"></div>
For U = 2<sup>2</sup>.<sup>2</sup><span class="roman">E</span><sub>6&124;g<sup><ont="/>
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 143
from <font size="+0">B</font> to 2.<font size="+0"  of  8712 , ,
for [U]andwecan find group 2< ">< java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 95
(Note that the class fusion from [U] to java.lang.StringIndexOutOfBoundsException: Range [109, 16) out of bounds for length 140
determined by the character tables of the two groups,
but the permutation character is unique.)

<div class="p"></div>

<pre>
    gap&#62"p></div>
    gap> tableUbar:= CharacterTable( "2.2E6(2)" );;
    gap> fus:= PossibleClassFusions( tableUbar, tableB );;
    gap&<br clear= >table"0 =100%"<>td
    >              map -> InducedClassFunctionsByFusionMap( tableUbar, tableB,
#
    gap<iv class="p"</div>
    1
    gap&62;:(,table2B pi )1];
    gap> mult:= List( Irr( table2B ),
    &#<pre<d/tr<table>
    gap> Maximum( mult );
    1
    gap> Positions( mult, 1 );
    [ 1, 2, 3, 5, 7, 13, 15, 17 ]

</pre>

<div class="p"></div>
2<sup+22<sup>. class=roman></pansub2</subfuse infontsize=+"<font>  ;
the character 1<sub>U</sub><sup>2.<font size="+0">B</font></supin theformercase
from <font size="+0">B</font> to 2.<font size="+0">B</font>,
for [U] = U / 〈a &#yle='color: green'>9002; =  2<sup>1+22</sup>.<span classgap&62;pos ,
a maximal subgroup of <    [147, 149 ]

<div class="p"></div>

<pre>
    gap> tableUbar:= CharacterTable wejava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    CharacterTable 2122C"")
    gap> fus:= PossibleClassFusions( tableUbar, tableB );;
    gap> pi:= Set( fus,
    >              map -> InducedClassFunctionsByFusionMap( tableUbar, tableB,
    >                         [ TrivialCharacter( tableUbar ) ], map )[1] );;
    gap62;  ;
    1
    gap#62;pi3: Inflated tableB,table2B, pi )[1];;
    gap> mult:= List( Irr( table2B ),
    &62;                 chi -#;ScalarProducttable2B, chi pi3 ;java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
    gap> Maximum( mult );
    1
    gap> Positions( mult, 1 );
    [ 1, 3, 5, 8, 13, 15, 28, 30, 37, 40 ]

</pre>

<div class="java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0
Next   ==span =roman>Fipansub23<>

<div class="p"></div>

<pre>
    gap> tableU:= do not fuse in <font size="+0">M< java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    gap>
    gap> pi:= Set( fus,
    &62;             map-#62 ( tableU ,
    >                         [ TrivialCharacter( tableU ) ], map )[1] < class">
    &62;Length  23java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    1
    gap&<
    gap> mult:= List( Irr( table2B ),
    >                 chi -> ScalarProduct( table2B, chi, pi4 ) );;
    gap&62;Maximum( mult )java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    1
    gap> Positions( mult, 1 );
    [ 1, 2, 3, 5, 7, 8, 9, 12, 13, 15, 17, 23, 27, 30, 32, 40, 41, 54,
      63, 68, 77, 81, 83java.lang.StringIndexOutOfBoundsException: Range [47, 44) out of bounds for length 51
       java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

</pre>

<div class""></div>
Next we consider U = <span class="roman">Th</span>.

<div class="p"></div>

<pre>
    gap&
    gap> fus:= PossibleClassFusions( tableU, table2B );;
    gap> pi:= Set( fus,
    >             map -> InducedClassFunctionsByFusionMap( tableU, table2Bgap&62;CollectedFactors(n ;
    >                         [ TrivialCharacter( tableU ) ], map )[1] );;
    gap> Length( pi );
    1
    gap> pi5:= pi[1];;
    gap> mult:= List( Irr( table2B ),
    >                 chi -> ScalarProduct(</re
    gapMaximum( mult
    2
    gap> Positions( mult, 1 <ivclass="p"></div>
    [ 1, 3, 7, 8, 12, 13, 16, 19, 27, 28, 34
      85, 89, 113, 114, 116, 129, 133, In orderprove that thetwoclasses   order23 in 2. size"+<class=p">!--<div
      188, 193, 195, 196, 201, do notfuse  font =+"M/java.lang.StringIndexOutOfBoundsException: Range [41, 38) out of bounds for length 90
      237, 242sup3<sup.
    gap> Positions( mult, 2 );
      java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 96

</pre>

<div class="p"></div>
For U = 2<sup>1+22</sup>.<span class="roman"
[<a href="#ctblpope" name="CITEctblpope">Breb</a>,Section "A permutation character of 2.<font size="+0">B</font>"].
We know that U is a subgroup of 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>,
and

<div gap : Positions java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 6
As a first step, java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0
to < class="<>
which can be&java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 50
then to inflate this character to 2<java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 14
then to induce this character to <font size="
and  thenecessary 2-part of the centralizer occurs/>

<div class="p"></div>

<pre>
    & : java.lang.StringIndexOutOfBoundsException: Range [39, 33) out of bounds for length 44
    gap> co2:= CharacterTable( "Co2" );;
    <java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
    gap> Length( fus );
    4
    gap> ind:= Set( fus    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } );
                                 TrivialCharacter( mcl ) ] map)1] );;
    #Length java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    1
    gap> bm2:= CharacterTable( "BM2" );
    CharacterTable(#   order23 from 2.  178 classes
    gap> infl:= Inflated( co2, bm2, ind );;
    gap> ind:= Induced( bm2, tableB, infl );;
    gap> infl:= Inflated( tableB, table2B, ind )[1];;

<pre

<divclass""</div>
a,
<>U/>2< java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 133
using that we can speed up these computations by prescribing
the permutation character induced gap&62 ( head , pi { []})
he #; #9002 2<=0"B/>

<div class="p"></div>
(We are lucky:
There is a unique solution, and its computation is quite fast< =p<java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<>

<62=;
    gap> centre:= ClassPositionsOfCentre( table2B );
    [ 1, 2 ]
    gap> pi:= PermChars( table2B, rec( torso:= [ 2 * infl[1], 0 ],
    >                             normalsubgroup:= centre,
    >                             nonfaithful:= infl ) );;
    gap> Length( pi );
    1
    gap> pi6:= pi[1];;
    gap#62; List(Irr table2B),chi-#2;ScalarProducttable2B, chi pi6))
    [ 1, 1, 2, 1, 2, 0, 2, 3, 2, 0, 0, 1, 4, 1, 2, 0, 3, 2, 0, 2, 0, 0,
      2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3, 2, 0, 6, 4, 0, 1, 1,
      0, 0, 0, 0, 3,[62 62]
      0, 4, 2, 4, 4, 3, 0, 2, 4pre
0,, ,,3 ,,4 ,1,,0 ,,0,gjava.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 21
      1, 4 0,3,2 3, , 3,0,1,3,0,2,2, 1,, 3,0 ,2,0,
      0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0,    228 ]
0 ,1 2,1,1,1,0,1, 1 1,1java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 57
      1, 3, 3, 0, 0, 0, 1, 1, 1,pre
      0, 0, 0, 0, 5, 2, 0, 0, 0, 1, )
      7,    [,1]  23  , ,1]
        deal  47.

</pre

< ">--<
Next we consider U = 2.<span class="roman">F
ize+"B<.

<div class="p"></div>

<pre>
    ;;
    gap> fus:= PossibleClassFusions( tableU, table2B g&#yle='color: green'>62;:=(OrdersClassRepresentatives  )  ;
    gap>pi= Set( us map-#62;InducedClassFunctionsByFusionMap228230java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
   )],map1 ;
    gap> Length( pi );
    2
    gap> pi:= java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 8
    gap> Length( pi );
    1
    gap> pi7:= pi[1];;
    gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi7 ) );
    [ 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 0, 2, 4, 1, 3, 0, 2, 1, 0, 0, 0, 0,
      2, 1, 0, 0, 2, 2, 1, 4, 0, 2, 1, 2 94,  java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
      0, 0, 1, 0, 0, 0, 1  47java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 57
      0 6 ,2,0 ,,,1  , ,4,0 ,0 ,1 ,0,
      0, 3, 2, 1, 3, 2, 0, 0, 4, 5, 3, 1, 0, 3, 0, 0, 1, 1, 2, 0, 0, 2,
      0, 2, 0, 3, 3, 3, 0, 4, 1, 0, 4, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 0,
<>
      2, 3, 1, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      1, 4, 4, 0, 0, 0, 3, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      0, 0, 1, 1, 4, 2, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1<& >A<>.
      5, 0, 1, 1, 2, 2, 4, 4, 0, 0, 3, 1, 1, 1, 0, 0, 4<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
      2, 5, 5, 0, 1 ]

</<pre

<div class="p"></div>
we =  "><java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 51

<div class="p"></div>

<pre>
    gap> tableUdiv class""</div>
    gap> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> pi:= Set( fus, map -> InducedClassFunctionsByFusionMap( tableU, table2B,
    >             [ TrivialCharacter( tableU ) ], map )[1] );;
    gap> Length( pi );
    1
    gap> pi8:= pi[1];;
    gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi8 ) );
    [ 1, 1, 2, 1, 2, 0, 3, 4, 2, 1, 1, 4, 4, 2, 1, 1, 3, 3, 1, 3, 0, 0,
      5, 3, 0, 0, 6, 4,    [ 97 ]
      0, 0, 2, 1, 3, 4, 7, 0, 0, 7, 3, 9, 5, 0, 0, 6, 4, 2, 13, 6, 0#    java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 57
      4, 12, 11, 16, 9, 7, 3, 11, 13, 12, 20, 5, 10, 6, 11, 13, 17, 4,
      10, 7, 19, 7, 7, 8, 10, 14, 18, 19, 5, 10, 12, 23, 7, 12, 6, 24, 6,
      4, 17, 16, 8, 9, 17, 11, 12, 23, 8, 24, 18, 26, 21, 29, 10, 18, 31,
      10, 24, 21, 17, 27, 35, 13, 14, 29, 19, 12, 7, 18, 26, 15, 34, 34,
      35,20, 14, 36, 14,39,8, 29, 2424,15, 0,13,9, 38, 17 35,
      32, 26, 26, 24, 22, 17, 31, 39, 29, 30, 30, 19, 44, 37, 37, 28, 30,
       ,40 ,56 ,30,,50  ,2 ,,4, 0,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
      0, 4, 6, 10, 10, 12, 8, 12, 0, 0, 2, 4, 16, 10, 0, 0, 2, 12, 10, 0,
      0, 0, 0, 0, 0, 28, 0, 0, 14, 34, 40, 2, 10, 10, 22, 40, 44, 44, 8,
      8 6 , 14, 16 ,8,46 ,28,,90 72,70,92,104 56,
      90 ]

</pre>

java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 28
Finally, we consider U = 2.<span class="roman">
There are two candidates for the permutation character (1<sub>U</sub><pre>
according to the possible class fusions.
One of the two characters is zero on the class of the central involution
of 2.<font size="+0">B<#Iafter order 13 elementfrom2. have classes
We know that U does not contain the central involution of 2.<font size="+0">B</font>,
hence we can decide which character is correct.

<div class="p"></div>

<pre>
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
    div class="p"></div>div>
    gap> pi:= Set( fus, map -> InducedClassFunctionsByFusionMap( tableU, table2B,
           java.lang.StringIndexOutOfBoundsException: Range [48, 40) out of bounds for length 66
    gap> Length( pi );
    2
    gap> pi:= Filtered( pi, x -> ClassPositionsOfKernel( x ) = [ 1 ] );java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> Length( pi );
    1
    gap>
    gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi9 ) );
    [ 1, 2, 3, 1, 4, 1, 5, 5, 5, 1, 1, 5, 8, 4, 4, 1, 7, 6, 0, 5, 0, 0,
107 ,10 6,,3 ,,10,, 511 ,4 19
      6, 6, 5, 0, 0, 0, 3, 6, 7, 11, 0, 0, 17, 2, 20, 9, 0, 0, 12, 8, 1,
      23, 11, 1, 8, 7, 23, 18, 27, 18, 12, 7, 22, 29, 21, 34, 6, 22    (  ){posjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
      22, 18, 33, 3, 19, 10, 34, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      40, 15, 25, 3, 40, 9, 6, 34, 25, 18, 21, 30, 21, 18, 43, 12, 45,
      39, 49, 38, 51,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      29,19, 40, 47,31 69, 69,65 42,35, 68,27 73, 20 53,46,38,
      75, 29, 24, 72, 50, 41, 72, 68, 58, 52, 54, 50, ble  29<h3< name"lements_29"
      65, 49, 85, 75, 75, 63, 68, 65, 63, 90, 87, 83, 118, 118, 74, 71,
      90, 109, 109, 2, 3, 6, 9, 8, 0,java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 28

      76, 11, 18, 18, 39, 77, 80, 77, 22, 22, 66, 27, 27, 33, 20, 20, 87,
      60, 60, 103, 175, 148, 152, 187, 215, 140, 201 ]

</pre>

<div class="p"></div>
Now we > x #  mod   ;

<div class=    &62 :=("Fi24");

<pre>
    gap> constit:= [ pi1, java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 53
    gap62&;.java.lang.StringIndexOutOfBoundsException: Range [41, 35) out of bounds for length 44

<pre

<div class="p"></div>

<h2><a name="tth_sEc4">
4</a>  The conjugacy <pre
</a>

<div class="p"></div>

     h3>aname="tth_sEc4.1"
4.1/a>nbsp;nbsp;Our strategyto the conjugacyclassesof fontsize=+0"></<h3>aname=strategy_classes>
</a>

<div classThusthe classes tt<> <>3A/> tt87A/>andtt/>
We know the order of <font size="+0">M</font> and its prime divisors.
Let us check whether this fits to our data div class=p></div>< ="><font> of  element of  29are 7,41 59  .java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78

<div class="p"></div>

<pre>
    gap> sizeM:= pi[1] * Size( table2B );
    808017424794512875886459904961710757005754368000000000
    gap> StringPP( sizeM );
    "2^46*3^20*5^9*7^6*11^2*13^3*17*19*23*29*31*41*47*59*71"
    gap> sizeM = Size( CharacterTable( "M" ) );
    true

</pre>

<div class="p">gap> head.SizesCentralizers{ poss };<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 21
we classify the conjugacy classes java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0
and use the java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 6
the classes of roots ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
are in <div class="p"
and that this bijection respects centralizer orders.

<div class="p"></div>

<div class="p"></div>
For each element x  ∈ <in <ontsize"0>M<
we will use the character
to establish <div class="p"><!->
In order not to count the same class several times,
we proceed by increasing p,
and collect only those classes of roots of x for which p is the smallest
rimedivisorof element.

<div class#62                     [ 2 ,5,11,,17,19,,29,31   ;
For elements x  ∈ <font size="+0">M</font> java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 21
it is not necessary to use the character table of N<sub><font size="+0">M</font></sub>(〈x 〉);
we will
and ad hoc arguments.

<div class="p"></div>

     <h3><a name="tth_sEc4.2">
4.<    ga>:FilteredCollectedFactorsheadSize)java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
</a>

< class"">!--<div
During the process of finding the conjugacy classes of <font size="+0">M</font>,
e our  aboutthe charactertable fontsize"0></
in a global <font face"helvetica">GAP</font> variable <tt>head</tt>,
which is a record with the following components.

<div class="p"></div>

< compact="ompact">
<dt><b><tt>Size</tt></b></dt>
<br/
    the group order |<font size="+0">M</font>|,</dd>
<dt><b><tt>SizesCentralizers</tt></b></dt>
<dd>  <br />
    helistof centralizer of the  classes
    established up to now,</dd>
<dt><b><tt>OrdersClassRepresentatives</tt>gap&62;cart: ( poss;java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
<dd  # = 29(, ;java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
    listjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
<dt><b><tt>fusions</tt
<dd>  <br />
the  known partial classfusions into< =+"<>
    eachentryis recordwith thecomponents
    <tt>subtable</tt> (the characterjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    and <tt><v ="p"></div>
    unknown positions are unbound).</dd>
</dl>

<div classgap#62 :  ,
java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 79
and that there is an    87 5133]

ivjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    &62head ( Size java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
    >                SizesCentralizers:= [ sizeM ],
    #;OrdersClassRepresentatives= [1]
    >                fusions:= [],
    &java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</pre>

<div class="p"></div>
on <></tt>takes
tt<>,
java.lang.StringIndexOutOfBoundsException: Range [44, 3) out of bounds for length 75
and an integer <tt>pos</tt> as its arguments,
where it is assumed that the <tt>pos</tt>-th class of <tt>s</tt>
an    orderp
such that N<sub><font size="+0">M</font><andthuswould benormalized hence centralized by  elementof
and such that <tt>head</tt> contains information only about
those classes of <font size="+0">M</font> whose elements<iv class="order 3, a .
that is smaller than p.

<div class="p"></div>

<pre>
    gap> ExtendTableHeadByRootClasses:= function( head, s, pos )
    >    local fus, orders, p, cents, oldnumber, i, ord;
    >
    >    # Initialize the fusion information.
    >    fus:= rec( subtable:= s, map:= [ 1 ] );
    >    Add( head.fusions, fus );
    >
    >    # Compute the positions of root classes of 'pos'.
    >    orders:= OrdersClassRepresentatives( s );
    >    p:= orders[ pos ];
    >    cents:= SizesCentralizers( s );
    >    oldnumber:= <divclass"p"></div>
    >
    >    # Run over the classes of 's'
    >    # are already contained in head
    >    for i in [ 1 .. NrConjugacyClasses( s ) ] do
    >
    >      if ord mod p = 0 and
    >         Minimum( PrimeDivisors( ord ) ) =    #   elementfrom .+.2  185 classes
    >         PowerMap( s, ord / p, i java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    >        # Class 'i' is a root class of 'pos' and is new in 'head'.
    >        Add( head.SizesCentralizers, centsThus we  <div class="p"></>
    >        Add( head.OrdersClassRepresentatives, java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    &#2 fus[iwith centralizer  order3 &183;29java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
    62;
    >    od;
    &62
   &62;Print"I  after" Identifier(s," found ",
    >           Length( head.java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
    >           " classes, now have ",
    >           Length( head.OrdersClassRepresentatives ), "\n" );
    >   end;

</pre>

<div class="p"></div>
Inseveralcases will a classgsupfont =+"M<y  an   4an  read off fromthefactorgroup
knowing the character table of a suitable subgroup of <font size="+0">M</font> to which
<tt>ExtendTableHeadByRootClasses</tt> #>8/><;/(java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
the root classes.
,mayjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 50
and

<div class="p"></div>pre>
We are a bit better off if we know the character table <tt>s</tt>
of a subgroup ofdivp>!-/
in this table which fuse to the class g<sup><font size="+0">M</font></sup>, because then we pre
store this information in the partial class fusion from <tt>s</tt>
that is java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 6

<div class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<pre>
    gap> java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
    >    local ord, fus, i;
    >
    &62;if
    >      ord    >                     x -&#2 (head. /  ) od =1 )
    #;       Length(ord1640163016
    >        Error( "    gap> List( good, Factors> List , Factors );
    >      fi;
    >      ord[[ 2, ,2 where 5 cannotdividetheorder.
    >    elif IsInt( s ) then
    >      ord:= s;
    #;    ;
    >    Add( head.SizesCentralizers, cent );
    >    Add( head.OrdersClassRepresentatives, ord );
    >
    >    Print( "#I  after order ", ord, " element" );
    #;    ifIsCharacterTable(s
    >      # extend the stored fusion from s
    >      fus:= First( head.fusions,
    >                   r -> Identifier( r.java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 28
    >      forthus normalizedby an  of order,
    >        fus.map[i]:= Length( head.SizesCentralizers );
    >      od;
    >      Print( " from ", Identifier( s ) );
    &
    >    Print( ": have ",
    >           Length( head.OrdersClassRepresentatives ), " classes\n" );
    #;    end;

</pre>

<div class="p"></div>
The permutation character 1<sub>H</sub><sup>G</sup>, where H  ≤ G are two groups,
has the property 1<sub>H</sub><sup>G</sup>(g) = |C<sub>G</sub    [
For g  ∈ H,
this implies that |C<sub>G</sub>(g)| = 1<sub>H</sub><sup>G</sup>(g) ·|H
can be computed from the character (1<sub>H</sub><sup>G</sup>)<sub>H</sub> </pre>
provided that we java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
The/>
in<t></ttby the data the g<supfont ="0"M<font>/>
where <tt>s</tt> is the character table of H,
<</tt 1>/java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 65
and <tt>poss</tt> is the list of positions
that fuse to g<sup><font size="+0">M</font></sup>.

<div class="p"></div>

<pre>
    gap> ExtendTableHeadByPermCharValue:= function( head, s, java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 41
    &#62    &62 ( head 41, 41,fail;,a .
    >
    >    pival:= java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    >    if Length( pival ) <> <div =p>
    >      Error( "classes cannot fuse" );
    >    fi;
    >
    &div =""<-->/iv>
    >    ExtendTableHeadByCentralizerOrder( head, s, cent, poss );
    #;end;

<

<div class="p"></div>

     <h3><a name="tth_sEc4.3">
4.3</a>  Classes of elements of even order</h3><a name="elements_2">
</a>

<div class="p"></div>
nbspjava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 58
we know we haveestablishedarationalclassof self
and that the involution centralizers have the structures
2.<font size="+0">B</font> (for the class <ttBy  above arguments,
2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</<iv ">

<div class="p"></div>
character of are
stored in the <font face="helvetica">GAP</font> Character Table Library are correct.
For 2.<font size="+0">B</font>, this follows from the correctness of the character table of <font size="+0">B</font>
in[a href#BMverify name="ITEBMverify">BMW20</a>] and the computations
in [].
For 2<sup>1+24</sup><sub>+</java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
in Section <a href="#sect:table_c2b">8</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<divclass=p"</div>
Thus we can determine the <font size="+0">M</font>-conjugacy classes of elements of even order
as follows.

<div class="p"></div>

<pre>
    gap> s:= CharacterTable( "2.B" )4.11<a nbsp;Classesof elements of order divisible  59<h3aname"elements_59">
    gap> ClassPositionsOfCentre( s );
    [1 2 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    gap> ExtendTableHeadByRootClasses( head, s, 2 );
    #I  after 2.B
    gap> s:= CharacterTablediv ="">!-></div>
    gap> ClassPositionsOfCentre( s );
    [1,2 ]
    gap> ExtendTableHeadByRootClasses( head, s, 2 );
    #I  after 2^1+24.Co1: found 91 classes, now have 134

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc4.4">Sylowsjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 61
4.4</a>  Classes of elements of order
</a>

<div class="p"></div> class="><pre>
We know that <font size="+0">M</font> has exactly three conjugacy classes of elements
of order 3,
and that their normalizers have the structures
3.<span class="roman">Fi</span><sub>24</sub> (for the class <tt>3A</tt>),
3<sup>1+12</sup><sub>+</sub>.2.<span classdiv class="p"</div>
Moreover,
the <the normalizer order of an element of order 71 divides
and have been verified, see [<a href="#BMO17" name="CITEBMO17">BMO17</a>].

<div class="p"></div>
We determine the <font size="+0">M</font>-conjugacy classes of elements of odd order
'sjust  order  71&java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

< class"p"></div>

<pre>
    gap> s:= CharacterTable( "3.Fi24" );;
    gap> ClassPositionsOfPCore( s, 3 self-centralizingelements  order59java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    [ 1, 2 ]
    &;ExtendTableHeadByRootClasses(head, s,2)
    #I  after 3.F3+.2: found 12 classes, now have 146
    &;s= CharacterTableDirectProduct( "h"),
    >                                     CharacterTable(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> ClassPositionsOfPCore( s, 3 );
    [ 1, 3 ]
    gap> ExtendTableHeadByRootClasses( head, s, 3 );
    #I  after ThxSym(3): found 7 classes, now have 153

</prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

<div class="p"></div>
The situation with the <tt>3B    #  order element: have 188 classes
Section <a href
of a downward extension of the structure 3<sup>1+12</sup><sub>+</sub<pre
of the
candidates for the character table of the < ="</div>

< ""</div>
It will turn out that each of these candidates leads to "the same"
root classes,
in the sense that the number of these classes, their element orders,
and their centralizer orders are equal.a
Note that the 3-
X ×N,
where X has order 3
and N  &8773; 3sup1+2<sup><sub>+/sub> such thatH/ N  &8773;6<spanclass=roman">2holds.
We are interested in the two "diagonal" factors, that is,
the factors of H by the one of the two normal subgroups of order 3 in H
are not equal   orZN)
(See the picture'   the   71&#;35.#  order     190

<div class="p"></div>
First we exclude the normal subgroup of order 3 that is contained in the
unique normal subgroup N of order 3<sup>13</sup>.

<div class="p"></div>

<pre>
    gap> exts:= CharacterTable( "3^(1+12):6.Suz.2" );;

    [ 2, 18, 19, 20 ]
    gap> order3_13:= Filtered( java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 4
    >        l -> Sum(gap&62; good:= Filtered( possord,
    [ [ 1 .. 4 ] ]
    gap> kernels:= Difference( kernels, order3_13[1] );
    [ 18, 19, .font size="+"B/ont>yields  rationalclasstt</>

</pre>

<div class="p"></div>
The classes in the subgroup X can be identified by the fact that
exactly one factor of H
class fusion from 2.<span class="roman">Suz</span>.2, and hence this must be the split
< ="">!-->/ivjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<div class="p"></div>

<div class="p"></div>

<pre>
    gap> facts    gap> s: CharacterTable("2.B );;
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 18 ]" ),
      CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ),
      CharacterTable 3(+12):6.uz/ ,20]  ]
    gap> f:= CharacterTable( "2.Suz.2" );;
    gap#;:=( facts
    >        x -> Length( PossibleClassFusions( f, xgap>ExtendTableHeadByCentralizerOrderhead71,71    #  after7elementfrom2. have 191 classes
    [CharacterTable 3(+12):6.Suz2/ 1, 19 ]" ),
      CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" ) ]

<    true

<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
We compute the<pre
For that,
we first create a copy <tt>head2</tt> of the information in <tt>head</tt>.

<div class="p"></div>

<pre>
    &a
    >        f -> Positions( SizesConjugacyClasses( f ), 2 ) );
      ,   java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    &h:java.lang.StringIndexOutOfBoundsException: Range [41, 35) out of bounds for length 45
    gap> java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
     3^12:.uz[,9 ]found12 ,nowhave165
    gap> ExtendTableHeadByRootClasses( head2, facts[2], 2 );
    #  3^112:.Suz2/ ,20 :found classes

<#18317 =119java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

<iv="p"<---</>
Weiv class="p"></div>
two character tables in the last fusion record.

<div class="p"></div>

<pre>
    gap&#;nams= RecNames head);
    [ ", "SizesCentralizers"
      "fusions" ]
    gap> ForAll( Difference( nams, [ "fusions" ] ),
    >            nam -> head.( nam ) = head2.( nam ) );
    true
    gap> Length( head.fusions );
    By additionalarguments wefindthat the<tt7</t>centralizerhasthe
    gap> ForAll( [ 1 .. 4 ], i -> head.
    true
    gap> head.fusions[5].map = head2.fusions[5].map;
    true

</pre>

<div class="p"></div>
We continue
The question which of the two above candidate tables belongs to a<div cl class""<-->/iv>
of <font size="+0">M</font> will be answered in Section <a href="#sect:natcharM">6</a>.

<div class="p"></div>

     <h3><a name="tth_sEc4.5prejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
4.5</a>  Classes
/

<=>
The group 2.<font size="+0">B</font> contains two rational conjugacy classes of java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 67
oforder ,
with different values in the permutation character (1<sub>2.<font size="+0">B</font></sub>)<sup><font size="+0">M<//re>

<div class="p"></div>

<pre>
    gap> s:= CharacterTable( "
    gap> pos:= Positions(     gap> ExtendTableHeadByCentralizerOrder head, u, 7^5 * Factorial7,[43 ] )java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
    [23,25 ]
    gap> pi{ pos };
    [ 1539000, 7875 ]

</pre>

<div ""</div>
This establishes two classes <tt>5A</tt>, <tt>5B</tt> of conjugacy of <font="0>M</font>java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
of  of 5in< ="0"M/ont,
with centralizer orders 5 |<span class="roman">HN</span>&# style='color: green'>124; and 5<sup>7</sup> |2.<span class="roman">J</span><sub>2</sub>|, respectively.

<div class="p"></div>

<pre>
    gap> cents:= List( pos,
    >      i -> pi[i] * Size( s ) / SizesConjugacyClasses( s )[i] );
    [ 1365154560000000, 94500000000 ]
    gap> cents = [ 5 * Size( CharacterTable( "HN" ) ),
    >              5^7 * Size( CharacterTable( "2.J2" ) ) ];
    true

</pre>

<divclass=""</div>
By [<a href="#Mverify" name="java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 28
we know that <font size="+0">M<
of order 5,
<tt>5A</tt> with centralizer 5 ×<span
(D<sub<re>
and <tt>5B</tt> with centralizer 5<gap#2 := ((
5<sup>1+6</sup><sub>+</sub>.4.<java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 44

<div class="p"></div>
The two classes aredivjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
becausethis );java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

<div java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0

<pre>
    gap> s:= CharacterTable( "MN5A" );
    CharacterTable( "(D10xHN).2" )
    gap&62; ClassPositionsOfPCore  5 )
    [ 1, 45 ]
    </a> nTpower  font="0">/>/<nameMpowermaps>
    #I  after (D10xHN).2: found 5 classes, now have 170

</pre>

<div class="p"></div>
The character table of 5<sup>1+6</sup><sub>+</sub>.4.<
with <font face="helvetica">MAGMA</font>, see Section <a href="#sect:table_N5B">10</a>,
thus we are allowed to use the character table from
the font face"elvetica"font character library

<divclass

<pre>
        #      java.lang.StringIndexOutOfBoundsException: Range [42, 39) out of bounds for length 44
    CharacterTable(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> 5core:= ClassPositionsOfPCore( s, 5 );
    [ 1 . 4 ]
    gap> SizesConjugacyClasses( s ){ 5core };
    [,4 37800 40320]
    gap> ExtendTableHeadByRootClasses( head, s, 2 );
    #I  after 5^(1+6):2.J2.4: found 3tial fusion.

</pre>

<div class="p"></div>

     <h3<pre>
4.6</a>  Classes of 5</a>  The power maps of nbsp; Thepowermapsof<size"0>/>/2< =java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 51
</a>

<div class="p"></div>
The group 2.<font size="+0">B</font> contains a rational class of elements of order 11.
The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></sup>)<sub>2.<font size="+0">B</font></sub
of order 11 with centralizer order 11 |M<sub>12</sub>&#e='color: green'>124; in <font size="+0">M</font>.

< element ofclass

<re
    gap#;s=CharacterTable("." );
    gap> pos:= Positions( OrdersClassRepresentatives( s ), 11 );
    [ 71 ]

</pre>

<div class="p"></div>
By the arguments in [<a href="#Mverify" name="CITEMverify">BMW24</a>],
<font size="+0">M</font> has no other classes of element order 11.

<div class="p"></div>

<pre>
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos );
    #I  after order 11 element from 2.B: have 174 classes

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc4.7">
4.7</a>  Classes of elements of the orders 17, 19, 23, 31, 47</h3><a name="elements_17">
</a>

<div class="p"></div>
The elements of the orders 17, 19, 23, 31, 47 in <font size="+0">M</font> lie#;if r.map[iin fus[]then
Sylow subgroups that First we initialize the class fusions  withthe definitions of

<div class=  classes as by thepartialwhichwehave.
The elements of order 17 and 19 are rational in 2.<font size="+0">B</font>
and hence  in font size="0>M</font.

<div class="p"></div>

<pre>
    gap> s:= CharacterTable( "2.B" );;
    gap#62;pos:=Positions( OrdersClassRepresentatives( s ), 17 );
    [ 118 ]
    gap> ExtendTableHeadByPermCharValue( head, s, pi&#le='color: green'>62;       ;
    #I  after order 17 element from 2.B: have 175 classes
    gap> pos:= Positions( java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 27
    [ 128 ]
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos );
have 176  classes

<

<div class="p"></div>
For elements g of order p  ∈ { 23, 31, 47 },
java.lang.StringIndexOutOfBoundsException: Range [40, 3) out of bounds for length 95
the nonidentity powers of g,
[>2<=0></</ub(#001g&9002;:<2.font=+0>B<subg  (p−1/
The equation |C<sub><font size="+0">M</font></sub>(g)&#e='color: green'>124; = |2.<font size="+0">B</font>&#le='color: green'>124; ·π(g) / |g<sup><font size="+0"    #62              Error("usion ")java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
implies

<
<table align="   119
|N<sub><font size="+    gap>: [;
/><java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 6

of elementsof until no further
fuse in <font size="+0">M</font> or not;
in the former case,
we have [N<sub><font size="+0">M</font></sub
|g<sup><font size="+0">M</font></sup> &g&62 java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 27
>:
[N<sub><font size="+0">M</#62;      for p in primes
&# &#2 for insafe_fusions
Thus we can >       : (r,p;
nd we can  findarguments thetwo62 res= TransferDiagram,.fus  )
do not fuse.

<div class="p"      start#             nam&;(am)l;62 then
First we deal with p = 23.

< class=p">-->/>

<pre>
    gap> s:= CharacterTable( "2.B" );
    CharacterTable( "2.B" )
    gap> ord:= OrdersClassRepresentativesI  round
    gap> classes:= SizesConjugacyClasses( s );;
    gap> p:= 23;;
    gap> pos:= Positions( ord, p );
    [ 147, 149 ]
    gap> n:= (p-1)/2 * Size( s ) * piLet us where power maps
    6072
    gap> Collected( Factors( n ) );
    [[ ,3 ] [3, 1 ] [ 11,1 ] [ 23,1]]

</pre>

<div class="p"></div>
In order to prove that the two classes<startingwith the 5-th power map.
do not fuse in <font size="+0">M</font>, it suffices to show that the centralizer order is
divisible by 2<sup>3</sup>.
We see<re

<div ="p></div>

<pre>
    gap> u:= CharacterTable( "MN2B" );
    CharacterTable( "2^1+24.Co1" )
    gap> upos:= Positions( OrdersClassRepresentatives( u ), p );
    [ 289, 294 ]
    gap> SizesCentralizers( u ){ upos } / 2^3;
    [ 23, 23 ]

</pre>

<div are java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
ThusFor of elementorders, is pairof  conjugate

<div class="p"></div>

<>
    gap> java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 28
    #I  after order 23 element from 2.B: have 177 classes
    #62;(head ,pi pos{2  )
    #Iafter  23 elementfrom 2.B: ave178classes

</pre>

<div class=" of   l - power
The case p = 31 is done analogously.
Herethenecessary 2part of the java.lang.StringIndexOutOfBoundsException: Range [22, 44) out of bounds for length 24

<div   decide java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 49

<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    gap62p31;
    gap> pos:= Positions( OrdersClassRepresentatives( s ), p );
    [ 190, 192 ]
    gap> n:= (p-1)/2 * Size( s ) * pi[ pos[1] ] / classes[ pos[1] ];
    2790
    gap> Collected( Factors( n ) );
      ,1]  ,2],  5, ,[ ,1]]
    gap> SizesCentralizers( s ){ pos };
    2 ]
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] }
    ent from 2B:have 179
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } );
    #  after  31  from 2B: have 180

</pre>

<div class="p"></div>
Finally, we deal with p = 47.

<div class="p"></div>

<pre>
    gap> p:= 47;;
    gap> pos:= Positions( OrdersClassRepresentatives( s ), p );
     java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    gap> n:= (p ,,311 ,313 ,315 ,( )]
    2162
    gap62;Collected Factors(n) );
    [ [ 2, 1 ], [ 23, 1 ], [ 47, 1 ] ]
    gap> SizesCentralizers( s ){ pos };
     94 94]
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] } );
    #I  after order 47 element from 2
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } );
    #I  after order 47 element from 2.B: have 182 java.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 14

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc4.8">
4.8</a>  Classes of elements of order 13</h3><a name="elements_13">
</a>

<;java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 70
The class <tt>13A</tt> of <font size="+0">M&62;         Gcdl,p 1
of order java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
We use the permutation character to enter the information about
the class <tt>13A</tt>.

<div class="p"></div>

<pre>
    gap> p:= 13;;
    gap> pos:= Positions( OrdersClassRepresentatives( s ), p );
    [ 97 ]
    gap> c:= Size( s ) * pi[ pos[1] ] / classes[ are determined uniquely except for the images of two classes
    73008
    &62;Factorsc)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    [ 2, 2, 2, 2, 3, 3, 3, 13, 13 ]
    gap> ExtendTableHeadByPermCharValue
    #I  after order 13 element from 2.B: have 183 java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 5

</pre>

<div class="p"></div>
The class <tt>13B    gap> istFiltered(head.,
Herewe just know the centralizer order 13<sup>3</sup> ·2<sup>3</sup> ·3.

< class="p"</div>

<pre
    gap&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> pos:= Positions( OrdersClassRepresentatives( c2b ), 13 );
    [ 220 ]
    gapExtendTableHeadByCentralizerOrder, c2b,13^  24  )
    #I  after order 13 element from 2^1+24.Co1: have 184 classes

</pre>

<div class="p"></div>

     <h3<aname=tth_sEc4.9">
4.9</a>  Classes of elements of order divisible by 29</h3><a name="elements_29">
</a>

< class="">!-->/iv>
The group 3.<span class="roman">Fi</span><sub>24</sub> contains a rational class of elements of order 29,
with centralizer order 3 ·29.

<div class="p"></div>

<pre>
    gap> u:= CharacterTable( "3.Fi24" );;
    [37, 132,133  ]
    [ 142 ]
    gap> SizesCentralizers( u ){ pos }java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 87 ]

</pre>

<div class="p"></div>
The list of classes of <font size="+0">M</font> collected up to now+"
elements of the orders 2, 3, 5, 11, 13, 17, 19, 23, 31, 47,
and 29 occurs as a factor of the centralizer order only for the
classes <tt>1A</tt>, <tt>3A</tt>, <tt>87A</tt>, and <tt>87B</tt>.
62( s 7 [342;
<div class="p"></div>

<pre>
    gap> poss:= PositionsProperty( head.SizesCentralizers,
    >                              x -> x mod 29 = 0 );
    [ 1, 135, 144, 145 ]
    gap> head.OrdersClassRepresentatives{ poss };
    [ 1, 3, 87, 87 ]
    gap> head.SizesCentralizers{ poss };
    [ 808017424794512875886459904961710757005754368000000000,
      3765617127571985163878400, 87, 87 ]

</pre>

<div class="p"></div>
Thusonly possiblejava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 73
in <font size="+0">M</font> of an element x of order 29 are 7, 41, 59, and 71.

<div class="p"></div>

<pre>
    gap> candprimes:= Difference( PrimeDivisors( head.Size ),
    >                      2, 3, 5,11,13,17,19,23, 29 31 47]);
    [ 7, 41, 59, 71 ]

</pre>

<div    #;         do
The centralizer order&62         Gcd l p ) =1 then
3 ·29 ·7<sup>i</sup> &#le='color: green'>183&62          if ( val p ) then
with  0  ≤ i  ≤ 6 and j, k, l  &#n style='color: green'>8712; { 0, 1 }.

<div class="p"></div>

<pre>
    gap> parts:= Filtered( Collected( Factors( head.Size ) ),
    >                      x -> x[1] in candprimes );
    [ [ 7, 6 ], [ 41, 1 ], [ 59, 1 ], [ 71, 1 ] ]
    &62 :=List(, l -#62;List  0. [2],  -&62 l[]i));
    gap> cart:= Cartesian( poss );;
    gap> possord:= 3 * 29 * List( cart, Product );;

</pre>

<java.lang.StringIndexOutOfBoundsException: Range [27, 4) out of bounds for length 28
Only 3 b we the sameresultfor candidates)
that is, |<font size="+0">M</font>| / |N<sub><font size="+0">M</font></sub>(〈x 〉)|  &#n style='color: green'>8801; 1  mod 29.
Note that we have [N<sub><font size="of element

<class">

<pre>
    gap> good:= Filtered( possord,
    >                     x ) )java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
    [ 87, 5133 ]
    gap> List( good, Factors );
    [ [ 3, 29 ], [ 3, 29, 59 ] ]

</pre>

<div class="p"></div>
we the   3&18329&83;59
by the fact that the Sylow 59 subgroup would be normal
and thus would be normalized and hence centralized by java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 0
order 3, acontradiction.

<div class="p"></div>

<pre>
2;Filtered(DivisorsInt( 5133  ) x -&#62 x mod59= 1 )
    [ 1 ]

</pre>

<div class="p"></div>
Thus haveestablished a rational class of elements of order 29,
with centralizer of order 3 ·29.

<div class="p"></div>

<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, u, 3 * 29, pos );
    #I  after order 29 element from 3.F3+.2: have 185 classes

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc4.10">
4.      ,,, 1,,,,1, , ,, , 1,, java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
/

<div class
that java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 126
The fact that an element x of order 41 in <font size="+0">M<    [ 37,,132,133  ]
by element  order 4 can readoff  the group
O<sub>8</sub><sup>−</sup>(3).

<div class="p"></div>

<pre>
    &;t=-->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    CharacterTable( "O8-(First we initializethe1+4.1"java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    gap> Length( Positions( OrdersClassRepresentatives( t ), 41 ) );
    10

</pre>

<div class="p"></div>
By the above arguments, the only possible odd prime divisors of
|N<sub><font size="+0">M</font></sub>(〈x 〉)|/41 are 5, 7, 59, 71,
where 5 cannot divide the centralizer order.
As in the case of p = 29, we apply Sylow's theorem,
andget
&    #62              ;

<div class="p"></div>

    gap> possord:= 2^2 * 41 * DivisorsInt( 2&62;         ;
    gap2&62
    >                     >  noti .i
    [ 1640, 163016 ]
    gap> List( good, Factors );
    [ [2, 2 2 5,41]     #;       ;

</pre>

<div class="p"></div>
Suppose that 71 divides |N<sub><font size="+0">M</java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 28
Then the 71 Sylow subgroup of N<sub><font size="+0">M</font></sub>(〈x 〉) is normal
thus normalized by an element of order 8,
and thus centralized by an involution, a contradiction.

<div class="p"></div>

<g&#;Position(poss[]map 163)
    gap> Filtered( DivisorsInt( good[2] ), x -> x mod 71 = 1 );
     java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

</pre>

<div class="p"></div>
Thus we have established a rational
of order 41.

<div class="p"></div>

<pre>
    gap&62;java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 6
    #I  after order 41 element: have 186 classes

</pre>

<div class="p"></div>

     <h3><a  java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 54
.>&n ofjava.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 91
</a>

<div"></div>
By the above arguments,
the normalizer order of an element of order 59 divides
58 ·7<sup>6</java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
Sylow's theorem admits just the normalizer order 59 ·29.

<iv=""</div>

<>
    gap> possord:= 59 * DivisorsInt( 58*7^6*71 );;
    gap> good:= Filtered( possord,
    >                     x -> ( head.Size / x ) mod 59 = 1 );
    [ 1711 ]
    gap> List( good, Factors );
    [ [ 29, 59 ] ]

</pre>

<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
Thus we have established a pair of Galois conjugate classes of
self-centralizing elements of order 59.

<div class="p"></div>

<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
    #I  after order 59 element: have 187 classes
    gap> ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
    #I  after order 59 element: have 188 classes

</pre>

<div class="p"></div>

     h3aname="tth_sEc4.">
4.12</a>  Classes of elementsgap&#; s2=   >         [p{poss }=possjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
</a>

<div class="p"></div>
By  arguments,java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
the normalizer#Error("inconsistency ) &62      java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
70 &#    &62;      fi
Sylow's theorem admits just the     &62;   od;

<ass=p></div>

<pre>
    gap> possord:= 71 * ,,,, 1,,,,,,,, 1,,,, 1,, 1,,,,1,,1, 1,,1,1,,1 java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
    gap> good:= Filtered( possord,
    #;                     -&62  .Size/ x) mod 71 =1 ;
    [ 2485 ]
    ap#; List(oodFactors)java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    [ [ 5, 7, 71 ] ]

</pre>

<div class="p"></div>
Thus we have established a pair of Galois conjugate classes of
self-centralizing elements of order 71.

<div class="p"></div>

<pre>
    gapFirst  initializethe classfusionfrom thefirstcandidate ...
    #I  after order 71 element: have 189 classes
    gap&62 ExtendTableHeadByCentralizerOrder( head,71, 71,fail);
    #I  after order 71 element: have 190 classes

</pre>

<div class="p"></div>

     <h3<a name"th_sEc4.13>
4.13</a> #;        ( i
/

<div class="p"></div>
The. =0>/java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 75
with centralizer order 7 ·|<span class="roman">He</java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 49

<iv=p>

<pre>
    gap> s:= CharacterTable( "2.B" );;
    gap&62;pos=Positions(OrdersClassRepresentatives s) 7 )java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
    [ 41 ]
    gap&#62
    #I  after order 7 element from 2.B: have 191 classes
    gap> Last( head.SizesCentralizers ) = 7 * Size( CharacterTable( "He" ) );
    true

</pre>

<div class="p"></div>
By additional arguments, we find that the <tt>7A</tt> centralizer has the
structure 7 ×<span class="roman">He</span>,
and the normalizer has the structure (7:3 ×<span class="roman">He</span>).2,
a subdirect product of 7:6 and <span class="roman">He</span>.2.

< classp"</div>
Since <span class="roman">He</span> has a pair java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
getpair Galois Since also the cubes of the concerned  elementorder 39
7 ·17 = 119termined,  questionis decided that

<&  iin 1 . ( r2 )]do

<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
ses
    gap> ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
    #I  after order 119 element: have 193 classes

</pre>

<iv class=p"</div>
The second class of elements of order 7, <tt>7B</tt>,
is established by the fact that the subgroup 3.<span class="roman">Fi</span><sub>24</sub> contains
two classes of elements of order 7, with different values of the
degreee 196 883 character χ of <font size="+0">M</font>.

<div class="p"></div>

<pre>
    gap> u:= CharacterTable(  &#;        fi
    gap> cand:= Filtered( Irr( u ), x -&#       fus2[i: r2.map[i];
    gap> rest:= Sum( cand{ [ 1, 4, 5, 7, 8 ] } );;
    gap> pos:= Positions( OrdersClassRepresentatives( u ), 7 );
    [ 41, 43 ]
    62;rest pos};
    [ 50, 1 ]

</pre>

<div class="p"></div>
Note that class 43 fuses to <tt>7B</tt> because the restriction of χ
to 2.<font size="+0">B</font> has the value 50 on <tt>7A</tt>.

< =p></div>

<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, u, 7^5 * Factorial(7), [ 43 ] );
    #I  after order 7 element from 3.F3+.2: have 194 classes

<pre>

<div class="p"></div>
e"0"M>java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96

<div class="p"></div>

<pre>
    gap> Sum( head.SizesCentralizers, x -&#62                        powermaps3]  l;#;fail
    true

</pre>

<div class="p"></div>
We initialize the character table head of <font size="+0">M</font>.

<div class="p"></div>

<pre>
    gap#62 :=ConvertToCharacterTableNC rec(
    >      UnderlyingCharacteristic:= 0,
    >      Size:= head.Size,
    >      SizesCentralizers:= head.SizesCentralizers,
    >      OrdersClassRepresentatives:= head.OrdersClassRepresentatives,
    >    ) );;

</pre>

<div class="p"></div>

<2><a name
5</a>  The power java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 5
</a>

<class"></div>
Using the element orders of the class representatives of the table head
f <font size="+0">M</font>, and the partial class fusions from the subgroups used in the
previous sections, we compute approximations of the p-th power maps,
for primes p up to the maximal element order in <font size="+0">M</font>.of  order 78.

<div class="pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Note that we have not yet determined which of the two possible character
tables of the <tt>3B</tt> normalizer belongs to a subgroup of <font size="+0">M</font>,
thus we exclude java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 5

<div class="</re>

<pre>
        gap&#62 pos78:= ( factsjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    >        -&62 not  subtable,[] ;
    gap> Length( safe_fusions );
    6

</pre>

<div class="p"></div>
irst we java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 5
s by the partial fusions which we have stored.

<div class="p"></div>

<pre>
    gap> for r in safe_fusions do
    >      fus:= InitFusion( r.subtable, m );
    >      for i in [ 1 .. Length( r.map ) ] do
    >        if ( r.mapi] ) then
    >          if IsInt( fus[i] ) then
    &java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    >              Error( "fusion problem<prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    >            fi;
    >          ;          elif IsInt( r.map[i] ) <divclass"p"</div>
Next we initialize approximations of the power maps of the table of <font size="+0<iv class=""><re>>

<pre>
    &62 : Maximum(head.OrdersClassRepresentatives );
    119
    gap&62;:=[];;
    gap> primes:= Filtered( [ 1 .. maxorder  gap>head.izesCentralizers]
    java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 8
    >      powermapsdiv ="p"<-->/>
    >      for r java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 6
    >        subpowermap:= PowerMap( r.subtable, p );
    >        if TransferDiagram( subpowermap, r.fus, powermaps[p] ) = fail then
    >          Error( "inconsistencygap> powermaps3{[163,,164]}java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    >        fi  ,184 ,[183, 184  ]
    >      od;
    >    od;

</pre>

<div class="p"></div>
We repeat applying the compatibility     [ ,16
java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 75

<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    gap> found:= true;;
    gap> res:= "dummy";;  # avoid a syntax warning
    gap> while found do
    >      Print( "#I  start a round\n" );
    >      found:= false;
    &62       inprimes do
    >        for r in safe_fusions do
    >          subpowermap:= PowerMap( r.subtable, p );
    >          res:= TransferDiagram( subpowermap, r.fus, powermaps[p] );
    62          if  =fail then
    >            Error( "inconsistency" );
    >          elif ForAny( RecNames( res ), nam -> res.( nam ) <> [] ) then
    >            found:= true;
    >          fi;
    >        od;
    >      od;
    >    od;
    #I  start a round
    #I  start a round
    #I  start a round
    #I  startCharacterTable 3(+):Suz2/ ,20 ")

</pre>

d java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
Let us see where the power maps are still not determined uniquely,
starting with the 5-th power map.

<div class="p"></div>

<pre>
    gap> pos:= PositionsProperty( powermaps[5], IsList );
    [ 157, 158, 163, 164, 187, 188, 189, 190, 192, 193 ]
    gap> head.OrdersClassRepresentatives{ pos };
    [15,15 39 39 59,59 71,71 ,119]

</haveapplying consistency .

<div class="p"></div>
The ambiguities for the classes of the element orders 59, 71, and 119
are understandable:
For each of these element orders, there is a pair of Galois conjugate
classes, and the subgroups whose class fusions we have used do not contain
these elements.

<div class="p"></div>
For each of the primes l  ∈ { 59, 71 },
the field of l-th roots of gap> poss:= Filtered( head
which is <font size="+0">Q</font>(√{−l}),
and the p-th power map, for p coprime to l,
java.lang.StringIndexOutOfBoundsException: Range [35, 5) out of bounds for length 47
the Galois automorphism that raises l-th roots of unity to the p-th power
fixes √{−l}.

#;java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 35
In the case of element order l = 119 = 7 ·17,
the field of l-th roots of unity contains the three quadratic subfields,
<   gap&62  [1&62 =java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 91
In order to decide which of them actually occurs,
we order 119.
The <tt>7A</tt> centralizer in <font size="+0">M</font> has the structure 7 ×<span class="roman">He</span>,
and the normalizer has the structure (7:3 ×<span class="roman">He</span>).2,
a subdirect product of 7:6 and <span class="roman">He</span>.2,
see Section <a href="#elements_7">4.13</a>.

<div class="p"></div>
The classes of element order 119 in the normalizer correspond to
the classes of this element order in <font size="+0">M</font>,
and the character values in the subgroup lie in the field <font size="+0">Q</font>(√{−119}).

<div class="p"></divto the class <tt>2B<tt.

<pre>
    gap> u:= CharacterTable( "(7:3xHe):2" );;
    gap> ConstructionInfoCharacterTable( u );
    [ "ConstructIndexTwoSubdirectProduct", "7:3", "7:6", "of <ontsize=+"M/ont>onthe classes  2< ="0><f,by &bsp;ahref
      [ 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129,
          130, 131, 132, 133, 134, 135, 207, 208, 209, 210, 211, 212,
          213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224,
          225, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307,
          308, 309, 310, 311, 312, 313, 314, 315 ], (), () ]
    gap&62 pos: Positions(OrdersClassRepresentatives  ) 119 ;
    [ 52, 53 ]
    gap#;f=Field(Rationals ( Irr u ,x -#62 x[os[]  );;
    gap> Sqrt(-119) in f;
    true

</pre>

<div class="">!-->/>
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

<div class="p"></div>

<gap&622{[118,120  :  177  ;java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
    gap> for l in [ 59, 71, 119 ] do
    >      val:= Sqrt( -l );
    >      poss:= Positions
    >      for p in primes do
    >        if Gcd( l, p ) = 1 then
    >          if GaloisCyc( val, p ) = val d ""!--<
    >            powermaps[p]{ poss }:= poss;
    >          else
    >            powermaps[p]{ poss }:= Reversed( poss );
    #;          ;
    >        fi;
    >      odod;
    >    od;

</pre>

<div class="p"></div>

are determined uniquely except for the images of two classes
of element order 39.
These classes had been found as roots of <tt>3B</tt> elements

<div class="p"></div>

<re
    gap> PositionsProperty( powermaps[17], IsList );
    [ 163, 164 ]
    gap> head.{ [ 163, 164 ] }java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
     39, 39
    gap> List( Filtered( head.fusions,
    >                    r -> IsSubset( r.map, [ 163, 164 ] ) ),
    >          r -> r.subtable );
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ) ]

</pre>

java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
In order to decide whether the p-th power map fixes or swaps
the two classes, we consider elements of order 78,
which are the square roots of the    #;powermaps2]52]
There are three classes of element order 78 in <font154
a rational class that powers to <tt>2A</tt>
and a pair of Galois conjugate classes that power to <tt>2B</tt>.

<div class="p"></div>

<pre>
    gap> 78pos:= Positions( head.<pre>
    [ 37, 132, 133 ]
    java.lang.StringIndexOutOfBoundsException: Range [37, 7) out of bounds for length 38
    CharacterTable( "2.B"     >                     SizesCentralizers( s )[i] = 3888 );
    gap> Intersection( 78pos, head.fusions[1
    [ 37 ]
    gap> s:= head.fusions[2].subtable;
    CharacterTable( "2^1+24.Co1" )
    gap> Intersection( 78pos, head.fusions[2].map );
    [ 132, 133 ]
    gapPositions( head.fusions[2].map, 132 );
    [342 ]
    gap> Positions( head.fusions[2].map, 133 );
    [ 344 ]
    gap> PowerMap( s, 7 )[342];
    344

</pre>

<div class="p"></div>
Since the <tt>3B</tt> normalizer in <font <re>
classes of element order     gap> r:= head.fusions[2];;
subgroup of order 3,
these two classes fuse to the non-java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 34
of order 78,
and their squares are the classes of element order 39 we are interested in.

<div class="p"></div>

<pre>
    gap&#62
    gap> List( poss, r -> r.subtable );    gap> rest:= List( knownpos, i -> java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 5
     ( 3^(+)6..2/ ,19 ])]
    &62;Position poss[1].ap 163)java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    173
    gap> Position( poss[1].map, 164 );
    174
    gap> List( facts, s -> Positions( OrdersClassRepresentatives( s ), 39 ) );
    [ [ 173, 174 ], [ 173, 174 ] ]
    gap> List( facts, s -> PowerMap( s, 7 )[173] );
    [ 174, 174 ]

</pre>

<gap&62 r2:=First .fusions  -#62;IsIdenticalObj rsubtable,facts[] ));
The field of character values on the two classes of <font size="+0">M</font> is equal
to the corresponding field of character values in the <tt>3B</tt> normalizer,
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 7
(Note that we have not yet decided which of the two candidate tables belong
to the <tt>3B</tt> normalizer,
but we get the same result for both candidates.)

<div class="p"></div>

<pre>
    62;=List
    >                   s -> Field( Rationals, List( Irr( s ),
    >                                                x -> x[173] ) ) );;
    gap> Length( Set( fields ) );
    1
    gap> Sqrt(-39) in fields[1];gap&62 O }
    true

</pre>

<div class="p"></div>
Now we can set the

<divclass=""><--->/div>

<pre>
    &;val: Sqrt -9  );
    gap> poss:= [ 163, 164 ];;
    gap> for p in primes do
    >      if Gcd( 39, p ) = 1 then
            if GaloisCyc( val, p ) = val then
     &62          powermaps[p] poss }= poss;
    >        else
    >          powermaps[p]{ poss }:= Reversed( poss );
    >        fi;
    >      fi;
    >    od;
    gap> List( powermaps, Indeterminateness );
    [ , 2048, 1536,, 4,, 2,,,, 2,, 9,,,, 1,, 1,,,, 1,,,,,, 1,, 1
      ,, 1,, 1,,,, 1,,,,,, 1,,,,,, 1,, 1,,,,,, 1,,,, 1,, 1,,,,,, 1,,,, 1,,
      ,,,, 1,,,,,,,, 1,,,, 1,, 1,,,, 1,, 1,,,, 1 ]

</pre&62        congr:= List divs,-#;chi[powermapsp]i  mod  );

<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 39
In the following,
we use the two candidates for the <tt>3B</tt> normalizer table
for answering most of the remaining questions about the power maps.
Again, the answers are equal for both candidate tables.

<div class="p"></div>
First we initialize the class fusion from the first candidate table ...

<div class="p"></div>

<pre>
    gap> r:= First( head&62      fijava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    gap> fus:= InitFusion( r.subtable, m );;
    gap> for i in [ 1 .. Length( r.map ) ] do
    >      if IsBound( r.map[i] ) then
62if i
    >          if fus[i] <> r.map[i] then
    >            Error( "fusion problem" );
    >          fi;
    >        elif IsInt( r.map[i] ) then
    >          if not r.map[i] in fus[i] then
    >            Error( "fusion problem" );
    &#62#|| ,|_(|=:java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 50
    >        else
     IsSubset(i,r.[i
    >            Error( "fusion problem" );
    >          fi;
    >        fi;
    >        fus[i]:= r.map[i];
    >      fi;
    >    od;
    gap> r.fus:= fus;;

</pre>

T java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 61
... and the class fusion from the second candidate table, ...

<div class="p"></div>

<pre&62    ;
    gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 27
    gap> fus2:= InitFusion( r2.subtable, m );;
    gap> for i in [ 1 .. Length( r2.map ) ] do
    >      if IsBound( r2.map[i] ) then
    >        if IsInt( fus2[i] ) then
    >          if fus2[i] <> r2.map[i] then
    >            Error( "fusion problem" );
    >          fi;
    >        elif IsInt( r2.map[i] ) then
    >          if not r2.map[i] in fus2[i] then
    >            Error( "fusion problem" );
    >          fi;
    >        else
    >          if not IsSubset( fus2[i], r2.map[i] ) then
    >            Error( "fusion problem" );
    >Now wedecide whichof the candidates for the charactertableof
    >        fi;
    >        fus2[i]:= r2.map[i];
    &62
    >    od;
    gap> r2.fus:= fus2;;

</pre>

<div class="p"></div>
. we  an  copy  the current approximations
of power maps, and apply the consistency conditions for class fusion and
power maps in the two cases.

<div class="p"></div>

<pre>
    gap> powermaps2:= StructuralCopy( powermaps );;
    gap> s:= r.subtable;
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" )
    gap> for p in primes do
    >      if TransferDiagram( PowerMap( s, p ), fus, powermaps[    #62 (  ;
    >        Error( "inconsistency" );
    >      fi;
    >    od;
     : java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 30
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
    gap> rest::=sol*cand;
    >      gap&62; i  1.  java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 47
    >        Error( "inconsistency" );
    >      fi;
    >    od;
    gap> powermaps = powermaps2;
    true
    gap> List( powermaps, Indeterminateness );
    [ , 32, 64,, 1,, 1,,,, 1,, 1,,,, 1,, 1,,,, 1,,,,,, 1,, 1,,,,,, 1,,,,
      1,, 1,,,, 1,,,,,, 1,,,,,, 1,, 1,,,,,, 1,,,, 1,, 1,,,,,, 1,,,, 1,,gap&62;map: r2.us;;
      , 1,,,,,,,, 1,,Which values are still missing?

</pre>

d =p>--/>
One open question is about the squares of the non-rational classes
of element order 78.

<div class="p"></div>

<pre>
    gap&#62 powermaps[{ [132,133] };
    [ [ 163, 164 ], [ 163, 164 ] ]
    gap> pos78:= Listgap> head.SizesCentralizers{ missing }
    >                  s -> Positions( OrdersClassRepresentatives( s ), 78 ) );
    [[235, 236] 235 236]]
    gap> fus{ [ 235, 236 ] };
    [ [ 132, 133 ], [ 132, 133 ] ]
    gap> fus2{ [ 235, 236 ] };
    [ [ 132,133],[ 132 133 ]]]

</pre>

<div class="p"></div>
We may identify one class of element order 78 in the <tt>3B</tt> java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 26
with the<pre>

<div class="p"></div>

<pre>
    gap&62; fus[235]:= 132;;
    gap> fus2[235]:= 132;the  character table that is stored in the <font face="helvetica">GAP</font> character table library.
    gap> TransferDiagram( PowerMap( s, 2 ), fus, powermapsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    true
    gap> TransferDiagram( PowerMap( s2, 2 ), fus2, powermaps2[2] ) <> fail;
    true
    gap> powermaps = powermaps2;
    true
    gap#;List powermaps [2 3 ],Indeterminateness )
    [ 8, 64 ]

</

<div class="p"></div>

<re
    gap> powermaps[3]{ [ 163, 164 ] };
    [ [ 183, 184 ], [ 183, 184 ] ]
    gap> TransferDiagram( powermaps[2], powermaps[3], powermaps[2] ) <> fail;
    true
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 16 ]

</pre>

<ivclass=p>!---><div>
The next open question is about the cubes of elements of order 93.
The classes of element order 93 -a pair of Galois conjugate classes-
have been found inside subgroups
of the type S<sub>3</sub> ×<span class="roman">Th</span>, and they do not occur in java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 35
we have considered.
Thus we may choose which of them cubes to the first class of element order
31.

<div class="    #I  |g| =C_M() =119:value 118modulo 119

<re>
    gap> poss:= Filtered( head.fusions,
    >                     r -> 93 in OrdersClassRepresentatives( r.subtable ) );;
    gap> List( poss, r -> r.subtable );
    [ CharacterTable( "ThxSym(3)" ) ]
    gap> pos93:= Positions( head.OrdersClassRepresentatives, 93 );
    [ 152, 153 ]
    gap> powermaps[3]{ pos93 };
    [ [ 179, 180 ], [ 179, 180 ] ]
    gap> powermaps[3][152]:= 179;;
    java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 75
    >                     powermaps[3] ) <> fail;
    true
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 4 ]

</pre>

<div class="p"></divd"<-->/>
The next open  we decide which java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 68
The classes of element order 69 -a pair of Galois conjugate classes-
have been found inside subgroups
."java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 98
we have considered.
Thus we may choose which of them cubes to the first class of element order
23.

<div class="p"></div>gap&62 r: ( head.fusions r -#62;IsIdenticalObj.ubtable,[1  ;java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89

<
    gap> poss:= Filtered( head.fusions ap&62 rest:= ( ,i &62 [ map[]]);
    >                     r -> 69 in OrdersClassRepresentatives( r.subtable ) );;
    gap> List( poss, r -> r.subtable );
    [ CharacterTable(gap&62 Length )
    gap> pos69:= Positions( head.OrdersClassRepresentatives, 69 );
    [ 142, 143 ]
    gap> powermaps[3]{ pos69 };
    [ [ 177, 178 ], [ 177, 178 ] ]
    gap> powermaps[3]{ [ 142, 143 ] }:= [ 177, 178 ];;
,
    >                     powermaps[3] ) <> fail;
    true
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 1 ]

</pre>

<div class="p"></div>
The next open question is about the squares of certain elements of order 46.
There are two pairs of Galois conjugate classes of element order 46,
and the 2nd power map is not yet determined for those classes which power
to the class <tt>2B</tt>.

<div class="p"></div>

gap#62;cand:=  (r2. , -#;x] <=chi[1 ;java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
    gap> pos46:= Positions( head.OrdersClassRepresentatives, 46 );
    [ 26, 27, 118, 120 ]
    gap> powermaps[2]{ pos46 };
    [ 177, 178, [ 177, 178 ], [ 177, 178 ] ]
    gap> powermaps[23]{ pos46 };
    [ 2, 2, 44, 44 ]

</pre>

<div class="p"">/div>>
We have defined the two classes of element order 23 as squares of those
two classes of element order 46 that power to <tt>2A</tt>,
and we have not yet distinguished the other two classes of element order 46.
Thus we may set the power map values.

<div class="p"></div>
Now just one value is left to The character table of the second candidate is equivalent to
the square of a class of element order 18 and centralizer order 3888.

<div class="p"></div>

<pre>
    gap> powermaps[2][78];
     155, 156 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    gap> head.OrdersClassRepresentatives[78];
    18
    gap> head.SizesCentralizers[78];
    3888

</pre>

<div class="p"></div>
There are two classes java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
both are roots of the generators of the normal subgroup of order 3 in
3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2, and the corresponding two classes in this subgroup
have the same square.

<div class="p"></div>

<pre>
    gap> Filtered( [ 1 .. Length( head.OrdersClassRepresentatives ) ],
    >              i -> head.OrdersClassRepresentatives[i] = 18 and
    >                   head.SizesCentralizers[i] = 3888 );
    [ 78, 79 ]
    gap> powermaps[3]{ [ 78, 79 ] };
    [52, 52 ]
    gap> powermaps[2][52];
    154
    gap> First( head.fusions, r -> 154 in r.map ).subtable;
    44
    gap> s:= facts[1];
    CharacterTable( "3^(1+12):6.Suz.2/[ 1,     s2=r2.subtable;
    gap> pos18:= Filtered( [ 1 .. NrConjugacyClasses( sjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    >                i -> OrdersClassRepresentatives( s )[i] = 18 and
    >                     gap> ForAll( Irr( s2 ), x -> IsInt( x[273] ) );
    [ 67, 83 ]
    gap> PowerMap( s, 2 ){ pos18 };
    [ 24, 24 ]
    gap> s:= facts[2];
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
    gap> pos18:= Filtered( [ 1 .. NrConjugacyClasses( s ) ],
    >                i -> OrdersClassRepresentatives( s )[i] = 18 and
    >                     SizesCentralizers( s )[i] = 3888 );
    [ 67, 83 ]
    gap> PowerMap( s, 2 ){ pos18 };
    [ 24, 24 ]

</pre>

<div class="p"></div>
We set the last missing value,
and improve the approximations of the class fusions we have used,
by applying the consistency criteria.

<div class="p"></div>

<pre>
    gap> powermaps[2][78]:= powermaps[2][79];;
    gap> for r in safe_fusions do
    &
    >                                  powermaps ) then
    >        Error( "inconsistent!" );
    >      fi;
    >    od;
    gap> r:= First( head.fusions, r -> IsIdenticalObj( r.subtable, factsand our currentknowledgedoes not allowus distinguish
    gap> TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
    >                         powermaps );
    true
    gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.subtable, facts[2] ) );;
    gap> TestConsistencyMaps( ComputedPowerMaps( r2.subtable ), r2.fus,
    >                         powermaps );
    true
    gap> SetComputedPowerMaps( m, powermaps );

</pre>

<div class="p"></div>

<h2><a name="tth_sEc6">
6</a>  The degree 196 883 character χ of <font size="+0">M</font></h2><a name="sect:natcharM">
<a>

<div class="p"></div>
We know the values of the irreducible degree 196 883 character χ
of <font size="+0">M</font> on the classes of 2.<font size="+0"

<div class="p"></div>
Thus we
that are known as images of the class fusion from 2.<font size="+0">B</font>.
This yields 111 out of the 194 character values.

<div class="p"></div>

<pre>
    gap> chi:= [];;
    gap> map:= r.fus;;
    gap> for i Now we can induce the irreducibles of 2.<font size="+0">B</font> to <font size="+0">M</font<">!>d>
    >      if IsInt( map[i] ) then
    >        chi[ map[i] ]:= rest[i];
    >      fi;
    >    od;
    gap> Number( chi );
    111

</pre>

<div div classp"!--<div
Also the restriction of χ to 3.<span class="roman">Fi</span><sub>24</sub> is known,
Section&bsp< "natural">2</a>.
This yields 29 more character values.

d java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    gap> r:= head.fusions[3];;
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 28
    CharacterTable( "3.F3+.2" )
    gap> cand:= Filtered( Irr( s ), x -> x[1] <= 196883 );;
    gap> rest:= Sum( cand{ [ 1, 4, 5, 7, 8 ] } );;
    gap> rest[1];
    196883
    gap&62;map:=r.us;;
    gap> for i in [ 1 .. Length( map ) ] do
    >      if IsInt( map[i] ) then
    >        if IsBound( chi[ map[i] ] ) and chi[ map[i] ] <> rest[i] then
    >          Error( "inconsistency!" );
    >        fi;
    >        chi[ map[i] ]:= rest[i];
    >      fi;
    >    od;
    gap> Number( chi );
    140

</pre>

Now we compute the restriction of χ to the <tt>2B</tt> normalizer.
There are only 13 possible irreducible constituents of this restriction.
We of 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 that swaps exactly these two classes.
for which the class gap> (217,218) injava.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 52
the value of χ on the image class is known.
This matrix has full rank, thus we can directly compute the java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 5
of the restriction into irreducibles,
and get 41 more character values.

d java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<>
    gap> r:= head.fusions[2];;
    gap> s:= r.subtable;
    CharacterTable( "2^1+24.Co1" )
    gap> cand:= Filtered( Irr( s ), x -> x[1] <= chi[1] );;
    gap&java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    gap> knownpos:= Filtered( [ 1 ..     gap> ForAll( ( s2 ),x -#62;IsInt(x[19]))java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
    #;                      -#;IsInt i )andIsBound  [i ]  ;
    gap> rest178
    gap&62 mat: List(cand, x -&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> Length( mat );
    13
    gap> RankMat( mat );
    13
    gap> sol:= SolutionMat( mat, java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 28
    w wethe java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 45
    gap> rest:= sol * cand;;
    gap> for i in [ 1 .. Length( map ) ] do
    >      if IsInt( map[i] ) then chi[ map[i] ]:= rest[i]; fi;
    >    od;
    gap> Number( chi );
    181

</pre>

<div class="p"></div>
arejava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

<div class="p"></div>

<pre1
    gap> missing:= Filtered( [ 1..194 ], i -> not IsBound
  [151 152, 153 ,169 170,,187 ,,190,192,
    gap> head.OrdersClassRepresentatives{ missing };
    [ 57, 93, 93, 27, 95, 95, 41, 59, 59, 71, 71, 119, 119 ]
    gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 57, 93, 93, 243, 95, 95, 41, 59, 59, 71, 71, 119, 119 ]

</pre>

<div class="p"></div>
For g  ∈ <font size="+0">M</font>, we have χ(g)  ≡ χ(g<sup>p
nd&124&967()&124;sup>2/sup>  &; &#124C<><java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 0
We apply     gap> poss= PossibleClassFusions( s, ,
In all cases except one, the centralizer orders are small enough
for determining the character value uniquely.

<div class="p"></div>

<&62                (knownirr java.lang.StringIndexOutOfBoundsException: Range [72, 69) out of bounds for length 72
    gap> for i in >                 : f  ;
    >      ord:= gap> List( possjava.lang.StringIndexOutOfBoundsException: Range [44, 42) out of bounds for length 45
    >      divs:= PrimeDivisors( ord );
    >      if ForAll divs,p -#2pre>
    >        congr:= List( divs, p -> chi[ powermaps[p][i] ] mod p );
    >        res:= ChineseRem( divs, congr );
    >        modulus:= Lcm( divs );
    >        c:= head.SizesCentralizers[i];
    >        Print( "#I  |g| = ", head.OrdersClassRepresentatives[i],
    >               ", |C_M(g)| = ", c,
    >               ": value ", res, " modulo ",java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    >        if ( res + 2 * modulus )^2 >= c and 2<>+24<><>/>< "><span<>/sub> <sup>+12<sup><ub><sub>2<spanclass
    := Filtered( res +[ -1 . 1 ]*modulus,a -&62 a^ l;c )
    &#62becausetheir restrictionsto thesubgroups   conditions
    &
    >          fi;
    >        fi;
    >      fi;
    >    od;
    #I  |g| = 57, |C_M(g)| = 57: value 56 modulo 57
    #I  |g| = 93, |C_M(g)| = 93: value 92 modulo 93
    #I  |g|java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    #I  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    #I  |g| = 95, |C_M(g)| = 95: value 0 modulo 95
    #I  |g| = 95, |C_M(g)| = 95: value 0 modulo 95
    #I  |g| = 41, |C_M(g)| = 41: value 1 modulo 41
    #I  |g| = 59, |C_M(g)| = 59: value 0 modulo 59
    #I  |g| = 59 gap&#;=java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 45
    #I  |g| = 71, |C_M(g)| = 71: value 0 modulo 71
#    ,|M(|=:  71
    #I  |g| = 119, |C_M(g)| = 119: value 118 modulo 119
    #I  |g| = 119, |C_M(g)| = 119: value 118 modulo 119
    gap> missing:= Filtered( [ 1..194 ], i -> not IsBound( chi[i] ) );
    [ 160 ]

</pre>

java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 28
The one missing#kjava.lang.StringIndexOutOfBoundsException: Range [24, 21) out of bounds for length 61
with the trivial character.

<div class="p"></div>

<pre>
    gap> diff:= Difference( [ 1 .. NrConjugacyClasses( m ) ], missing );;
    gap> classes:= SizesConjugacyClasses( m );;
    gap> sum:= Sum( diff, i -> classes[i] * chi[319
    -650349175263480459970863415322722279882752000000000
    gap>  is spanned by>lllr<java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    2

</pre>

<div class="p"></div>
we decide whichthetwo  java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 68
the <tt>3B</tt> normalizer is the correct one.
firstjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 50
cannot be decomposed into irreducibles.

<div class="p"></div>

<pre>
    &62 : First(head.usions, r -> IsIdenticalObj( r.subtable, facts[1] ) );;
    gap> map:= r.fus;;
    gap> knownpos:= Filtered( [ 1 .. Length( map ) ], i -&#='color: green'>62; IsInt( map[i] ) );;
    ; chi[ i]];
    gap> cand:= Filtered( Irr( r.subtable ), x -> x[1] <= chi[1] );;
    gap> mat:= List( cand, x -> x{ knownpos } );;
    gap> Length( mat );
    #;     coeffs    gap&#2 Append induced,
    gap> RankMat( mat );
    88
    gap> SolutionMat( mat, rest );
    fail

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

<div class="p"></div>
The second candidate admits a decomposition.

<div class="p"></div>

<pre>
    gap> r2:= First( head2.fusions, java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 0
    gap> map:= r2.fus;;
    gap> knownpos:= Filtered( [ 1 .. Length( map ) ], i -&#='color: green'>62; IsInt( map[i] ) );;
    gap> rest:= List( knownpos, i -> chi[ map[i] ] )    gap&; java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 45
    62 java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 80
    gap> mat:= List( cand, x -> x{ knownpos }<iv class="p"></div>
    gap> Length( mat );
    95
    gap> RankMat( mat );
    88
    gap> SolutionMat( mat, rest );
    [ 0, 0, 0, 1, 0, 0, 0N wejava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 62
      0,0,,0 ,0,, 0,0 0 ,0 0,0, 0 ,0 0,0, 0, 0,0
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      ,,0 ,, ,
    gap> Add( safe_fusions, r2 );

</pre>

<div class="p"></div>
The character table of the second candidate is equivalent to
the   java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 97

<div class="p"></div>

<pre>
    gap> TransformingPermutationsCharacterTables( r2.subtable,
    >        CharacterTable( "MN3B" ) ) <> fail;
    java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8

</pre>

<div class="p"></div>

     &62;      extlen:= Length( extmat );
7</a>  The irreducible characters of <font size="+0">M</java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 55
<a>

<div class="p"></div>
We will not compute the irreducibles of <font size="+0">M</font> from scratch but verify the
irreducibles from the  A<font size="-2">TLAS</font> character table of <font size="+0">M</font>,
in the sense that we use java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
o.
For that,
we compute first a bijection between the columns of our character table head
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 91
This is done by using the following invariants:
element orders, centralizer orders,
the values

<div class="p"></div>

<pre>
    gap> invs:= TransposedMat( [
    >      OrdersClassRepresentatives( m ),
    >      SizesCentralizers( m ),
    &62      java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    >      CompositionMaps( chi, PowerMap( m, 2 ) ) ] );;
    gap> invs_set:= Set( invs );;
    gap> Length( invs_set );
    172
    gap> atlas_m:= CharacterTable( "M" );;
    gap> invs_atlas:= TransposedMat( [
    >      OrdersClassRepresentatives( atlas_m ),
    >      SizesCentralizers>a t>
    >      Irr( atlas_m )[2],
    >      CompositionMaps( Irr( atlas_m )[2], PowerMap( atlas_m, 2 ) ) ] );;
    gap> invs_atlas_set:= Set( invs_atlas );;
    gap> invs_atlas_set = invs_set;
    true

</pre>

<div class="p"></div>
In particular, we see that the sets of invariants are equal for the two
tables.

<div class="p"></div>
Note that we cannot get a better choice#2        :  rowtrans  {[1. java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 79
sincethere &;     fi
and our current knowledge does not allow us to distinguish
the classes of each pair.

<div class="p"></div>
Now we compute a permutation that maps the classes of the  A<font size="-2">TLAS</font> table
to suitable classes of our table head
permute the irreducibles of the  A<
and create the "oracle" list.

<div class="p"></div>
(The explicit permutation (32,33)(179,180) makes sure that the power maps
of the  A<font size="-2">TLAS</font> table and of our table are compatible.)

<--<

<pre>
    #;pi1 ();java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    gap> pi2:= SortingPerm( invs_atlas );;
    gap> pi:= pi2 / pi1 * (32,33)(179,180);;
    gap> oracle:= List( Irrjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</pre>

<div class="p"></div>
In order to prohibit that <font face="helvetica">GAP</font> tries to compute table automorphisms
of our
irreducible characters),
java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 40
<tt>AutomorphismsOfTablea matrix representation of H/X  ≅ 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>,
this will be revised as soon as the irreducibles are known.

<div class="p"></div>

<pre>
    gap> java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

</pre>

<div d class="p"></div>

<div class="p"></div>
First we compute candidates for the class fusion from 2.<font size="+0">B</font><re>
starting from the approximation we have already computed.
Before we apply <font face="helvetica">GAP</font>'s criteria for computing possible class fusions,
we decide about the images of four classes of element orders 40 and 44.

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28

<pre>
    #;r= [1];
    gap> s:= r.subtable;
    CharacterTable( "2.B" )
    gap> pos:= [ 217, 218, 222, 223 ];;
    gap> r.fus{ pos };
    [ [ 110, 111 ], [ 110, 111 ], [ 88, 89 ], [ 88, 89 ] ]
    &62 (  {pos }
    [ 40, 40, 44, 44 ]

</pre>

<div class="p"></div>
The java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 32
Note that their elements are roots java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 28
and the classes 110 and 111 correspond to non-rational elements
of order 40 in the <ttdiv class="

d =<--/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<pre>
    gap
    44
    gap> r2:= safe_fusions[2];;
    gap& &62;TransformingPermutationsCharacterTables,)<&62 java.lang.StringIndexOutOfBoundsException: Range [82, 83) out of bounds for length 82
    CharacterTable( "2^1+24.Co1" )
    gap> Position( r2.map, 110 );
    273
    &62;ForAll( rr( s2 ) -&62;;IsInt(273) )
    false

</pre>

<he tensor square ψ<sup>2</sup> belongs to the group H/E,
e java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 90
because there is a table automorphism of 2.<font size="+0">B</font> that swaps exactly these
.

<div class="p"></div>

<pre>
    gap> (217,218) in AutomorphismsOfTable( s );
    true
    gap> r.fus{ [ 217, 218 ] }:= [ 110, 111 ];;

</preHa  .

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28
With the same argument,
the  java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 72
both <div cl=><--<d>

<div class="p<iv class=""<-->/ivjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<   such that java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    gap> r.fus[ PowerMap( s, 22 )[ 222 ] ];
    44
    gap> Position( r2.map, 88 );
    178
    gap> div class="p"></div>
Now the remaining open questions java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 5
can be answered by <font face="helvetica">GAP</java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0

<div class="p"></div>

<pre>
    gap> knownirr:= [ TrivialCharacter( m ), chi ];;
    gap> poss:= PossibleClassFusions( s, m,
    >               rec( chars:= knownirr, fusionmap:= r.fus ) );;
    gap> List( poss, Indeterminateness )<li>
    [ 1l   First java.lang.StringIndexOutOfBoundsException: Range [58, 55) out of bounds for length 63

</pre>

<div class="p"></div>
Now we can induce the irreducibles of 2.<font size="+0">B</font> to <font size="+0">M</font>.

<div class="p"></div>

<pre>
    gap> induced:= InducedClassFunctionsByFusionMap( s, m, Irr( s ), java.lang.StringIndexOutOfBoundsException: Range [0, 77) out of bounds for length 30

</pre>

<div div classp!->/
Next we compute candidates for the class fusions from the subgroups
2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>, 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2,n  tojava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 71
Here we enter also the characters of <
because their restrictions to the subgroups provide additional conditions.

<div class="p"></div>
The fusion from 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><      java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Range [21, 2) out of bounds for length 22
We extend the list of known induced the extraspecial group N = 3<sup>1+12</sup><sujava.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 94

<div class="is   3java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 70

<pre>
    gap> poss:= PossibleClassFusions( s2, m,
    >               rec( chars:= Concatenation( knownirr, induced )java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 0
&62 fusionmap=java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
   &;   ;
    [ 1, 1, 1, 1 ]
    gap> Length( RepresentativesFusions( AutomorphismsOfTable( s2 ), poss,
    #62;               Group )  ))java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    1
    #;( java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    >        InducedClassFunctionsByFusionMap( s2, m, Irr( s2 ), poss[1] ) );

</pre>

<divclass=p>!-->/iv>
In order to compute the fusion from 3<sup>1+12</sup><sub>+</sub>.2.<<divclass=p">!-->/ivjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
we have to consider two classes of element order 56 first.
They are a pair of Galois conjugates both in 3<sup>1+12</sup><sub>+</sub>.2.<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
and we may freely choose their fusion because there is a table automorphism
of 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 that swaps exactly these two classes.

<div class="p"></div>

<pre>
    gap> r:= safe_fusions[7];;
    gap> s:= r.subtable;
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
    gap> pos:= Positions( OrdersClassRepresentatives( s ), 56 );
    [ 250java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gapThis implies that  are not more class splittings w. r. t. the
    [[125,126 ,[125  ] ]
    gap> r.fus[ PowerMap( s, 28 )[ 250 ] ];
    44
    gap> Position( r2.map, 125 );
    319
    ap#62 ( (s2 ,x &62 (x319]]))
    false
    #;(250 251 inAutomorphismsOfTable ;
    true
    gap> r.fus{ [ 250, 251 ] }:= [ 125, 126 ];;

</pre>

<div class="p"></div>
Now the class fusion to <font size="+0">M</font> is determined uniquely up to table    That is, we create a character table head for H together with
of the subgroup,
and we extend the list of induced characters.

<div class="p

<pre>
    gap> poss:= PossibleClassFusions( s, m,
    of degree 3<up></sup>
    >                    fusionmap:= r.fus ) );;
    gap> List(    we get one preimageclass in H.
    [ 1, 1 ]
    gap>    the    in  /X.
    <div class"p">!----</div>
    1
    gap> Append( induced,
&;java.lang.StringIndexOutOfBoundsException: Range [60, 49) out of bounds for length 79

</pre>

<div class="p"></div>
The fusion from 3.<span class="and  of degree 46
of the subgroup table.
We extend the list of known induced of degree 1 599 669 = 1 594 323 + 5 346,

<div class="p"></div>

<pre>
    gap> r:= safe_fusions[3];;
    gap> s:= r.subtable;
    CharacterTable( "3.F3+.2" )
    gap> poss:= PossibleClassFusions( s, m,
    >               rec( chars:=    of the character table of H.
    >                    fusionmap:= r.fus ) );;
    gap> List( poss, Indeterminateness );
    [ 1 ]
    gap> Append( induced,
    &62;InducedClassFunctionsByFusionMap( , Irr(s,poss] ) )

</pre

<div class="p"></div>
Next we induce the irreducible characters of cyclic subgroups.

<div class="p"></div>

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 5
    gap> Append( induced,
    >      InducedCyclic( m, [ 2 .. NrConjugacyClasses( m ) ], "all" ) );

</pre>

<div class="p">     <3><a name"tth_sEc9..1">
Now we reduce the induced characters with the two known irreducibles of <font size="+0">M</java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 71
and apply the LLL algorithm to ion described in Section <a href="#computeHtable">9.4</a>
this yields four new irreducibles.

<div class="p"></div>

<pre>
    gap> red:= Reduced( m, knownirr, induced In order to describe a reproducible construction of the character table
    gap> Length( red.irreducibles );
    0
    gap> lll:= LLL( m, red.remainders );;
    gap> Length( lll.irreducibles );
    4

</pre>

<div class="p"></div>
We extend the list of known irreducibles,
reduce the induced characters,
andthe extraspecial group N = 3<sup>1+12</sup><sub>+</sub> by 6.<span class="roman">Suz</span>.2,

<div class="pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<<iv class="p"></div>
    gap> knownirr:= Union( knownirr, lllogap   3(
    gap> red:= Reduced( m, knownirr, induced );;
    gap>  />Picture Omitted
    0
    gap> lll:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> Length( lll.irreducibles );
    0

</pre>

<div class="p"></div>
Now we use the irreducibles of the  A<font size="-2Then M =X&N anormal subgroupX of order 3java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
as follows.
Whenevercharacter  oracle  belongs   f size"0><font>-lattice
that is spanned by <tt>lll.remainders</tt>
then we regard this character as verified,
since we can compute the coefficients of the <font size="+0">Z</font>-linear combination,
form the character, and check that it has indeed norm 1.

<div class="p"></div>

p>
    gap> mat:= MatScalarProducts( m,  java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 58
    gap> norm:= NormalFormIntMat( mat, -which java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 88
    gap> rowtrans:= norm.rowtrans;;
    gap> normal:= norm.normal{ [ 1 .. norm.rank ] };;
    gap> one:= IdentityMat( NrConjugacyClasses( m ) );;
    #62  i in   .java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 28
    >      extmat:= Concatenation( java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 42
    #62;      extlen: Length(  )java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    62 extnorm=NormalFormIntMat extmat, ;
    >      if extnorm.rank = Length( extnorm.normal ) or
    >         extnorm.rowtrans[ extlen ][ extlen ] <> 1 java.lang.StringIndexOutOfBoundsException: Range [0, 71) out of bounds for length 28
&62;;        coeffs:=failjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    >      else
    >        <div class="p"></div>
We take the generators of the <font size="+0">Z</font>-lattice and some symmetrizations
of the known irreducibles,
reuce them with the known irreducibles,
and apply LLL again.

<div class="p"></div>

<pre>
    gap> red:= Reduced( m, knownirr, lll.remainders );;
    gap> Length( red.irreducibles );
    0
    gap> sym:= Symmetrizations( m, knownirr, 2 );;
    gap> sym:= Reduced( m, knownirr, sym );;
    gap> Length( sym.irreducibles );
    0
    gap
    gap> Length( lll.irreducibles );
    0

<>
6
<div class="p"></div>
Weuse above  again,for  <ont size=+0><font>ljava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76

<    gap&62; orb_large:=First orbs,x-&>62;;   x );

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 5
    gap> mat:= MatScalarProducts( m, oracle, lll.remainders );;
    gap>Length( ) 3^;
    gap> rowtrans:= norm.rowtrans;;
    gap> normal:= norm.normal{ [ 1 .. norm.rank ] };;
    gap&    gap> orb_large: SortedList java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 50
    gap> for i in [ 2 .. Length( one ) ] do
    >      extmat:= Concatenation( normal, [ one[i] ] );
    >      extlen:= Length( extmat );
    & 859230155080499200
    >      if extnorm.rank = Length( extnorm.normal ) or
    >         extnorm.rowtrans[ extlen ][not java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 49
    >        coeffs:= fail;
    >      else
    >        coeffs:= - extnorm.rowtrans[ extlen ]{ [ 1 .. extnorm.rank ] }
    >                   * rowtrans{ [ 1 .. extnorm.rank ] };
    >      fi;
    >      if coeffs <> fail and java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 28
    >        Add( knownirr, coeffs * lll.remainders );
    >      fi;
    >    od;
    gap> Length( knownirr );
    194

</pre>

<div class="p"></div>
Noware .
As stated in thea faithful representationofH,
we unbind the stored trivial value for <tt>as a group of 38 ×38 matrices over the field with three elements.

<">!--><d>

<pre>
    e matrices are blockdiagonal matrices  blocksof the lengths
    gap> ResetFilterObj( m, HasAutomorphismsOfTable );
    gap> java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 28
    true

</pre>

<div class="p"></div>

<h2>div class="p"></div>
8</a>  Appendix: The character table of 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub></h2><a name="sect:table_c2b">
</a>

<div class="p"></div>
TheFor that, we&62                         &;ForAll  1 .  ,
has the structure 2<sup>1+24</sup&#2                                       &62  [j  and
which   constructedas follows.

<div class="p"></div>
Consider a subdirect product H of two groups H/X
and </pre
where X is a cyclic group
N is an extraspecial group 2<sup
H/N is the double cover of <span class="roman">Co</span><sub>1</sub>, and
H/X is an extension of 2<sup>1+24</sup><sub>+</sub> by <span class="roman">Co</span><sub>1</sub>.

<div class="p"></div>
The centre of H is a Klein four group E whose order two subgroups
are X, Y = Z(N), and a third subgroup D.
We have C  ≅ H/D.

<div class="p"></div>

<div class="p"></div>

<center>
<a href="ctblm1.png">Figure</a>

b java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
</center>

<div class="p"></div>
From the 2-local construction of a matrix representation for <font size="+0">M</font>,
we know the following faithful representations
given -/

<div class="p"></div>

<ul>
<li>   A monomial permutation     gap> Comm( mats[3], mats[3]&#2 ([3],mats[]mats[]) =Inverse [4])
   of degree 98&<iv class=">!--<java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<div class="p"></div>
</li>

<li>
   a matrix standard generators of the factor group 2.<span class="roman">Suz</span>.2,
   of dimension 4 096 over the field is sufficient to form the diagonalproduct of our representation of
<div class="p"></div>
/>

<li>
   a matrix representation of
   of dimension 24 over the field with three elements.
<div class="p"></div>
</li>
<>

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28

<""!--><div
We proceed in the following steps.

<div class="p"></div>

<ul>
<li>   First we compute conjugacy class representatives of H/E.
<div class="p"></div>
<   6

<li>
   A faithful permutation representation of H/The first  of  generators are   of6.s
   2 ·196 560 = 393 120 points is obtained by
   glueing the permutation generators of H/E
   (n 2&183;98&bsp;80=196&;560 )
   & java.lang.StringIndexOutOfBoundsException: Range [25, 22) out of bounds for length 57
   (the smallest permutation representation of 2.<span class="roman">Co</span><sub>1</sub>,
   on 196 560 points) together.

<div class="p"></div>
   The character table of this permutation group can be computed
   directly with <font face="helvetica">MAGMA</font> in about 16 hours of CPU time.
d p>-->/iv>
</li>

<li>
   Starting from the character table of the factor group H/E,
    3-odular  matrixrepresentationofdimension&096of H/isused
   to compute necessary class splittings from H/E to H/X, as follows.

<div class="p"></div>
   This representation lifts to characteristic zero because its restriction
   to the extraspecial group M/X
   is the unique faithful irreducible 3-modular representation of M/X,
   becausethis representation extends to the full automorphism group
   of the extraspecial group.
   Thus we can compute the Brauer character values of java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 0
   on the 3-regular classes of H/X,
   and interpret the values as those of an ordinary character ψ, say.
   The tensor square ψ<sup>2</sup> belongs to the group H/E,
   andthe known valuesof &968; sufficeto determine decomposition
   of ψ<sup>2</sup> into irreducibles of H/ENow #2;gensHmodX   m-&#an style='color: green'>2;homHtoHmodX;;
   and thus to compute also the values of ψ<sup>2</sup> on 3-singular classes.
   Taking square roots, we get all values of ψ, up to signs.
   (We cannot distinguish which of the two values belongs to which of the two
   preimage classes, which just means that we are defining these classes
   by choosing the positive value for one of them,&62 java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 28
   for the other one.)
div=">-/>
</li>

<li>
   From now on, we argue character-theoretically.

<div class="p"></div>
   missing  characters H are computed as
   tensor products of ψ with the irreducible charactersand then show that elementssatisfy the java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 28
   the factor group H/M  ≅ <span class="roman">Co</span><sub>1<

<div class="p"></div>
   Note that this procedure yields enough new irreducible characters
   such that the sum of degree squares of all known irreducibles of H/X
group.
   This implies that there are not more class splittings w. r. t. the
   fusion from H/X to H/E than the splittings forced by the values
   of ψ.
<div class="p"></div>
</li>

<li>
   two  class splittingsfromH/and H/ to H/,
   we compute es:( prog,gensHmodX);
   That is, we create a character table head for H together with
    fusionsto H/ and H/,
   assuming that not more columns occur than is forced by H/X and H/Y:
   For each class of H/E that splits in both H/X and H/Y,
   we get four preimage classes in H.
   For each class that splits in exactly one of H/X and H/Y,
   we get two preimage classes in H    gap&#62 Order( gensHmodX2 ;
   For each class that splits in none of H/X and H/Y,
   we get one preimage class in H.

<div class="p"></div>
   Thenwe take thoseirreducible charactersof H/X and H/Y,
   respectively, that do not have E/X or E/Y, respectively,
   in their kernel;
   we form tensor products of them, which yields characters with kernel D,
   nd  the LLL algorithm to them.

<div class="p"></div>
    yields allmissingirreducibles of :
   The degree squares of all now known irreducibles sum up to the order
   of H, which means that no more class splitting occurs.

<div class="p"></div>
   Finally,we compute the power maps (and thus the element orders)
   of the character table of H.
        tablethat permutation
   thetable  isstoredin< ""GAP/>s tablejava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
<div class="p"></div>
</li>
</l

<div class="p"></div>

<h2><a name="tth_sEc9">
9</a>  Appendix: The character table of 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2</h2><a name="sect:norm3B">
</a>

<div class="p"></div>

     <h3><a name="tth_sEc9gap&&#62 gensH:    .  ,  -#62 [i *shiftedgensi ;
9java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>
This character table was usedin Sectionnbsp;href"elements_5>5<a.
Its character table has been computed by Richard Barraclough and
Robert A. Wilson, see [<a href="#BW07"<iv class""<--><div>
In order to describe a reproducible construction of the character table
that does not assume the character table of <font size="+0">M</font>,
we recompute this table.

<div class="p"></div>

<div class="p"></div>
Our approach is similar to that in [<a href="#BW07" name dt><href""BCP97<>dt<>
question  a  group ofthe split
the extraspecial group N = 3<sup>1+12</sup><sub>+</sub> by 6.<span class="roman">Suz</span>.2,
and we<div =p"!--<div>

<div class="p"></div>

<div class="p"></div>

<center> <a href="ctblm2.png">Figure</a>

<br />Picture Omitted<br />
</center>

<div class="p"><<div class="p"></div>
We have H / N  ≅ 6.<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Let M be the normal subgroup of H above N such that
H / M  ≅ 2.<span class="roman">Suz</span>.2 holds.
Then M = X ×N for a normal subgroup X of order 3,
where the extension of M / X by H / M is split.
Let N<sub>2</sub> be the normal subgroup of H above N such that
H / N<sub>2</sub>  ≅ 3.<span class="roman">Suz</span>.2 holds.
Then M<sub>2</sub> = M N<sub>2</sub> has the property H / M<sub>2</sub>  ≅ <span class="roman">Suz</span>.2.

<div class="p"></div>
Let Y =
Then E = X ×  111-130 MR 4135540
with diagonal normal subgroups D<sub>1</sub> and D<sub>2</sub>.
It<re>
are non-split.

<div class="p"></div>
H / Y is a subdirect product of H / N and
H / E  ≅ 3<sup>12</sup>.2.<span class="roman">Suz</span>.2,
we have H / Y  ≅ (3 ×3<sup>12</sup>).2.<span class="roman">Suz<<td> &an style='color: green'>2;;     C(3(+)6Suz java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

<div class="p"></div>
The group <pre>

d =""<-->/>
The <tt>3B</tt> normalizer in <font size="+0">M</font> is isomorphic to
one of the two factor groups H / D<sub>1</sub>, H / D<sub>2</sub>.
(The decision which of the two groups occurs as a subgroup of <font size="+0">M</font>
appears in Section <a href="#sect:natcharM">6</a>.)

<div class="p"></div>
We use the following approach to compute the character table of H.

<div class="p"></div>

<ul>
<li>    Compute permutation generators <tt>gensHmodX</tt> of H / X,
    of degree 3<sup>13</sup>,
    see Section <a href="#HmodXpermgens">9.2</a>.
    The first two generators are standard generators of 2.<span class="roman">Suz</span>.2,
    the third generator lies in M / X.
<div class="p"></div>
</li>

<li>
    Fetch permutation generators <tt>gensHmodN</tt> of H / N<sub>2</sub>,
    of degree 5 346,
    and compute permutation generators <tt>gensH</tt> of H,
    of degree 1 599 669 = 1 594 323 + 5 346,
    see
    The first two generators are standard generators of 6.<2><a name="tth_sEc10>
    the third generator lies in N.
dclass=p">!--<java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
</li>

<li>
    Let <font face="helvetica">MAGMA</font> compute the character table of H from <tt>gensH</tt>,
;"">.<>
<div class="p"></div>
</li>
</ul>

<div class="p"></div>
<Remark<bjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
In an earlier construction (in September 2020),
<><    ofH/,
java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 78
from H to H / Y and then determining the full character table head of H.
The computation of the missing irreducible characters of H was then
not difficult, using character theoretic methods.
trying the  input files in 2024showedmany
some of the required conjugacy tests did not finish in reasonable R.nbsp;.java.lang.StringIndexOutOfBoundsException: Range [104, 105) out of bounds for length 104
Luckily, the automatic computation described in Section <a href="#computeHtable">9.4</a>
worked.

<div class="p"></div>

     h3>aname=tth_sEc9.2>
9.2</a>  A&62;fi;
</a>

<div class="p"></div>
The  A<font size="-2">TLAS</font
a faithful representation of H,
as a group of 38 ×38 matrices T<sub>font size="-1">E</font></sub>X
The generating matrices are called <tt>M3max7G0-f3r38B0.m1</tt>, ...,
<tt>M3max7G0-f3r38B0.m4</tt>.
These matrices are block diagonal matrices with blocks of the lengths
24 and 14.

<div class="p"></div>

<pre>
    gap> info:= OneAtlasGeneratingSetInfo( "3^(1+12):6.Suz.2", Dimension, 38 );;
    gap> gens:= AtlasGenerators( info ).generators;;
    gap> Length( gens );
    4
    gap> ForAll( gens,
    >            m -> ForAll( [ 25 .. 38 ],
    >                         i -> ForAll( [ 1 .. 24 ]
    >                                      j -> IsZero( m[i,j] ) and
    #62;                                          IsZero m[i,j] ) ) ))
    true

</pre>

<div class="p"></div>
Here3682583
which generate the factor group H / X  ≅ 3<sup>1+12</sup><sub>+</sub>:2.<span class="roman">Suz</span>.2.
(Note that the 12-dimensional irreducible representation of 2.<span class="roman">Suz</span>.2
over <dt><ahref"CITEBMO17 name=BMO17[]/a/<java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Range [89, 3) out of bounds for length 96
-which   p. 21.3682588
yields a construction of the semidirect product 3<sup>1+12</sup><sub>+</sub>:2.<span class="roman">Suz</span>.2
as a group of 14 ×14 matrices.)

<div class="p"></div>
Laterwe will construct a faithful permutation representation of H
as a subdirect product of H / X and H / N  ≅ 6.<span class="roman">Suz</span>.2.
Let <tt>G</tt> be the group generated by the 14 ×14 matrices.

<div class="p"></div>
When one deals with H / X, the fourth generator is redundant,
we will leave it out in the following.

<div class="p"></div>

<pre>
    =  ,x -62;x{[25.. 38  { [25.   });
    gap> List( mats, ConvertToMatrixRep );;
    gap> Comm( mats[3], mats[3]^mats[2] ) = Inverse( mats[4] );
    true
    gap> mats:= mats{ [ 1 .. 3 ] };;

</pre>

<div""<---</div>
We use just the following facts.
The <tt>G</tt>-action on GF(3)<sup>14</sup> has six orbits,
from which we get a faithful permutation representation of H / E  ≅ 3<sup>12</sup>.2.<span class="roman">Suz</span>.2 on 196 560 points
and a faithful representation <tt>homHtoHmodX</tt>
of a group of the structure 3<sup>1+12</up><><sub..<span =r"Suz</span>2  1&bsp;594&bsp;23 .
Since the image contains a subgroup 2.<span class="roman">Suz</span>.2,
the image is the split extension H / X of 3<sup>1+12</sup><sub>+</sub> by 2.<span class="roman">Suz</span>.2.

<div class="<br /><table =l" border"0">tr>td width="50"java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56

<div class="p"></div>

<pre>
    tt>><t><java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    gap> orbs:= ShallowCopy( OrbitsDomain( G, GF(3)^14 ) );;
    gap> Length( orbs );
    6
    gap&62;SortBy( orbs, Length );
    gap> List( orbs, Length );
    [ 1, 2, 196560, 1397760, 1594323, 1594323 ]
    gap v:  0,0 ,0 ,0,0,,0,0 ,0,0,  ] (3^;java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
    gap> orb_large:= First( orbs, x -> v in x );;
    gap> Length( orb_large );
    1594323
    gap> Length( orb_large ) = 3^13;
    true
    gap> orb_large:= SortedList( orb_large );;
    gap> homHtoHmodX:= ActionHomomorphism( G, orb_large );;
    gap> represHmodX:= Image( homHtoHmodX );;
    gap> Size( represHmodX );
    2859230155080499200
e<java.lang.StringIndexOutOfBoundsException: Range [14, 9) out of bounds for length 76
    true

</pre><iv class=p"<---/div>

<div class="p"></div>
We conclude that the . L. Griess Jr., U, &java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 90
H / E.
As for the action on 1 594 323 points,
we show that it has a nonabelian normal subgroup of the order 3<sup>13</sup>.

<iv class="p"></div>

<div class="p"></div>

<pre>
    gap> gensHmodX:= List( mats, m -&#>62 S.java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 96
    gap n: represHmodX,
    <dl>
    gap> Size( n )
    true
    gap> IsAbelian( n );
    alse

</pre>

<div class="p"></div>

<div class="p"></div>
Next we show that the first two elements from the generating set
are standard generators of 2.<span class="roman">Suz</span>.2.
For that, we first show that these elements are preimages
of standard generators of <span class="roman">Suz</span>.2,
by computing that the words in question lie in the centre of 2.<span class="roman">Suz</span>.2,
and then show that the elements satisfy the conditions of
standard generators of 2.<span class="roman">Suz</span>.2,
that is, the second generator has order 3,
see the page on <span class="roman">Suz</span> in the  A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+</sup></a>].

<div class="p"></div>

<div class="p"></div>

<pre>
    gap> slp:= AtlasProgram( "Suz.2", "check" );;
    gap> prog:= StraightLineProgramFromStraightLineDecision( slp.program );;
    gap> res:= ResultOfStraightLineProgram( prog, gensHmodX );;
    gap> List( res, Order );
    [ 2, 1, 2, 1 ]
    gap> ForAll( gensHmodX{ [ 1, 2 ] }, x -> ForAll( res, y -> x*y = y*x ) );
    true
    gap> Order( gensHmodX[2] );
    3

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc9.3">
9.3</a>  A permutation representation of H</h3><a name="Hpermgens">
</a>

<div class="p"></div>
The group H is a subdirect product of H / X and H / N<sub>2</sub>  ≅ 3.<span class="roman">Suz</span>.2,
w.r.t. the common factor group H / M<sub>2</sub>  ≅ <span class="roman">Suz</span>.2.
Since we know that our generators for H / X are compatible with
standard generators of the factor group 2.<span class="roman">Suz</span>.2,
it is sufficient to form the diagonal product of our representation of
H / X and a representation of 3.<span class="roman">Suz</span>.2 on standard generators.

<div class="p"></div>

<pre>
    gap> 3suz2:= OneAtlasGeneratingSet( "3.Suz.2", NrMovedPoints, 5346 );;
    gap> 3suz2:= 3suz2.generators;;
    gap> omega:= [ 1 .. LargestMovedPoint( 3suz2 ) ];;
    gap> shifted:= omega + LargestMovedPoint( gensHmodX );;
    gap> pi:= MappingPermListList( omega, shifted );;
    gap> shiftedgens:= List( 3suz2, x -> x^pi );;
    gap> Append( shiftedgens, [ () ] );
    gap> gensH:= List( [ 1 .. 3 ], i -> gensHmodX[i] * shiftedgens[i] );;
    gap> NrMovedPoints( gensH );
    1599669

</pre>

<div class="p"></div>
The first two of the generators are standard generators of 6.<span class="roman">Suz</span>.2.
Note that we know already that they are preimages of standard generators
of <span class="roman">Suz</span>.2,
it remains to show that they are elements C, D where D has order 3
and CDCDD has order 7.

<div class="p"></div>

<div class="p"></div>

<pre>
    gap> Order( gensH[2] );
    3
    gap> Order( Product( gensH{ [ 1, 2, 1, 2, 2 ] } ) );
    7

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc9.4">
9.4</a>  Compute the character table of H</h3><a name="computeHtable">
</a>

<div class="p"></div>
Now we let <font face="helvetica">MAGMA</font> do the work.

<div class="p"></div>

<pre>
    gap> H:= GroupWithGenerators( gensH );;
    gap> if CTblLib.IsMagmaAvailable() then
    >      mgmt:= CharacterTableComputedByMagma( H, "H_Magma" );
    >    else
    >      mgmt:= CharacterTable( "3^(1+12):6.Suz.2" );
    >    fi;

</pre>

<div class="p"></div>
This computation needed about three weeks of CPU time.

<div class="p"></div>

<div class="p"></div>
The result verifies the character table of H
that is available in the library.

<div class="p"></div>

<pre>
    gap> IsRecord( TransformingPermutationsCharacterTables( mgmt,
    >        CharacterTable( "3^(1+12):6.Suz.2" ) ) );
    true

</pre>

<div class="p"></div>
This character table was used in Section <a href="#elements_3">4.4</a>.

<div class="p"></div>

<h2><a name="tth_sEc10">
10</a>  Appendix: The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2</h2><a name="sect:table_N5B">
</a>

<div class="p"></div>
The normalizer of a <tt>5B</tt> element in <font size="+0">M</font> has the structure
5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2,
generators for this group as a permutation group of degree 78 125
are available in the  A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+</sup></a>].
<font face="helvetica">MAGMA</font> [<a href="#Magma" name="CITEMagma">BCP97</a>] can compute the character table from the group
within a few minutes,
the result turns out to be equivalent to the table that is available in
<font face="helvetica">GAP</font>'s character table library.

<div class="p"></div>

<pre>
    gap> g:= AtlasGroup( "5^(1+6):2.J2.4" );;
    gap> if CTblLib.IsMagmaAvailable() then
    >      mgmt:= CharacterTableComputedByMagma( g, "MN5B_Magma" );
    >    else
    >      mgmt:= CharacterTable( "5^(1+6):2.J2.4" );
    >    fi;
    gap> IsRecord( TransformingPermutationsCharacterTables( mgmt,
    >        CharacterTable( "5^(1+6):2.J2.4" ) ) );
    true

</pre>

<div class="p"></div>
This character table was used in Section <a href="#elements_5">4.5</a>.

<div class="p"></div>

<h2>References</h2>

<dl compact="compact">
<dt><a href="#CITEMagma" name="Magma">[BCP97]</a></dt><dd>
W. Bosma, J. Cannon, and C. Playoust, <em>The Magma algebra system. I.
  The user language</em>, J. Symbolic Comput. <b>24</b> (1997),
  no. 3-4, 235-265. MR 1484478

<div class="p"></div>
</dd>
<dt><a href="#CITEAtlas2017" name="Atlas2017">[BGH<sup>+</sup>17]</a></dt><dd>
M. Bhargava, R. Guralnick, G. Hiss, K. Lux, and P. H. Tiep (eds.), <em>Finite
  simple groups: thirty years of the Atlas and beyond</em>, Contemporary
  Mathematics, vol. 694, Providence, RI, American Mathematical Society, 2017.
  MR 3682583

<div class="p"></div>
</dd>
<dt><a href="#CITEBMO17" name="BMO17">[BMO17]</a></dt><dd>
T. Breuer, G. Malle, and E. A. O'Brien, <em>Reliability and reproducibility
  of Atlas information</em>, in Bhargava et al. [<a href="#Atlas2017" name="CITEAtlas2017">BGH<sup>+</sup>17</a>],
  p. 21-31. MR 3682588

<div class="p"></div>
</dd>
<dt><a href="#CITEBMverify" name="BMverify">[BMW20]</a></dt><dd>
T. Breuer, K. Magaard, and R. A. Wilson, <em>Verification of the ordinary
  character table of the Baby Monster</em>, J. Algebra <b>561</b> (2020),
  111-130. MR 4135540

<div class="p"></div>
</dd>
<dt><a href="#CITEMverify" name="Mverify">[BMW24]</a></dt><dd>
<br /><table align="left" border="0"><tr><td width="50">
<hr />
</td></tr></table>
, <em>Verification of the conjugacy classes and ordinary character
  table of the Monster</em>, submitted, 2024.

<div class="p"></div>
</dd>
<dt>[]</dt><dd>T. Breuer, <em>Constructing the ordinary character tables of some Atlas
  groups using character theoretic methods.</em>, <a href="https://export.arxiv.org/abs/1604.00754">arXiv:1604.00754</a>.

<div class="p"></div>
</dd>
<dt><a href="#CITEctblpope" name="ctblpope">[Breb]</a></dt><dd>
<br /><table align="left" border="0"><tr><td width="50">
<hr />
</td></tr></table>
, <em>Permutation Characters in <font face="helvetica">GAP</font></em>, <a href="https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/doc2/manual.pdf"><tt>https://www.math.rwth<tt>-</tt>aachen.de/</tt>
<tt><tt>~</tt>Thomas.Breuer/</tt>
<tt>ctbllib/</tt><tt>doc2/</tt>
<tt>manual.pdf</tt></a>.

<div class="p"></div>
</dd>
<dt><a href="#CITEBW07" name="BW07">[BW07]</a></dt><dd>
R. W. Barraclough and R. A. Wilson, <em>The character table of a maximal
  subgroup of the Monster</em>, LMS J. Comput. Math. <b>10</b> (2007),
  161-175. MR 2308856

<div class="p"></div>
</dd>
<dt><a href="#CITEGAP" name="GAP">[GAP24]</a></dt><dd>
<em><font face="helvetica">GAP</font> - Groups, Algorithms, and Programming,
  Version 4.13.1</em>, <a href="https://www.gap-system.org"><tt>https://www.gap<tt>-</tt>system.org</tt></a>, Mar 2024.

<div class="p"></div>
</dd>
<dt><a href="#CITEGMS89" name="GMS89">[GMS89]</a></dt><dd>
R. L. Griess Jr., U. Meierfrankenfeld, and Y. Segev, <em>A uniqueness proof
  for the Monster</em>, Ann. of Math. (2) <b>130</b> (1989), no. 3,
  567-602. MR 1025167

<div class="p"></div>
</dd>
<dt><a href="#CITEAGRv3" name="AGRv3">[WWT<sup>+</sup>]</a></dt><dd>
R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. A. Parker, S. P. Norton,
  S. Nickerson, S. Linton, J. Bray, and R. Abbott, <em>ATLAS of Finite Group
  Representations</em>, <a href="http://brauer.maths.qmul.ac.uk/Atlas/v3"><tt>http://brauer.maths.qmul.ac.uk/</tt>
<tt>Atlas/v3</tt></a>.</dd>
</dl>

<div class="p"></div>

<div class="p"></div>

<br /><br /><hr /><small>File translated from
T<sub><font size="-1">E</font></sub>X
by <a href="http://hutchinson.belmont.ma.us/tth/">
T<sub><font size="-1">T</font></sub>H</a>,
version 3.59.<br />On 22 May 2025, 15:04.</small>
</html>

Messung V0.5 in Prozent

¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.160Angebot ¤

*Eine klare Vorstellung vom Zielzustand

Wurzel

Suchen

PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

Haftungshinweis

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

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.