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             <title>Some steps in the verification of the ordinary character table of the Monster group</title>
<h1 align="center">Some steps in the verification of the ordinary character table of the Monster group</h1>
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<h3 td .height}

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<h3 align="center">December 12th, 2024 </h3>

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We show thespanroman{ontfamily  stylenormal font-eight:normal}

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<h1>Contents </h1><a href="#tth_sEc1"
>1  Overview</a><br /><a href="#tth_sEc2"
>2  Some restrictions of the natural character span. {osition:relative  left 8em-.ex}
>3  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></a span.overacc1 {position:<style
>4  Theconjugacy classes of<font size="+0">M</font></a><br />    <a href="#tth_sEc4.1"
>4.1  Our strategy to describe the conjugacy classes of <font size="+0">M</font></a><br />    <a href="#tth_sEc4.2"
>4.2  Utility functions</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>.&bsp;Classes of elementseven order/a<br> nbsp &bsp<ahref"#tth_sEc4.4"
>4.4  Classes of elements of order divisible by 3</a><br />    <a href="#tth_sEc4.5"
>4.5  Classes of java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>4.=p>
>4.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>4.8&bsp Classes of elements of order 13</a><br />   &n>nbsp;<a href="#tth_sEc4.9"
>0"
>4.10 Classesof elements oforder divisible by 41</a><br />  &span>nbsp; <a href="#tth_sEc4.11"
>4.11  Classes of elements of order divisible by 59</a><br />  &span>nbsp; <a href="#tth_sEc4.4.&bsp;  strategy   the conjugacyclasses   =+"M</font>/>br/&bsp;&bsp;&;nbsp<href"tth_sEc42"
>4.12  Classes of elements of order divisible by 71</a>&bsp  character of s>+<sub<6."java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 160
>4.3n;Classes  elements of order divisibleby 7<a>br >a ="tth_sEc5"
>5  The power maps of <font size="+0">M</font></a><br /><a href="#tth_sEc6"
>&; 196; java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 106
>7  The irreducible characters of <font size="+0">M</font></a><br /><a href="#tth_sEc8"
>8  Appendix: The character table of 2<to the subgroups 2.<font size="0>/>and3<class="oman</<></><sup>&8242< 3.< =""Fi<span>sub>24/ub>,
>9  Appendix: The degree 196 883 character of <font size="+0">M</font>
>9.1  Overview</a><br />    <a java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
>9.2  A permutation representation of H / X</a><br />    <a href="#tth_sEc9.3"
>9.3  A of the subgroups2<font size=+"B/ont>, 2sup124<sup<+/ub><class="roman">Co</span>>/>3< ="java.lang.StringIndexOutOfBoundsException: Range [138, 137) out of bounds for length 166
thecharacter table of H<a<br ><a ="#th_sEc10"
>10n;Appendix: Thecharacter tableof 5<sup>16/>sub>+/..spanclass=roman"J/span><ub></sub>2/>br

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

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

<h2><a name
1<a &Overview/2>

<div class="gap#62;LoadPackage( "tbllib,false ;
The &bsp[a="#"name""BMW24/> isverify
of the Monster     >LoadPackage "tlasrep"  );
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
in the form of an explicit < ="">AGMA/ systemn;[a ="#agma"nameCITEMagma">CP97<a> will needed
<ont face=helvetica>GAP/font> [<a href="#GAP" name="CITEGAP">GAP24</a>] session protocol,
the relevant computations that are needed in that paper.

<div If  following returns<ttfalse/>
We then thesethese  will .

<iv class=""<-->/>
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
  character of degree 196 883 of <font size="+0">M</font>
tothe subgroups.font =+"><font>and 3<span class"></>sub24/><up&8242;/>( 3.spanclassroman>Fi/pan<><sub>,
as stated in [<a href="#Mverify" name="CITEMverify">BMW24

<div class="p"We set the linelength 72  in   testfiles.
Section;a href""3/ thedecompositionsjava.lang.StringIndexOutOfBoundsException: Range [70, 71) out of bounds for length 70
the transitive constituents of the permutation character
of the action of C<sub><font size="+0">M</font></subgap&62 (  72]);;
where a is a <tt>2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<
n;a href=#><a>and&bsp< href#">5/
2</a/ nbsp;ome restrictions java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 110
the lists of conjugacy class lengths, element orders, and power maps.

<div class="p"></div>
Section&<div class=""</div>
java.lang.StringIndexOutOfBoundsException: Range [29, 6) out of bounds for length 57
and decides the isomorphism type of the degree&;883 ofthe  groupfont size"0><font>

<div
this  and with(alreadyverified  tables
of the we computetherestriction #967 to 2.size=+"B<font>java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
3<sup>1+12</sup>< onlyfaithful 196nbsp;  of 2.fontsize=+"B/font>
computing  charactersof< ="0></  then ;
this corresponds 1a + 4371a + 96255a 4371a+96255a 96256 as claimed [< =# =""BMW24/>&bsp1.
and is done in Section <a href="#sect:irreduciblesM">7</a follows thefollowing dataabout 2.font size"0><font.

<div class
final < href":able_c2b"8/>,
<agap#;table2B: ( ".");
    gap&# cand: Filtered((  )  &#1 lt=196883 ;java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76

<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 73
We will use the <font face="helvetica">GAP</java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 21
and the interface to the   gap PrintArray( List( cand, x -&#tyle='color: green'>62; x{ Concatenation( [ 1 ], inv ) } ) );
thus we load these <font face="helvetica">GAP</font> packages    [ [       11       1,      1,      ,       1 ]java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63

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

<>
    gap> LoadPackage( "ctbllib", false );
    true/prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    gap> LoadPackage( "atlasrep", <div class="p"></div>
    true

<because it is the  faithful ,

<iv class="p"></div>
The <font face="helvetica">MAGMA</font> system [<a href="#Magma" name="CITEMagma">BCP97</a>] will be needed
for computing some character tables
and for many conjugacy tests.
If  following  returns<>false</tt>
then these steps will not work.

<div classThus the values of&967;<sub>2.fontsize="+0">B</font></sub> on the classes 4 and 5 differ by 4096.

<pre>
    gap&f 96255a would <bnot</b>thenthe of &967;sub>.font size=+0>B</font>/sub
    true

</pre>

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

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

<pre>
    gap> SizeScreen( [ 72 ] );;

</pre>

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

<h2><a name="tth_sEc2">
2</a>&bsp;&bsp;restrictions thenatural character of <font size="+0">M</font></h2><a name="natural">
</a>

<div class="p"></div>
Thus96255amustexactlyjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
of degree
<divclass="p"div>

<div class="p"></div>
First we  involutions  also 4371amustoccurjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44

<=""></div>
The only faithful 4371 and 275, respective.
that has at most two different values on involutions is
1a
This follows from the<div class="p"></div>

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

<>
    gap&#    gap#62;Sum( cand) inv };
    gap#62; cand=Filtered ( table2B ), x -#; x[1]<= 196883 );;
    gap> List( cand, x -> xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 1, 4371, 96255, 96256 ]
    gap> inv:= Positions( OrdersClassRepresentatives( table2B ), 2 );
      [2, 3,,5, 7 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    gap&62 ( List(cand,x&62 { Concatenation([ 1 ,inv)}))
    [[       ,       1       ,      ,1        ,
      [   4371    ,    -93,    275     ,     19],
      [  96255   96255,    4863,,    2047,2047     255]java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
      [   96256,  -96256,       0,    2048   ,0  java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64

<

< =p
Note that 96256gap62 (mat
because it is the only faithful candidate,
ly because only4371 +  #;
or 4371 ·1a + 2 ·96256a would be possible decompositions,
which have more than two values on involution classes     79      ,
Thus the values of χ<sub>2.<font size="+0">B</font></[64584,  72]

<div class="/>
If 96255a would <b>div class="p">
onthe  3 and   differ by 512 times the multiplicity of 4371,
but 65659 ·1a + 8 &#characters,that is the  constituents  1,8671a a,
Thus 96255a must

<div class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
he of 96255a 96256a has different values
n , hencealso4371 must .

<div class="p"></div>
We seejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
4371 275,.

<div and too large largefor the possible  275java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

<invol are   tothevalues first,
    gap&and    1a,
    [4371,4371 ,275,275]

</pre>

<div class="p"></div>
The restriction of χWe seefrom theratioof  valueonthe identity element
as follows.

<div class="p"></div>
Exactly seven irreducible characters of 3.<spanthat constituents of degree 57477 or 2 &#yle='color: green'>183;64584 exist.
of the restriction of &967;

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

<pre>
    gap> java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 10
    gap&62;cand=Filtered (table3Fi24prime  &62 []l;=196883 )java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
    gap> inv:= Positions[1., 24.7037, 49.6776,9.91139, 9.91139,, 47.7692,47.692]
    [ 4, 7 ]
    gap> mat:    gap#62;Float( (196883 -2* 64584)/(4371 -2 *1352 ;
    gap> PrintArray( mat );
    [ [      1,      1,      1 ],
      [   8671,   351,    33  ]java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
      [  57477,   1157,    133 ],
      [    783,     79,     15 ],
      [    783,     79,     15 ],
       1352      ,
      [  64584,   1352,     72 ] ]

</java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0

<div class="p"></div>
Since χ is rational, we need to consider only tuents of degree 57477
charactersjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
783ab, and 64584ab.

<divthe on the class of involutions be275,

<pre>
    gap><div class=p"</div>
    *E(3), 64584*E)^2 ]

</pre>

<div class="p"></div>
We see that     gap> ma>mat[3 +mat[6[7];
since all values of the possible constituents are positive
and too large for the other possible value 275.

<div class="p"></div>
Since the
involutions are  equaltothevalues the  class
and equal onlyThe second involutionclass forces oneconstituentdegree8671
we which the candidate can contribute negative),

< class=p></div>
We see from for the degrees ofits  is1+1566.
and on the first class of involutions
that constituents of degreeWe get the

< class=""</div>

<pre>
    #;Float 196883/4371 ;
    45.043
    List mat, v -#62; Float [  v[]))java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    [ 1.,     gap&62 Sum(mat )
    gap> Float( ( 196883 - 2 * 64584 ) / ( 4371 - 2 * 1352 ) );
    40.6209
    &;   196883-57477   ( 4371  1157 );
    43.3746
    gap> Float( ( from 3< class"></><ub>24</sub><sup>#;/> to 3<span class=r"</<24/>
    39.8294
    gap> In order to decompose the restriction of χ to 3.<span class="roman">Fi</ have  which from3span"oman"<<></subsup#8242<sup occur
    27.1689

</pre>

<div class="p"></div>
Firstclass"">
The above
must occur.

<div class="p"></div>#;:=Filtered((table3Fi24  &62 x1 l; 196883);
However, then the gap&62 := ( OrdersClassRepresentatives) 2 ;
hence value the  classof involutionscannot 275,
a contradiction.

<div class="p"></div>gap#;PrintArray );
hismeans botha and  occurwith multiplicity onejava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

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

<pre>
    gap> mat[3]         57477    ,    133    5865     233 ,
    [ 186645, 3861, 277 ]

</>

<div class="p"></div>
The second involution class forces one constituent of degree 8671
(which is the only candidate that can contribute a negative value),
and then a java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
The only solutiondiv class="p"</div>
We get the decomposition
+ 57477 +783ab+ 64584ab,
as claimed in [<a href="#Mverify" name=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

<pre>
    gap&62; Sum( mat)
    [ 196883, 4371, 275 ]

</pre>

<div class="p"></div>
The characters of the degrees 1, 8671,
java.lang.StringIndexOutOfBoundsException: Range [24, 4) out of bounds for length 116
java.lang.StringIndexOutOfBoundsException: Range [84, 2) out of bounds for length 96
we have to determine which extensions from 3.<span class="roman"Fi<span<sub>24<sub<sup>&le='color: green'>8242<sup>occur.
The following irreducible characters of 3.<span class="roman">Fi</span><sub>24</sub> can occur as
constituents of the restriction of χ.

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

<pre>
    gap> table3Fi24:= CharacterTable( "3.Fi24" );;
    gap> cand= Filtered( Irr( table3Fi24 ), x -> x[1] <= 196883 );;
    gap&#2 inv= Positions(  )2 ;
    [ 3, 5, 172, 173 ]
    #62;:= List ,x-#;x{());
    gap> PrintArrayAccording  < href#MS89 name"ITEGMS89">MS89<aTablesVII,IX,
    [the restriction of  permutationcharacter 1<sub2.<font size=+">B/font></sub><sup><font size="+0">M</font></sup> to 2.<font size="+0">B</font>
      [       1,      1,      ,      1 - ,
      [    8671,     351,     -33,    1495,     -41 ],
      [   8671,,351 -33,  -495,     41 ],
      [   57477,    1157,     133,    5865,     233 ],
      [   57477,    1157,     133,   -5865,    -233 ],
      [    1566,     158,      30,       0,       0 ],
      [  129168,    704,    144,       0,       0,      0 ] ]

</pre>

<div class="p"></div>java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
We get decomposition
1a + 8671b + 57477a + 1566a + 129168a claimed in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 2].

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

<pre>
       gap&62 ( mat [1 4,5,, 8]})
    [ 196883, 4371, 275, 4371<>td<>B/><td<><up>22/><class""Co/>sub2/>  <td<align=r>11707448673375</>/r

</pre>

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

<h2><a name="tth_sEc3">
<a&bsp&;he character(1sub.font =""Bfont>/><size+0""><<sup><sub>.font "0></font>/></h2< name="suborbits"java.lang.StringIndexOutOfBoundsException: Index 180 out of bounds for length 180
</a>

<div class="p"></div>
According to [<a href="#GMS89" name="CITEGMS89">GMS89</a>,Tables VII, IX],
the restriction of the permutation character 1<sub>2.<font size="+0">B</font></sub><sup<><>5A</><td<dspan=r></>             /><alignright30434513446055706624 />tr
java.lang.StringIndexOutOfBoundsException: Range [24, 10) out of bounds for length 106
with the point stabilizers U listed in Table <a href="#suborbitsTable">1</a>.

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

<div class="p"></div>
<a name="tth_tAb1">
</> center
</a>

<>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<tr><td>a c  ∈     </td><In , we  theninecharacters1subU<sub<>size"0>B/font<sup>,
<<>tt1<tt>/><td>.fontsize=0>/            <td>alignr"> /td>/>
<tr><td><tt>2A</tt> </td><td>Note that   8712;G<sub>,csub>holds
<tr<td><t>2B/tt> </td><td>2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>  </td><td align="right">11707448673375 </td></tr>
<tr><td><tt>3A</tt> </td><td><span class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<r<td><tt>3</><td<td>span class"roman"Thspan             </d<align"right>91569524834304000 /td></tr>
<tr><td><tt>4A</tt> </td><td>2<sup>1+22</sup>.<span class="roman">McL</span>   </td><td align="right">1102935324621312000 </td></tr>
<tr><td><tt>4B</tt> </td><td>2.<span class="roman">F</span><sub>4</sub>(2)       < subgroup 2sup2+22<sup>.spanclass"roman"Co/>sub2/> thepreimage maximal
<tr><td><tt>5A</tt> </td><td><span class="roman">HN</span>             </td><td align="right">30434513446055706624 </td></tr>
<tr><td><tt>6A</tt> </td><td>has  describedin [ahref#BMverify name=CITEBMverify">BMW20</a>].

</center>

<div class="p"></div>
ere denotes central  2.font ="0">B/>
e< =+"M/- classof ,
and c  ∈ a<sup><>pi1/>,<></tt,.. tt>pi9/>

<div class="p"></div>
Inthis section,we  the nine characters 1<sub>U</sub><sup>2.<font size="+0">B</font></sup>,
where U is one of the above point we have 1<sub>U</sub><sup>2.<font size="+0</font></sup> = 1<sub>2.<ont size=+"B/><sub>
Note that a  &#<div class""</div>
(and thus the character is an  gap#62;pi1 TrivialCharacter( table2B;java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
if and< class"></div>
hisexactlyfor firstthree orbitsjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

<div class="p"></div>
All subgroups U except 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> and 2<sup>1+22</sup>.<span class="roman">McLthe  1<><sub>sup2.<font size+"B/>sup>  theinflation  1<>#63717;]/>sup>font size=+0"<>/up>
are  A<font size="-2">TLAS</font> groups whose character tables have been verified.
The subgroup 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub> is the preimage of a maximal subgroup
2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub> of <font size="+0">B</font> under the natural epimorphism from 2.<font sizefrom <ontsize=+>font  2.fontsize"+0"<,
and the computation/verification of the character table of 2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub>
has been described in [<a href="#BMverify" name="CITEBMverify">BMW20</a>].
It will turn out that we do not need the character table of 2<sup>1+22</sup>.<span class="roman">McL</span>.

<div class="p"></div>
The nine characters will be stored in the variables
<tt>pi1</tt>, <tt>pi2</tt>, ..., <tt>pi9</tt>.

<div class="p"></div>
For U = 2.<font size="+0">B</font>,
we have 1<sub>U</sub><sup>2.<font size="+0">B</font></sup> = 1<sub>2.<font size="+0">B</font></sub>.

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

<pre>
    gap> pi1:= TrivialCharacter( table2B );;

java.lang.StringIndexOutOfBoundsException: Index 13 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</sub>(2),
the character 1<sub>U</sub><sup>2.<font size="+0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
from <font size="+0">B</font> to 2.<font size="+0">B</font>,
for [&#gap&#62 fus=PossibleClassFusions  ;
(ote that  classfusion from[#63717;U] to < ="+0></font>is  uniquely
>mapInducedClassFunctionsByFusionMap, tableB
but the permutation >                  ( tableUbar)]  )1 );

<java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 26

<pre>
    #;tableB:= CharacterTable( "B" );;
    gap> tableUbar:= CharacterTable( "2.2E6(2)" );;
    gap> fus:= PossibleClassFusions( tableUbar, tableB    #62;                  -&#pan style='color: green'>62;; ScalarProduct , chi pi2  ;
    gap> pi:gap&62 (  1 )
    >              map -> InducedClassFunctionsByFusionMap 1 ,3,5 7,13,15,17 ]
    >                         [ TrivialCharacter(<prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    gap> Length( pi );
    1
    gap> pi2:= Inflated( tableB, table2B, pi )[1];;
    &#62 := List(Irr(table2B)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    &62;                 chi&62 ( table2B,chi,  ;java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
    gap&for [63717;]=U&9001a&9002;=2sup1+<sup>< class="Co/span>sub>2/>,
    1
    gap> Positions( mult, 1 );
       [ , 2 3,5 ,13, 15 17]

</pre>

<div class="p"></div>
For U = <re>
the character 1<sub>U</sub><sup>2.    gap&62;tableUbar=CharacterTable B" ;
from <font size="+0">B</font> to 2.<font size="+0">B</font>,
for [U] = U / &#    #;map#;InducedClassFunctionsByFusionMap,
java.lang.StringIndexOutOfBoundsException: Range [39, 1) out of bounds for length 47

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

<pre>
    gap> tableUbar:= CharacterTablegap&#; : List Irr(table2B,
    CharacterTable( "2^(1+22).Co2" )
    gap> fus:= PossibleClassFusions( tableUbar, tableB );;
    gap> pi:= Set( gap&#62 (mult;
    >              map -> java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 5
    >                         [ TrivialCharacter( tableUbar ) ], map )[1] );;
    gap> java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 6
    1
    gap> pi3:= Inflated( tableB, table2B, pi )[1];;
    #; mult=List Irr ,
    >                 chi -> ScalarProduct( table2B, chi, pi3 ) );;
    gap> Maximum( mult );
    1
    gap> Positions ap#;tableU=( "i23" );
    , 13,15 28,30,37, 40]

</>

<div class="p"></div>
Next    #;                        TrivialCharacter(tableU )] map[1 ;

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

<pre>
    gap> tableU:= CharacterTable( "Fi23" );;
    gap> fus:= PossibleClassFusions( tableU, table2B );;
    gap&#;pi= Set( fus,
    >              map -> InducedClassFunctionsByFusionMap( java.lang.StringIndexOutOfBoundsException: Range [0, 74) out of bounds for length 5
    >                         [ TrivialCharacter( tableU ) ], map )[1] );
   gap&62 ( pi);
    1
    gap> pi4:= pi[1];;
    gap> mult:=
    >                chi-#62;( table2B,chi   ;java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
    #62;Maximum mult )
    1
    gap#; Positions(mult, 1)java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    [ 1, 2, 3<iv ="<!->/>
      63, 68,
      ,220]

</pre>

<div classgap#62;fus: PossibleClassFusions( tableU table2B);
Next we consider U = <span class="gap#62 pi:=Set( fus,

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

<<;Length ;
    gapjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    ap#;fus: PossibleClassFusions( tableU table2B );;
    gap> pi:= Set( fus,
    >              map -> InducedClassFunctionsByFusionMap( tableU, table2B,
                             [TrivialCharacter(tableU ) ], map )[1] );;
    gap> Length( pi );
    1
    gap> pi5:= pi[1];;
    gap mult:=List( Irr( table2B ),
    &62;                 chi -#62; ScalarProduct table2B,  ));
    gap> Maximum( mult );
    2
    gap> Positions( mult, 1    [1, 3, 7,8, 12, 13,16 , 27 28,34,38,41,57 68,70,77 ,
    [ 1, 3, 7, 8, 12, 13, 16, 19, 27, 28, 34, 38, 41, 57, 68, 70, 77, 78,
      85, 89, 113, 114, 116, 129, 133, 142, 143, 145, 155, 156, 185, 187,
      188, 193, 195, 196, 201, 208, 216, 219, 225, 232, 233, 235, 236,
      237,
    gap#;Positions( mult, 2)java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    [ 62 ]

</pre>

<java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 28
For U = 2<sup>1+22</sup>.<span class="roman">McL</span>, we carry out We know U  subgroup 2sup>2+</>.spanclass""Co/>sub2/>,
[<a href="#ctblpope" name="CITEctblpope">Breb</a>,Section "A permutation character of 2.<font size="+0">B</font>"].
We  that U is asubgroup of 2<sup>2+22</sup>.<span class="roman">Co</span><sub>2</sub>,
andthat &9001;U a &9002 has thestructure 2<>2+<sup.<pan =r"McL</pan>

< class""></div>
As a first step, we induce the trivialcharacterof < class"oman"McL> tospan=roman>Co<span<>2/>,
to 2.<font size="+0">B</font>,
roman">McL</span> to <span class=roman">o<spansub2</>,
then to inflate this then to induce this character  characterto font ="+"B/ont,
   thischaracterto < size="+0"></ont>
and then to inflate this

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

<pre>
    gap> mcl:= CharacterTable( "McL" );;
    &62;co2:=CharacterTable( "Co2" );;
    gap> fus:= PossibleClassFusions( mcl, co2 );;
    gap> Length( fus );
    4
    gap> indgap&#62 Length( fus )java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    #;                               [ TrivialCharacter( mcl ) ], map )[1] );;
    gap> Length( ind );
    1
    gap> bm2:= CharacterTable( "BM2" );
    CharacterTable( "2^(1+22).Co2" )
    ap#;infl= ( co2,bm2  ;java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
    gap> ind:= Induced( bm2, tableB, infl&62 : ( co2,bm2,ind )java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
    gap> infl:= Inflated( tableB, table2B, ind )[1];gap&yle='color: green'>62 :=Inflated(,table2B, )1;;

</pre>

<div class="p"></div>
As a second step,
wecompute<U>2<java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 133
using that we can speed up these computations by prescribing
the permutation character induced from the closure of U with
the normal subgroup 〈a 〉 of 2.<font size="+0">B</font>.

< class"">!--<div
(We are
There uniquesolution and isfast

<div class=p></div>

<pre>
    gap> centre:= ClassPositionsOfCentre( table2B );
    [ 1, 2 ]
    gap> pi:= <div class="p"></div
    >                             normalsubgroup:= centre,
    &62;                            := infl ) );;
    gap> Length( pi );
    1
    gap> pi6:= pi[1]    &62;                            normalsubgroup:= centre,,
    gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi6 ) );
    [ 1, 1, 2, 1, 2, 0, 2, 3, 2, 0, 0, 1, 4, 1, 2, 0, 3,    1
      2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3,1 , ,1 ,0 ,3, 2, 0,0,,4 ,2,0,,,2 ,,
      0, 0, 0, 0, 3, 0, 1, 0, 0, 5, 0, 5, 2, 0, 0, 2, 0, 0, 4, 1, 0, 2,
      0,,2,4 4,3,0,2, 4, 2, 4 0,3,0,3,2,5,0,1,0,3 1,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
      0, 1, 1, 2, 5, 3, 1, 1, 4, 5, 1, 1, 0, 3, 0, 0, 3, 2, 1, 1, 2, 1,
      1, 4, 0, 3, 2, 3, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 3, 0, 0, 2, 0, 0,
      0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0,
      2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1,
      1,3 , 0, 0,0, 1,1,1 ,2 2 , ,2,4 3 ,2 ,
      0, 0, 0, 0, 5, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 1, 7,
      7, 0, 0, 0, 1, 6, 4, 5, 0, 0, 3, 0, 0, 0, 0, 0, 4,2 1,1 0 ,0 0 1,2,1 ,1 ,1 ,1 1,,1 ,2 ,
      2, 2, 5, 0, 1 ]

</pre>

<div class0,0,0,0 , 2 0, 0,0, 1 1,0, 0 0,0 0,0,7 ,0 1,7
Next we consider U = 2.<span class="roman">F</span><sub>4</sub>(2).
We know that U2,2, 5, 0, 1 ]

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

<pre><prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    gap> tableU:= CharacterTable weconsider U 2.span class="roman">F</span><sub>4</sub>(2).
    gap&#62We know that U does not contain the central involution of 2.<font size="+0">B</font>.
    gap> pi:= Set( fus, map -> InducedClassFunctionsByFusionMap( tableU, table2B
    >             [ TrivialCharacter( tableU ) ], map    gap#2;tableU:=CharacterTable( "2.F4(2)" );;
    gap> Length( pi );
    2
    gap> pi:= Filtered( pi, x     gap> i: Set fus, map -#62; InducedClassFunctionsByFusionMap( tableU, table2B,
    gap> Length( pi );

    gap> pi7:= pi[1];;
    gap#62;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,
       ,0,0,2, 2,1, 4 0,2, ,2,0,0, 3, 2, 0, 4 4,0,0,0,
     ,0  ,1 ,0 2 ,3 ,0,0 3 , ,4 0 ,3
      0, 6, 0, 3, 2, 0, 0, 1, 4, 1, 4, 2, 6, 1, 4, 0, 4, 0, 1, 1, 2, 0,
      0, 3, 2, 1, 3, 2, 0, 0, 4, 5, 3, 1, 0, 3, 0, 0, 1, 1, 2, 0, 0, 2,
       ,, ,3,3 ,4, 1,0,,1 , , ,,2 ,1,2 3,0,
      0, 2, 2, 0, 5, 5, 3, 0, 1, 5, 1, 4, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0,
      ,3,,0,2,,0 ,1,,1 0 ,1 0, 0 ,1 ,0,,2
      1, 4, 4, 0, 0, 0, 3, 1, 1, 1, 2, 2, 2, 0, 0, 1, 2, 3, 3, 3, 1, 2,
      0 ,,1 4, 2 0 0, 0 3,2 0,0,0, 0 0,0,4 ,0 1,5,
      5      0,3, 2 1 ,2 ,0,4 ,3 ,0 ,,0 ,1 ,0 ,
      2, 5, 5, 0, 1 ]

</pre>

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

<div,,4,0,0,0,3 ,1 , ,,2, ,0 ,2 ,3 ,1, 2

<pre>
    gap&#62 : ( "");
    gap> fus:= PossibleClassFusions( tableU, table2B );;
    #;pi: (fus,map#;( tableU ,
    >             [ TrivialCharacter( tableU ) ], map )[1] );;
    gap> Length( pi );
    1
    gap> pi8:= pi[1];;
    gap> List( Irr( < =p></div>
    [ 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, 5, 6, 1, 7, 4, 7, 0, 0, 3, 8, 2java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 0
      0,0, 2,1,3,4 7,0,0 7,3 ,5 ,0,6,4 2,13,6 0 4,
      4 ,11 ,9, 7,3 ,13 12 20 ,10,11 ,17 ,
      10, 7, 19, 7, 7, 8, 10, 14, 18, 19, 5, 10#62              (   , map)1 ;
      4, 17, 16, 8    gap&62; List( Irr  )  -62 ( table2B chi pi8) )
      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, 24, 15, 40, 13, 9, 38, 24, 17, 35,
      32, 26, 26, 24, 22, 17, 31, 39, 29, 30, 30, 19, 44, 37, 37, 28, 30,
      31, 29, 42, 40, 40, 56, 56, 30, 30, 42, 50, 47, 2, 2, 4, 6, 4, 0,
      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, 36, 14, 14, 16, 8, 8, 46, 28, 28, 58, 90, 72, 70, 92, 104, 56,
      90 ]

</pre>

<div class="p"></div>
Finally, we consider U = 2.<span class="roman ,20 14 ,,,829 24 , 40 13 9 ,24 ,35
There are two candidates for the permutation character (1<sub>U</sub>)<sup>2.<font size="+0">B</font></sup>,
according       32, 26 26 24,,22 17 ,39,,37 37 28 30
One of the two characters is zero on the class of the central involution
java.lang.StringIndexOutOfBoundsException: Range [50, 51) out of bounds for length 48
We know thatU does not   central involution  2. size=0"B</
hence we can decide which character is correct.

<div class="p">      0, 0, 0, 0, 0, 28, 0, 0, 14, 34, 40, 2, 10, 10, 22, 40, 44, 44, 8, <div class="p"><div>

<pre>
    gap&#      8,36,14, 14 16,, 8, 8, 46, 28 28,,58 90, 72,70 92, 104, 56
    gap> fus:= PossibleClassFusions( tableU, table2B );<pre>
    gap> pi:= Set( fus<divclass""</div>
    &( tableU )]  )1 );
    There  two for  character<>U)>.=0><<java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
    2
    , x -62 (  ) [1  ;;
    #;(  )
    1
    gap&#62hence can  java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 47
    gap> List( Irr( table2B ), chi -> ScalarProduct class="></div>
    [ 1, 2, 3, 1, 4, 1, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      10, 7, 0    gaptableU:= CharacterTable( "2.Fi22" );;
      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 ,22 7,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
      22,18, 33,3,19 10, 34,12,12, 15,17, 28,34 34,7 20,26
     ,,25 ,40,,6 ,25,,21, ,21,18 43 ,45
      39, 49, 38, 51, 18, 32, 63, 19, 42, 41, 33, 48, 64, 27, 29, 52, 38,
      29, 19, 40, 47, 31, 69, 69, 65, 42, 35, 68, 27, 73, 20, 53, 46, 38,
      75, 29, 24, 72, 50, 41, 72, java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
      65, 49,85 ,75 63,68,65 , 90, 87,83 ,118,74,71
      90, 109, 109, 2, 3, 6, 9, 8, 0, 0, 7, 10, 18, 16, 22, 12, 23, 0, 0,
      2,6 ,19,0 0, , 16,18 ,0 0, 0 0, 0,52,1,1 26, 59,
      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 can form the restriction of (1<sub>2.<font size="+0">B</font></sub>)<sup><font size="+0">M</font></sup> to 2.<font size="+0">B</ 22,18 33, ,19,10 ,12 ,15 ,28,34,34 ,20 26,

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

<pre>
, pi7 ,pi9];
    gap> pi:= Sum( constit );;

</pre>

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

<h2><a name="tth_sEc4">
4</a>  The conjugacy classes of <font size="+0">M</font></h2><a name="Mclasses">
</a>

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

     <h3><a name="      ,6, 28 19,0 0,5 16, 18 ,0, 0,0 ,0,52 1 , 26,59,
4.1</a>  Our strategy to describe the conjugacy classes of <font size="+0">M</font></h3><a name="strategy_classes">
</a>

<div class="p"></div>
We know the order of <font size="+0">M</font> and its prime divisors.
Let us check whether this fits to our data computed up to now.

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

<
    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"></div>
For each prime p<div class=p></div>
we classify the conjugacy classes <>< name="th_sEc4"
and use the facts that for each thefacts that for each such representative x
/a
are in bijection with the corresponding classes in <div=p>--<>
and that this bijection respects java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 30

<div>

<div class=djava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
For each element x  ∈ <font size="+0"> java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 62
we will use the character table of N<sub><font java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
to establish <font size="+0">M</font>-conjugacy classes of roots of x.
java.lang.StringIndexOutOfBoundsException: Range [15, 2) out of bounds for length 51
we proceed2^*^205^7611^*3317*9*2329*141475971java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
andjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
prime of java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

< class""
For    classes  ofpjava.lang.StringIndexOutOfBoundsException: Range [61, 59) out of bounds for length 84
    N> ""/<(9001;&#;)
we will use the permutation character values (1<sub>2.<font size="+0">B</font></sub>)<sup><font size="+0">M</font></sup of x / x
hocjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0

     > =t2>
4.2</a>  Utility functions</h3><a name="functions">
</a>

<div  order  to   ,
During the process of finding the conjugacy    roots   whichp the
we record our knowledge about the character table of <font size="+0">M</font>
in a global <font face="helveticaFor elements x  ∈ <font size="+0">M</font> of prime order p  >  5,
   recordthe java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 48

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

<dland hoc
<dt><b><tt>Size</tt></b></dt>div =p>!--<div
<dd>  <br /     <><nametth_sEc4>
    the grouporder #124<size=0><>124;<dd
<dt><b><tt</>
<dd>  <br />
    the list of centralizer orders of the  the process of finding the conjugacy classes=+></,
      tonow<ddjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<dt><b><tt>OrdersClassRepresentatives</tt></b></dt>
<dd>  < /
    the list java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<dt><b><tt>fusions</tt></b></dt>
<>   /
    list  thecurrently      < =+"<font>
    each entry is a record with the components
    <tt>subtable</tt> (the character  <dt>b>tt>SizesCentralizerstt>/></t
     tt</t>(he of ;
      .dd>
</dl>

We initialize this variable, java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 13
and    identity .

<a  collectscurrently java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 97

pre
    &62;head= (:=,
    >                java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    >                OrdersClassRepresentatives:= [
       &62;               java.lang.StringIndexOutOfBoundsException: Range [36, 32) out of bounds for length 38
    >              );;

</pre>

<div
The tt>java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 56
the object <tt>head</tt>,
the character table <tt>s</tt> of a#;               SizesCentralizers=[sizeM]
and an integer <tt>pos</tt> as its arguments,
where it is assumed that the <    # );
contains an element x of prime order p
such that N<sub><font size="+0">
and that <tt>ead<tt contains information only about
those classes of <font size="+0">M</font> whose elements have order divisible by a prime
that is smaller than p.

<div   tt></>of  subgroup of< =+0"M/ont>

<pre>
    gap#62;: (head,s,pos java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    #;    fus p ,,ijava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    >
    >    # Initialize the fusion information.
    62;fus:= ( subtable:= s :   ;
    >    Add( head.fusions, fus );
    >
    &62      the  of rootof ''java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    >    orders:= OrdersClassRepresentatives(    gap&62 ExtendTableHeadByRootClasses:= function( head , pos java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    >    p:= orders[ pos ];    &62;      the information.
    &#2 cents: SizesCentralizerss )java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
    >    oldnumber:= Length( head.OrdersClassRepresentatives );
    >
    >     java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 42
    >    # are already contained in head
    >    for i in [ 1 .. NrConjugacyClasses( s ) ] do
    >      ord:&62 java.lang.StringIndexOutOfBoundsException: Range [42, 40) out of bounds for length 57
    >      if ord mod p = 0 and
    >         Minimum( PrimeDivisors( ord ) ) = p and
    >         PowerMap( s, ord / p, i ) = pos then
    >        # Class 'i' is a#;fus.[i]=(java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 63
    >        Add( head.SizesCentralizers, cents[i] );
    &adOrdersClassRepresentatives, orders[] );
    >        fus.map[i]:= Length( head.SizesCentralizers );
    >      fi;
    >           "  have "
    #;
    >    Print>   ;;
    >           Length( head.OrdersClassRepresentatives ) - oldnumber/>
    >           " classes, now have ",
    >           Length( head.OrdersClassRepresentatives ), "\n" );
    >    end;;

<pre

< =p>
In several cases, we will establish a conjugacy the root classesjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
knowing the character table of a suitable java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 36
<tt>ExtendTableHeadByRootClasses</tt> can be applied, where gdiv=">!-->/divjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
the root classes.
That is, we may know just element order <tt>s</tt>
and centralizer order<tt></tt>.

<div class="p"></div>
We are a bitin thistable whichfuse to  classg<><fontsize"0"M/font</up becausewe can
of subgroupof <fontsize="0"M</font andthe list<tposs/t ofall those classes
in this table which fuse to the class g<sup><font size="+0">M</font></sup>, because then we can
store this information
that is stored in <tt<div class="p">!----</div>

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

<pre>
    gap> ExtendTableHeadByCentralizerOrder:= function( head, s, cent, poss )
    >    local ord, java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 68
    >
    >    if IsCharacterTable( s ) then
    >      ordives( s ){ poss } );
    >      if Length( ord ) <> 1 then
    >        Error( "classes cannot fuse" );
    >      fi;
    #62;      ord:= ord];
    >    elif IsInt( s ) then
    >      ord:= s;
    >    fi;
    >    Add( head.SizesCentralizers, cent );
    >    Add( head.OrdersClassRepresentatives, ord );
    >
    >    Print( "#I  >    Print( "#I  after ", ord  element ;

    >      # extend the stored fusion &62;      #extend the  fusion froms
    >      fus:= First( head.fusions,
    >                   r -> Identifier( r.subtable ) = Identifier( s ) );
    &;      for i in poss do
    >        fus.map[i]:= Length( head.SizesCentralizers );
    >      od;
    >      Print( " from "#62;      ;
    #62    java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    &62    ("have ",
    &), " classes\n" );
    >    end;;

</pre>

<div class="p"></div>
The permutation character1<></<>G/>  H &8804;  are groups
has the property 1<sub>H</sub>The permutationcharacter1sub<sub><>sup   #804;G  groups,
For g  ∈ H,
this  impliesthat <subG</sub>(g)&#>124; = 1subH/sub><sup><sup(g &183;;#124;H&124;/ #124g<up</sup &8745;|
can be computed For g  ∈ ; H,
provided that know classes H fuse into<supG</sup>
The functionttExtendTableHeadByPermCharValue> extendsthe information
in <tt>head</tt> by the data for the  H  into <>G/>.
where <tt>s/t  the charactertable ofH,
<tt>pi_rest_to_s</tt> is (1<sub>H</sub><sup>G</sup>)<sub>H</sub>,
and<ttposs</>  the list positions of those classes  <tt><>
that fuse to g<sup><font size="+0">M</where <tt>s</tt> is the character table of H,

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

<pre
    gap> ExtendTableHeadByPermCharValue fuse gsup>font ="0"></ont<sup.
    >    local pival, cent;
    >
    >    pival:= Set( pi_rest_to_s{ poss } );
    >    if Length( pival ) <> 1 then
    >      > ExtendTableHeadByPermCharValue( head , pi_rest_to_s, poss )
    >    fi;
    >
    >    cent:= pival[1] &#2    pival: Set( pi_rest_to_s{poss })java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    >    ExtendTableHeadByCentralizerOrder( head    #62;     Error( "lasses cannot fuse")
    >    end;;

<>

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

     <h3#;   ;
4.3</a>  Classes of elements <pre
</a>

<div class="p"></div>
By [<a href="#Mverify" name="CITEMverify">BMW24</a>],
we know that <font size="+0">M</font> has.</>nbsp;nbsp;Classeselements even order/< namee"
and that the involution centralizers have
.font "0>B</font (for  class <>2A</>)and
2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub> (for the class <tt>2B</tt>), respectively.

<div class="p"></div>
Moreover,thecharacter ofthesesubgroups are
stored in the <font face="helvetica">GAP</font> 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><><sub><span =roman></span><sub><sub>(ortheclass<>2B/t) .
For 2.<font size="+0">B</font>, this follows from the correctness<div class"">!--><>
as,   tables these subgroups are
in [].
For 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>, the recomputation of the For 2.<font size"+0""></font, this follows from the correctness of the character table of <font size="+0">B</font>
in Section <a href="#sect:table_c2b">8</a>.

<div class="p"></div>
Thus we can determine the <font size="+0">M</font>-conjugacy java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 48
as follows.

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

<pre>
    gap> s:java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    gap> ClassPositionsOfCentre( s );
     ,2 ]
    gap> ExtendTableHeadByRootClasses( head, s, 2     gap&#le='color: green'>62; s:=CharacterTable( "2.B" );;
    #I  after 2.B: found 42 classes, now have 43
    gap> s:= CharacterTable( "MN2B" );;
    gap> ClassPositionsOfCentre( s );
    [ 1  java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    gap> ExtendTableHeadByRootClasses( head, s, 2 );
    #   2124C: found 91 , now have 134

</pre>

< p></div>

     <h3><     <h3><a ^+24Co1 found91 , have134
4<pre
</a>

<div class="p"></div>
We  that <font size"+"></font> has exactly three conjugacy classes of elements
of order 3,
their have structures
3.<span class="roman"a>
3<sup>1+12</sup><sub>+</sub>.2.<div class""</div>
and S<sub>3</sub> ×<span class="roman">Th</span> (for the class <tt>3C</tt>), respectively order,

<div class="p"></div>
Moreover,
the <face"">GAPfont   of3.span =r><span<>24<sub>and <pan=roman"></>   <font size=-"TLAS/> tables
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
that are roots of <tt>3A</tt> or <tt>3C</tt> elements, as follows.

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

<pre>
    gaps:= CharacterTable ".i24");
    gap> ClassPositionsOfPCore( s, 3 );
    [ 1, 2 ]
    gap> ExtendTableHeadByRootClasses( head, s, 2 );
    #I  after 3.F3+.2: found 12 classes, now have 146
    gap#62 s:=CharacterTableDirectProduct( CharacterTable( "h ,
    >                                     CharacterTable( "Symmetric",
    gap> ClassPositionsOfPCore( s, 3 );
    [ 1, 3 ]
    gap> xtendTableHeadByRootClasses( head s 3);
    #I      gap&62;s= CharacterTable( ".Fi24");java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

</pre>

<div class="p"></div>
The situation with the <tt>3B</tt> normalizer    &62;s: CharacterTableDirectProductTh )
Section;a href#:norm3B9/> documentstheconstruction ofthe character table
of a downward extension of the structure 3<sup>1+12</sup><sub>+    &; ,3)
of, and gives
candidates for the character table of the <tt>3B</#Iafter ThxSym(: found 7 classes,now have153

<div class="p"></div>
It will turn out ---<div
root classes,
in thesense that the number of these classes, their element orders,
and their centralizer orders are equal.
Notethatthe - of H <sup1+12/sup>sub><sub:.span class"roman"Suz<span>.2 hasthe structure
of a downward extension of the structure 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2
where X has order 3
and N  ≅ 3<sup>1+12</sup><sub>+</sub> such that H / N  ≅ 6.<span class="roman">Suz</span>.2 holds.
We are interested divclass=""</div>
the factors of H by the one of the two normal subgroups of order 3 in H
that are not equal to X or Z(N).
(See the picture in Section <a href="#sect:norm3B">9</a> for the details.)

<div class="p"></div>
Firstexclude 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:= Note that the 3-core of H = 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2 has the structure
    gap> kernels:= Positions( SizesConjugacyClasses( exts ), 2 );
    [ 2 18, 19, 20 ]
    gap> order3_13:= Filtered( ClassPositionsOfNormalSubgroups( exts ),
    >        l -> Sum( SizesConjugacyClasses( exts ){ l } ) = 3^13 );
    [ [ 1 .. 4 ] ]
    gap> kernels:= Difference( kernels, order3_13[1] );
    [ 18, 19, 20 ]

</pre>

<div class="p"></div>
The classes in the subgroup X can be identified by the fact that
ormal order3
class fusion from 2.<span class="roman">Suz</span>.2, and hence this must be the split extension
ofjava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 79

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

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

<>
    gap&u
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 18 ]" ),
      CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ),
      java.lang.StringIndexOutOfBoundsException: Range [47, 20) out of bounds for length 54
    gap#; f: CharacterTable( "2.Suz.2" );;
    gap> facts:= Filtered( facts,
    >        x -> Length( PossibleClassFusions( f,     [ [ 1 .  ]]
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ),
      CharacterTable(3^112:.Suz2/ 1,20 ] ) ]

</pre>

<div class="p"></div>
We compute the root classes for both candidates.
For that,
we first create a copy <tt>head2</tt> of the information in that

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

<pre>
    gap&#62 kernels:=List facts
    >        f -> Positions( SizesConjugacyClasses( f ), 2 ) );
    [ [ 2 ], [ 2 ] ]
    gap> head2:= java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 6
    dTableHeadByRootClasses
    #I  after 3^(1+12):6.Suz.2/[ 1, 19 ]: found 12 classes, now have 165
    ap62; ExtendTableHeadByRootClasses(head2,facts[]  ;
I ^)62[1  : 12,  165

<>

<div class="p"></div>
We observe java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
two character       CharacterTable "^1+2):6Suz.2/[ 1, 20 ]" ) ]

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

<pre>
    gap> nams:= RecNames( head );
     "ize" ", "izesCentralizers
      "fusions" ]
    gap> ForAll( Difference.&bsp  functions/>br>&We.
    >            nam -> head.( nam ) = head2.( nam ) );
    true
    gap>Length.usions;
    5
    &62;ForAll  1 .4]  ->headfusions[i =head2f[i )
    true
    &;eadfusions[]ap=.usions5map
    true

</7&nbs Theirreducible  of< =+"M/<a>br ><a href=tjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92

<ivclass">!---/>
We continue  establishing  conjugacyclasses of< =+"M</>
The question which of the two above >9.4  Compute the character table of H</ href"t"
of<anbsp&bsp</>

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

     <h3><a name="tth_sEc4.5">
4.5<We java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 22
</a>

<ivclass=""</div>
group< +B>contains tworational classes of elements
java.lang.StringIndexOutOfBoundsException: Range [28, 2) out of bounds for length 11
with different values in the permutation character (1<sub>2.<font size java.lang.StringIndexOutOfBoundsException: Range [31, 27) out of bounds for length 56

<div class=    gap&62 nams:= RecNames( head )java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

<pre>
    gap> s:= CharacterTable( "2.B" );;
    gap> pos:= Positions( OrdersClassRepresentatives( s ), 5 );
    [ 23, 25 ]
    gap> pi{ pos };
    [ 1539000, 7875 ]

</pre>

<iv class="p">!---></iv>
This establishes two classes <tt>5A</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
of elements of order 5 in lists of  class lengths,element , and  maps
ithcentralizerorders5 &;span class="roman"HN/pan&;and 5<7<sup&124;.spanclass"oman"><span>sub2</>&124,respectively

<div class="p"

<pre>
    gap> cents:= List( pos,
    >      We continue with establishing the
    [ 1365154560000000, 94500000000 ]
    &62;cents =[5* Size CharacterTable HN ) )
    >              5^7 * Sizecharactertables of subgroups of <font size="+0">M<


</re>

< class=p</div>

class>--</
By the  returnstt<tt
we know java.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 0
of order 5,
<tt>5A</tt> with centralizer 5 gap#62;izeScreen<iv=">java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 28
(D<sub
and <tt>5B</tt> with centralizer assume theexistence ofanordinary irreducible  &967java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
5<sup>1+6</sup><sub>+</sub>.4.<span class=" ,25]

<classp!----></div>
The classes are
sethis is  forintersectionswith 2.fontsize="0java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 70

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

<>
    gap> s:= CharacterTable( "MN5A" );
    CharacterTable( "(D10xHN).2" )
    gap>       [   96255,   96255,  ,2047 255]
    [1 45]
    gap> java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
    #I  after (D10xHN).2: found 5 classes itis the onlyfaithfulcandidatejava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

</

<div class="classe 3    differ by512 times the multiplicityof 4371java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
The ofaand ahas  values
<face"elvetica"MAGMA>  &;a=":able_N5B"<ajava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
thusareallowed to usethe character table
the <font face="helvetica">GAP</font> character table

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

<pre>
    gap#;s CharacterTable("MN5B");
    CharacterTable "^16).2java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 11
    gap &2;
    [ 1 .. 4 ]
    gap> SizesConjugacyClasses( s ){ 5core 4584   1352     ]java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    [ 1 4,37800 40320]
    gap
    #  after^16:..:found3 ,now 173

</pre>

<classp</div>

     <h3><a name="    gap> s: CharacterTable( "MN5Ainvolutions areat most to values the class,
4.6</a>  java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
</>

< class=""</div>
The group 2.<font [ 1., 24.7037, 49.91139 ., 477692,477692 ]
The permutation character <>2
  java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 93

< class""</div>

<pre>
    gap&#62     thatboth 64584 andaoccurwith multiplicity one.
    gap> pos:= Positions( OrdersClassRepresentatives( s
    [ 71/>

</pre>

<div class="p"></div>
By the arguments in [<a as claimed in [<a href="#Mverify"
<font<pre

<

<>
     .
    #I  after order[1 4 37800 (Irr   #;x1 l=196883)

java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 6

<div [   ,351,     33 1495,    -1]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

     <h3>          1566 158,     30 0,       ],
4.7</a>  Classes ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</a1a + 8671b + 57477 +1566a +129168aaclaimed in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 2].

<div class="p"></div>
The elements of the orders 17, 19, 23, 31, 47 in <font size="+0">M</font> lie
Sylowsubgroups that appear already in 2.font ="0"B/ont>.

<div class="p"></div>
The to[ =#GMS89 =Cjava.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 74
and hence also in <font size="+ order11with centralizer  order 11 #124M<>12/sub>&;in font size="+0">M</font>.

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

<pre>
    gap>
    gap> pos:= Positions( OrdersClassRepresentatives ]
    [ 118 ]
    gap> java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
    #I  after order 17 <tr><td><tt>2B</tt> </td><td>2>2+22<sup><class="oman">o/<Bythe arguments in;[<a ="Mverify"="CITEMverify"BMW24a
    gap> pos( s ), 19 );
    [ 128 ]
    gap> ExtendTableHeadByPermCharValue( headtrtd<>A/t <td<>2>122/>< class="oman"McLspan   /< align"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    #I  after order 19 element from 2.B: have 176 classes

</pre

<div class="p"></div>
For elements g of order p<tr>><td>tt6A/tt> <td>td2<span /re>
<
the  #;a>font=+"<>/up>java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 30
which means that [N<sub>2.<font size="+0">B</font></sub>(〈g &#n style='color: green'>9002;):C<subwhere U is one of the above point stabilizers G<sub>a,c</sub>.
class>>/
implies

< clear""/<border" "00"<<>
<table align="center" cellspacing="0"  cellpadding="2"><tr><td nowrap="nowrap" align="center">
|N<sub><font size="+0">M</font></sub>(〈g 〉)| = [N
</td></tr></table>

java.lang.StringIndexOutOfBoundsException: Range [16, 4) out of bounds for length 85
fuse nine   in gap> ExtendTableHeadByPermCharValue( head ;
in the former case,
we java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 120
|g<sup><font size="+0">M<ivclass""></div>
whereas we have
[N<sub><font size="+0">M</font></sub>(〈g 〉):C<sub><font size="+0">M</font></sub>(
#g< size+pre
Thus we can compute |N<sub><font size="+0">thecharacter 1<sub>U</sub><sup>2.<font =p>
  then  arguments2ont=""/ java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 95
donot java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<div class="p"></div
java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 27

<ivclass"p"</div>

<<>table
    gap&#java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
    For U =>2<.spanclass"oman"Co/><></sub  <font "0>/font> or not;
    gap> classes:= SizesConjugacyClasses( s );;
    gap> p:= 23;;
    #62;:=Positions( ord p);
     ,java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    gap> n:= (p-1)/2 * Size( s ) * 9001; #;:() java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    6072
    gap> Collected( Factors( n>Length(pi)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    [&62                 &62 ( table2B,  ));

</pre>

<v weU= "oman"</>sub23<sub>
In order
do notdo not  fuse.
divisible by 2<sup>3</sup>.
We see&;             map -#62;InducedClassFunctionsByFusionMap, table2B

<divgap&#62  =23java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

<pre>
    gapgap& mult;
    CharacterTable( "21
    gap> upos:= Positions( OrdersClassRepresentatives(62; := OrdersClassRepresentatives( s;java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
    [ 289, 294 ]
    gap=p</div>
    [ 23, 23 ]

</;map java.lang.StringIndexOutOfBoundsException: Range [68, 66) out of bounds for length 84

<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 26
Thus we have#;Maximum

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

<pre>
    &;( ,s ,pos{ [1 })
    #I  after order
    gap> ExtendTableHeadByPermCharValue<ivclass=">
    #I  after order 23 element from 2.B: have 178 classes

</pre>

<div class="p"><div
The case p = 31 is done analogously.
  java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 90

<=java.lang.StringIndexOutOfBoundsException: Range [36, 33) out of bounds for length 44

<pre>
    gap> p:= 31;;
    gap&#62[   , )]);
    [ 190, 192 ]
    gap> n:= (p-1)/2 * java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 43
    2790
    gap> <pre
    [ [ 2/pre>
    gap> SizesCentralizers( s ){As  second step
    [ 62, 62 ]
    #62;ExtendTableHeadByPermCharValue,s,pos [  ;
    #I  after order 31 element from 2.B: have 179 classes
    gap&#62
    #I  after order 31 <pre&#;p: 31java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 6

<     &62 ( )  -#2 ( table2B,  );
Finally, we deal with p = 47.

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

<>
    &62 : 47;
    gap> pos:= Positions( OrdersClassRepresentatives ,0 ,,13 ,,3 ,2,, ,3 3 ,0 ,0,0,
    [ 228,, 230 ]
    gap> n:= (p-1,0 ,2,1 ,1 ,1, ,1 I  31  from.: have180classes
    2162
;
    [ [2 1 ], [23,, 1], [47  ]]
    gappre
    [ 94, 94 ]
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [1] ize="0>/font>.
    #I  after order 47 element from 2.B: have 181;;
    gap> ExtendTableHeadByPermCharValue( head62;:f, -62;InducedClassFunctionsByFusionMap ,
    #I  after order 47[ TrivialCharacter(tableU ,map )[1] );

</pre>

<    &62 (  ){pos;

     <h3><ajava.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 71
4.8</a>  Classes of > ExtendTableHeadByPermCharValue,      , 6 ,,2     ,  1,  1  ,0
</a>

<div class="p"></div>
The class <tt>13A</tt> of <font size="+0">M</font>       0, 2, 2, 0, 5, 5, 3, 0, 1, 5, 1, 4, 0, 1, 0, 1
f13 in.font ="0"B/>.
We use the permutation character to enter the information about
thejava.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for length 23

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

<
    gap>       consider U<spanclass="roman">N/span>.
    gap> pos:= tt>.
    [java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> c:= Size( s ) * java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 5
    73008
    gap> Factors( c );
    [ 2, 2, 2, 2, 3, 3, 3, 13, 13 ]
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos );
    Iafter order13element from 2.B: have 183 classes

</pre>

<div class="p"></div>
The class <tt>13B</tt> intersects the <tt>2B</tt> centralizer.
Here we just know the centralizer order 13<sup>3</sup> ·2<sup>3</sup> ·3.

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

<pre>
    gap> c2b:= CharacterTable( "MN2B" );;
    gap> pos:= Positions( OrdersClassRepresentatives( c2b ), 13 ) 31, 29,42,40,40,56,56,,30 30,42, 50,47,2 2,4, 6 4,0
    [ 220 ]
    62 ExtendTableHeadByCentralizerOrder,c2b13^ 24, pos ,14 14 16,8 8 46,28 2858 ,72,,70 92 104 ,,
    #I  after order 13 element from 2^1+24.Co1: have

<>

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

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

<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 60
The 3            TrivialCharacter tableU]  )[1] );
with centralizer order 3 ·29.

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

<pre>
    gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap& = Positions OrdersClassRepresentatives( u10 , 0,0 ,6,6 3,,14 10 110,,5,,2,4,19
    [ 142 ]
gap#;SizesCentralizersu{  };
    [ 87 ]

</pre>

<div class="p"></div>
The list of classes of <font size="+0">M</font> collected up to now covers all roots of
elements of the orders 2, 3, 5, 11, 13, 17, 19<a
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>.

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

<pre
    gap> poss:= java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
                                  -&62;x 29 = 0)
    [ 1, 135, 144, 145 ]
    gap#;pos=((u,29 ;
    [ 1, 3, 87, 87 ]
    gap#2 ead.{ poss };
<>
      3765617127571985163878400, 87

/>

<div class4<a&bsp;Our  describethe conjugacy classes of < size"0M/ont>/< ="
theclasses <>1Att t</t, <>87<tt>and tt>ttjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
in size+Mfont x 29  7,41 ,and71.

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

<pre>
    gap> candprimes:= Difference( PrimeDivisors( headtrue
    >                     [ 2, 3, 5, 11, 13, 17, java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 20
    [ 7, 41, 59, 71 ]

</pre>

<></div>
The centralizer order of x
3 ·29 ·7    gapjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 35
with  0  ≤ i  ≤ 6 and j, k, l  &#n style='color: green'>8712; { 0, 1 }.

<div class use

<pre>
    &62parts=   . ))java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
    >                      x -> x[1] in candprimes )div="">-->/>
    [ [ 7w recordourknowledge the  of< =+"Mfont>
    gap> poss:= List( parts, l -> List( [ 0 .. l[2] ], i -> l[1]^i ) );;
    gap> cart:= Cartesian( poss );;
    <dd>   /

</pre>

<div class="p"></div>
Only 3 <> =  29*ListP );
that is, |<font size="+0">M    the  of corresponding java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
Note that we have [N<    a list that collectscurrentlyknown partialclass fusionsinto fontsize"0>M/font>

< classp></div>

pre
    &62;good:=Filtered( possord,
    >                     x -> ( <iv class=""</div>
    [ 87, 5133 ]
     (, < class=p</div>
    [     #62                     x &62  head

<[ , ,&;java.lang.StringIndexOutOfBoundsException: Range [53, 51) out of bounds for length 60

<div class="p"></div>
Nowthe object<>head<ttjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
byby the fact the  59 subgroup benormal
    and  an java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
3 contradiction

<div

<pre>
    gap> Filtered( DivisorsInt( 5133 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 1 ]

</pre>

<div class=/pre>
havejava.lang.StringIndexOutOfBoundsException: Range [24, 18) out of bounds for length 28
3183;java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

<#;   ( #   ,Identifier  ) :"

<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, u, 3 * 29, pos );
    I   F+2  185 java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

</pre>

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

     <h3><a name="tth_sEc4.10">
4.10</a> <re>
</a>

<div class="p"></div>
We assume that <font size#;   ;
The fact that an element
element    from java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 62
sub/><>#<up3.

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

<pre
    gap&#62
    CharacterTable( "O8-(3)" )
    gap> Length( Positions( OrdersClassRepresentatives( t ), 41 ) );
    10

</pre>

<div class="p"></div>
By the above arguments, the only possible &62     IsCharacterTable(s)then
|N<sub><font size="+0">M</font></sub>(〈x 〉)|/41&tyle='color: green'>62;if Length     [1640  163016
5   centralizerjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
As in the case of p = 29, we apply    &62;    fi
and
n "+></>/>(#9001x &62     IsCharacterTable()

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

<>
    gap> possord:= 2^2 * 41 * DivisorsInt(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    gap&#62 62    ;
    >                     x -> ( head.Size / java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    [ 1640, 163016 ]
    gap> List( good, Factors );
    [ [ 2, 2, 2, 5, 41 ], [ 2, 2, 2, 7, 41, 71 ] ]

</re

<div class="p"></div>
Suppose that 71 divides |N<sub><font size="+0">M</font></sub  41.
hen  71Sylowsubgroup of<sub>font size="+0
thus normalized by an element of order 8,
and thus centralized by an nvolution contradiction

<divclass=p"</div>

<pre>
    gap> Filtered( DivisorsInt( good[2] ), x -> x &62    ;
    [ 1 ]

</pre>

<div class="By [<a href="#Mverify" name="CITEMverify">BMW24</a>],
Thuswe   a    -centralizing elements
of order 41.

<iv class=""</div>

<Moreover,the tables these subgroupsthat are
    gap> ExtendTableHeadByCentralizerOrder( head, 41, 41, failSylow'theorem admits just the normalizer order 59 ·29.
    #I  after order 41 element: have 186 classes

</pre>

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

     <h3><a name="tth_sEc4.11">
./>;  oforder by 59/>< =">
</a[,2 ]

<classp>!--</
By the , java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
the normalizer order of an element of order 59 divides
3; #;.
's theorem admits java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div=p<pre>

<pre>
    gap> possord:= 59 * DivisorsInt( 58*7^6*71 );;
    gap&#62</re>
    >                     x -> ( head.Size / x ) mod 59 = 1 );
    [ 1711 ]
    gap> List( good, Factors );
    [ [ 29, 59 ] ]

</pre>

<div class="p"><that theorem  the  order 71 &8335java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
Thus we have established a pair of Galois java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0
elf  oforder .

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

<pre>
    &62 ExtendTableHeadByCentralizerOrder, 59, 59,fail )
    #I  after<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
    Iafter  59 : 188

<>

<divclass=p></div>

     <h3><a name="tth_sEc4.12">
4.12</a>  Classes of elements of order java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 13
</>

<div class="p"></div>
By N  #8773 <>1+2<sup><sub   N&8773 . ="Suz</span>. java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
the normalizer  an of71divides
70 ·7<sup>5</sup> ·71.
Sylows theorem admits justthenormalizerorder 71  &183;35.#I order71 :have 190

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

<pre>
    gap&#62java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   &;good (,
    &<div class"p"></div>
    [ 2485 ]
    gap> List( good, Factors );
    [ [ 5, 7, 71 ] ]

</pre>

<classp>-->/>
Thus we have established a pair of Galois conjugate classes of
self-

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

<pre>
    gap> ExtendTableHeadByCentralizerOrder    gap&#62 pos:=Positions(   ), ;("^112)6..2[1  ")]
    #I  after order 71 element: have 189 classes
    62 ( , 71,71    Iafter   2:have 191classes
    #I  after order 71 element: have 190     [("^112:6..2/ 1 19]" )

</pre>

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

     <h3><a name="tth_sEc4.13">
4.13</a>  Classes of
<>

<div class="p"></div>
e""<>    gap#62 =(headjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
with centralizer order 7 ·|<span class="roman">He</span>|.

<div class="pSince <span class="></>has  pair      Iafter3(+)..[,9] 12  165

java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 5
    gap> s:= CharacterTable( "2.B" );;
    gap> pos:< class"">!->/iv
    [ 41 ]
    gap> ExtendTableHeadByPermCharValue
    #  after  7 from 2B
    gap> Last    &62 : (
    true

<>

<div class="p"></div>
  ,    <>7A/t
structure 7 ×<span class="roman">He</span>,
and the normalizer has the structure (7:3 ×<span class="roman">HeThe second class ofelements of
a subdirect product of 7:6 and <span class="roman">He</span>.2.

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
< class="roman">He<  a     oforder
we get also a pair
7 ·17 = 119.

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

<pre>
    gap> ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
    #I  after order 119 element order5java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    gap> ExtendTableHeadByCentralizerOrder( head, 119, 119, fail );
    #I  after<iv class="p"></div>

<pre>

<div class="p"></div>
The second class of elements of order 7div class="java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 6
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     [23,25]

<div class="p"

<pre>
    gap> u:= CharacterTable( "3.Fi24" );;
    div classp>!--<divelements5   java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 51
    gap> rest:= Sum( cand{ [ 1, 4, 5, 7, 8 ] }
    gap> pos:= Positions( OrdersClassRepresentatives( u ), 7 );
    [ 41, 43 ]
    gap> rest{ pos };
    [ 50, 1 ]

</

<div class="p"></div>
Note that class 43 fuses to <tt>7B</tt> because the restriction of χ
to 2.<font size="+0"div =">!->/

<div class="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>

< class="p"></div>
Now the sum of class lengths in <tt>head</tt> is equal to the order of <font size="+0">     >   );

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

<pre>
    gap> Sum( head.SizesCentralizers, x -> head.Size / x ) = head.Size;
    true

</pre>

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

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

<>
    gap> m:= ConvertToCharacterTableNC( rec(
    &#;      : 0,
    >      Size:= head.Size,
    >      SizesCentralizers:= head.SizesCentralizers,
    >      OrdersClassRepresentatives:= head.OrdersClassRepresentativesN that we  not  java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 14
    >    ) );;

</pre>

<

<h2><a name="tth_sEc5">
5</a>&bsp;nbsp   of font =+"M<font>/>aname=    gap> afe_fusions:= Filtered( head.fusions,
</a>

<div class="p"></div>
Using the  orders the  representatives of the tablehead
of <font size="+0<re>
previous sections&62 :  2B ;
for primes p up tothe     the fusions   stored.

<div class="p"></div>
Note that we have not yet determined which of the two possible character
tables of the<pre>
thus weexcludethe   .

< class""></div>

<pre>
    gap> safe_fusions:= Filtered( head.fusions,
    >        r -> not IsIdenticalObj( r.subtable, facts[1] ) );;
    &;(safe_fusions )
    6

</pre>

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [29, 5) out of bounds for length 73
he givenbythe  fusions which   storedjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

<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 IsBound( r.map[i] ) then
    >          if IsInt( fus[i] ) then
    >            if fus[i] <> r.map[i] then
    >              Error( "fusion problem" );
    >            fi
    >          elif IsInt( r.map[i]  176
    >            if not r.map[i] in
    >              Error( "fusion problem" );
    >            fi;
    >          else
    >            if not IsSubset( fus[i], r.map[i] ) then
    &62;              (f problem ;
    >            fi;
    >          fi;
    >          fus[i]:= r.map
    >        fi;
    >      od;
    >      r.fus:= fus;
    >    od

</pre>

<div class= &#2         TransferDiagram ,.fus[])=fail
Next we initialize approximations of the power maps of the table of <font    &#;          Error("java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 44
and improve them using the compatibility of these maps with the power maps
of the subgroups w. r.&    #62    od;
Note that also the knowledge about the class fusions increases this way.

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

<pre>
    gap> maxorder:= Maximum( head.OrdersClassRepresentatives );
    119
    We repeat  the compatibilityconditions no
    gap are.
    gap>
    &62;;      powermapsp: ( m p)java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
    >      for r in safe_fusions do
    >        subpowermap:= PowerMap( r.subtable, p );
    >        if TransferDiagram( subpowermap, r.fus, powermaps[p] ) = fail then
    >          Error( "inconsistency" );
    >        fi;
    >      od;
    >    od;

</pre

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
improvements are found.

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

<pre>
    gap> found:= true&#62           res fail then
    gap#;res=";   avoid a syntax warning
    ap#62; while found do
    >      Print( "#I  start a round\n" );
    &      ;;
    & do
    #;        r  do
       subpowermap:PowerMap .subtable, )
    &;         =TransferDiagramsubpowermap .fuspowermapsp])java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    >          if res = fail then
    >            Error( "inconsistency" );
    #;         ForAny(res)  -&62 .n )&t&62;[ then
    >            found:= true;
    >
    >        od;
    >      java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 5
    >    od;
    #  starta
    #I  start a round
    #I  start a round
    #I  start a round

</pre>

<div class="p"></div>
Let us see where the power maps are java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 33

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

<pre>
    &#62 = java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 61
    [ 157, 158, 163, 164, 187, 188, 189, 190, 192, 193 ]
    gap> head.OrdersClassRepresentatives{ pos };
    [ 15, 15, 39, 39, 59, 59, 71, 71, 119, 119 ]

<>

<div class="p"></div>
The ambiguities for the classes of the element orders 59, 71, andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
are understandable:
For each<re
classes, and the subgroups whose class fusions we java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
.

<div class="p"></div>the field of lthroots of unity contains a unique java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
For each of the primes l   ap&62 ExtendTableHeadByPermCharValue , s , pos{[2]})
the field of l-th roots of unity contains a unique quadratic subfield,
which is <font size="+0">Q</font>(√{−l}),and the p-hpower java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
and the p-th power map, for p coprime to l,
fixes a class of element order l if and only if
the Galois automorphism that raises l-th roots of unity to the p-th power
fixes √{−l}.

<div class="p"></div>
In the case of element order l = 119 = 7 ·17,
the field ofHere the necessary 2- ofthe centralizer occurs already in 2.<font size="+0">B</font>.
<font size="+0">Q</font>(√{−7}), <font size="+0">Q</font>(√{17}), and <font size="<ivclass"</div>
In to which  actuallyoccurs,
welooka containsjava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 58
java.lang.StringIndexOutOfBoundsException: Range [45, 3) out of bounds for length 113
and the normalizer has the structure (7:3 ×<span class=T  gap#62;:= 31;
a subdirect product of 7:6 and <span class="roman">He</span>.2,
see Section&bsp<ahref="elements_7">4.13</a>.

<div    [[2 1 ][3  ], [ 5,1] [31   ]
The classes of element order 119 in the normalizer correspond to
the classes of this     #I  after order 31 elem .: 179classes
and the character values in the subgroup lie in the field <font size="+0">Q</#I  after order31element2.: have180 classes

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

<pre>
    &java.lang.StringIndexOutOfBoundsException: Range [34, 31) out of bounds for length 49
    gap> ConstructionInfoCharacterTable130, 131 132 133 134,135 ,208,209 210 ,212
    [ "[ 228, 230]
      [ 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 , ,301 ,303,305 306 3079494java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 14
          308, 309, 310, 311, 312, 313, 314, 315 ], (), () ]
    gap> pos:= Positions( OrdersClassRepresentatives( u ), 119 );
    [ 52, 53 ]
    gap> f:= Field( Rationals, List( Irr( u ), x -> x[pos[1]] ) );;
    gap> Sqrt(-119) in f;
    true

</pre>

<div class="p"></div>
We insert the relevant power map values.

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

<pre>
    gap> for l in [ 59, 71, 119 ] do
    >      val:=
    >      poss:= Positions( head.OrdersClassRepresentatives, l );
         for pin primes
    #;if( ,p) =1then
    #;           GaloisCyc, p)= val
    >            powermaps[p]{ poss }:= poss;
    >          else
    >            powermaps[p]{ poss }:= Reversed( poss );
    >          fi;
    >        fi;
    >      od;
    &#62dof charactervalues the  of< size="0></font is equal

</pre>

<div class="p"></div>
Now p-th power       both.
are determined java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [10, 10) out of bounds for length 0
These classes had been found as roots of <tt>3B</tt> elementsgap&62;fields List< class"

< class""</div>

<pre>
    gap> PositionsProperty( powermaps[17], IsList );
    [ 163, 164 ]
    gap> head.OrdersClassRepresentatives{ [ 163, 164 ] };
    [ 39, 39 ]
    gap&62;List ( head,
    &62                    -#;IsSubsetmap163,wecan  possibleorder3&183; #8359
    >          r -> r.subtable );
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ) ]

</pre>

<div class   5133) #;   1;
In order to decide whether the p-th power map fixes or swaps
the two classes, we consider elements of order 78,
which are we java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 66
three ofjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
a rational class that powers to <tt>2A</tt>
and a pair of >       ;

< classp>!--</

<>
    java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 126
    [37 132 133 ]
    gap> head.by an  of 4  be offfrom factorgroup
    CharacterTable( "2.B" )
    gap> Intersection( 78pos, head.fusions[1].map );
    [ 37 ]
    gap> s:= head.gap#62 : ->div
1+.1"
    gap> Intersection( 78pos, head.fusions[2].map );
    [ 132, 133 ]
   62 .[2., 132 )
    [ 342 ]
    gap> Positions( head.fusions[2].map, 133 );
    [ 344 ]
    gap> PowerMap( s, 7 )[342];
    344

</pre>

<div class="#;        ( r.[i] ) then
Since the <tt>3B</tt> normalizer in <font size="+0">M<pre>
classes of element order 78 which power to the generators of the normal
java.lang.StringIndexOutOfBoundsException: Range [5, 2) out of bounds for length 59
these two      [, ,5  ,    &2 fi
of order 78,
andtheir squares theclasses  element  39   are interested in.

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

<pre>

    & (   -;r
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ) ]
    ap62 Position(1., 163 ;
    173
    gap> Position( poss[[ 1]
    174
    gap&gap> us2: InitFusion( r2subtable,m;;
    [ [ 173, 174 ], [ 173, 174 ] ]
    gap> List( facts, s -> PowerMap( s, 7 )[173] );
    [ 174, 174 ]

</pre>

<div class="p"></div>
The field of character values on the two classes of <font size="+0"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
to the corresponding
whichis <fontsize="+0">Q</font>(√{−39}).
(Note that we have not yet decided which 411</a&bsp;&bsp;Classesof elements of order divisible by 59</h3><a name#62;          ;
to the <tt>3B</tt> normalizer class=p"</div>
but we get the same result for both candidates.)

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

<pre>
    gap> fields:= List( factspre>
    >                   s -> Field( Rationals, List( Irr( s ),
    >                                                x -> x[173] ) ) );;
    gap> Length( Set( fields ) );
    1
    gap>Sqrt(-9)  fields1;


</pre>

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

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

<
    gap> val:    gap> forp do
    gap> poss:= [ 163, 164 ];;
    gap> for p in primes do
    &62;     java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 39
     62    <>12
   62java.lang.StringIndexOutOfBoundsException: Range [30, 28) out of bounds for length 47
    >        else
    >          powermaps[p]{ poss }:= Reversed( poss );
    >        fi theabove,
   62      fijava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    >    od;
    gap> List( powermaps, Indeterminateness );
    [ ass="
      ,, 1,, 1,,,, 1,,,,,, 1,,,,,, 1,, 1,,,,,, 1,,,, 1,, 1,,,,,, 1,,,, 1,,
      ,,,, 1,,,,,, 1,, , 1,,,, ,, 1]

</pre>

<div class="p"></div>,,,, ,, , ,, ,  62( java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 35
Inthefollowing
we use the two candidates for the <tt>3
for answering most of the remaining questions about the power maps.
Again, the answers One open question is a the  of  non- classes

<div class="p"></div>
we   from   table.

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

<pre>

       62 :( r, m);
    gap> for i in [ 1 .. Length( r.map ) ] do
    >           <h3><a name=>a ="th_sEc413"
    &#62        ifIsInt( fus[] )then
    >          if fus[i</>
    >            Error( "fusion problem" );
    >          fi;
    >        elif IsInt( r.mapThe subgroup 2.font size"+0"><font> yields a rational class <tt>7A</tt<pre
    >          if not r.map[i] in fus[i] then
    >            Error( "fusion problem"
    >          fi;
               else
    >          if not java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
    >            Error( "fusion problem" );
    >          fi;
    >        fi;
    >        fus[i]:= r.map[i];
    >      fi;
    &62;    od;
    gap> r.fus:= fus;;

</pre>

<ivp>!--<div
... and the class fusion from the java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 13

<div class<iv=">

<pre>
    we  also a  of java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 63
    gap> fus2:= InitFusion( r2.subtable, m );;
    gap#62;fori in [1 .Length.map
    >      if IsBound( r2.map[i] ) then
    >        if IsInt( fus2[i] ) then
    >          if fus2[i] <> r2.map
    >            Error( "fusion problem" );
    >          fi;
    >        elif IsInt( r2.map[i] ) then
    >          if not r2.map[i] in fus2< class=></div>
    >            Error( "fusion problem" );
    >          fi;
    >        else
    >          if java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
    >            Error
    >          fi;
    62;
    #2        ]:r2map
    >      fi;
    &62;;   ;
    gap> r2.fus:= fus2;;

</pre>

<div class="p"></div>
... then we create an independent copy of the current approximations
of <pre>
inthetwo 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[p] ) = fail =+0></font.
    >
    >      fi;
    >    od;
    gap> s2:= r2.subtable;
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
fjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 31
    >      if
    >        Error( "inconsistency" );
    >      fi;
    >    od;
    62  =;

   &  , )java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
    [ , 32, 64,, 1,, 1,,,, 1,, 1,,,, 1,, 1,,,, 1,,,,,, 1,, 1,,,,,, 1,,,,
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      , 1,,,,,,,, 1,,,, 1,, 1,,,, 1,, 1, <2>a

</pre>

<<div="p"</div>
One open question is about the squares of the non-<java.lang.StringIndexOutOfBoundsException: Range [13, 8) out of bounds for length 89
element .

ss>

<pre8 ]
    gap> powermaps[2]{ [ 132, 133 ] };
    [ [ 163, 164 ], [ 163, 164 ] ]
    gap&62; = List,
    >                  s62;       r#;not IsIdenticalObj( r.subtable facts1 ));
    [ [ 235, 236 ], [ 235, 236 ] ]
    gap> fus{ [ 235, 236 ] };
    [ [ <div class="">!---</iv>
    gap> fus2{ [ 235, 236 ] };
    [ [ 132, 133 F java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 5

</pre>

<div class="p"></div>
We may identify one class of element order 78 in the <tt    62        IsBoundm
with the#java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

=>--<

<pre>
    gap&62 fus[235]=132;
    gap> fus2[235]:= 132;;
    gap> TransferDiagram( PowerMap( s, 2 ), fus, powermaps[2] ) <&#pan style='color: green'>62; fail;
    true
    gap> TransferDiagram( PowerMap( s2, 2 ), fus2, powermaps2[2] ) <> fail;
    true
    gap> powermaps = powermaps2;
    true
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 64 ]

</pre>

</
Since also the    gap&62;maxorder=  java.lang.StringIndexOutOfBoundsException: Range [67, 64) out of bounds for length 67
are still    gap&62;powermaps:=[];java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
2nd and the 3rd power map commute.

<classp"!---<div>

<pre>
    &62; powermaps[] [163   ;
    [[183,184],[183,184  ]]
    gap> TransferDiagram( powermaps[2], powermaps[3], powermaps[2] ) <> fail;
    true
    gap> List( powermaps{ [ 2, 3 ] }, java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 0
    8 16 ]

</pre>

<div class="p"></div>
The next open question is about the cubes of elements of order 93.
The classes of element order 93 -<divclass="p"></div>
have been
of the type S<sub>3</sub> ×<span class="roman">Th</span>, and they do not occur in other subgroups
we have considered.
Thus we may choose     [ 52, 2 ]
31.

<div gap&62;( head., r&;       java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 33

<pre>
    gap> poss:= Filtered( head.fusions,
    >                     r -> 93 in OrdersClassRepresentatives( r.subtable ) );;
    #62; List poss,r -#;r. )java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
    [ CharacterTable( "ThxSym(3)" ) ]
    gap     , 83]
    [ 152, 153 ]
    gap> powermaps[3]{ pos93 };
    [ [ 179, 180 ], [ 179, 180 ] ]
    gap> powermaps[3][152]:= 179;;
    gap> TransferDiagram( PowerMap( poss[1].subtable, 3 ), poss[1].fus,
    >                     powermaps[<diviv class="p"></div>
    true
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 4 ]

</pre>

<div class="p"></div>
The next open question is about the cubes of elements of order 69.
The classes of element order      15 ,39,39,59 ,71,71,119  java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
haveby applying consistencycriteriajava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
of the type 3.<span class="roman">Fi</span><sub>24</sub>, and they do not occur in other subgroups
we have considered.
Thus we may choose which of them cubes to the first class of element order
23.

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

p
    .fusions,
    >                     r -> 69 in OrdersClassRepresentatives( r.subtable ) );;
    gap> List( poss, r -> fixes a class of element order l if and only if
    [ CharacterTable( "3.F3+.2" ) ]
    gap> pos69:= Positions( head.OrdersClassRepresentatives, 69 );
    [ 142, 143 ]
    ap&62; powermaps[3]{ pos69 };
    [ [ 177, 178 ], [ 177, 178 ] ]
    gap> powermaps[3]{ [ 142, 143 ] }:= [ 177, 178 ];;
    gap&62;TransferDiagram(( PowerMap( oss[gap#62; r2:= First head2.usions,r-#62;IsIdenticalObj(r. [2] ));
    >                     powermaps[3] ) <> fail;
    true
    gap> List( powermaps{ [    &62;                         powermaps ;
    [ 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 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
and
t2<>

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

<pre>

    [ 26, 27, 118, 120 ]
    gap> powermaps[2]{ pos46 };
    [ 177, 178, [ 177, 178 ], [ 177, 178 ] ]
    gap> powermaps[23]{ pos46 };
    [ 2, 2, 44, 44 ]

ga#;pos=Positions((u ), 119);

<div class=    gap&62 : (,List (u ) x &62;[os1]));;
We have defined the two classes of element order 23 as squares of those
two classes of element order 46 that power to <gap&62;rest: Sum  );
and we have not yet distinguished the other two classes of element order 46.
Thus ""<--<div

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

<pre>
    &62; powermaps[2]{ [ 118,120 ]}=[ 177,178];
    gap> Indeterminateness ( powermaps[2] );
    2

</pre>

<div class="p"></div>
Now just one value is left to be determined,
the square of a class of element order 18 and centralizer order 3888.

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

<pre>
    java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 30

    gap><
    18
    gap&#by &bsp<a href="natural"2<a>java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    3888

<>

<div class="pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
There are two classes with this property in <font java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 34
both are  of generators of the normal subgroup of order 3 in
3<sup>1+12</sup><OrdersClassRepresentatives;
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 )div class="p"></div>
    [ 78, 79 ]
    gap> powermaps[3]{ [ 78, 79 ] };
    [ 52, 52 ]
    gap&62 [2]52]
    154
    gap> First( head.fusions, r -> 154 in r.map ).subtable;
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" )
    gap> s:= facts[1];
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" )
    gap> pos18:= Filtered( [ 1 .. NrConjugacyClasses( s ) ],
    >                i -> gap> head.fusions[1].subtable;
62java.lang.StringIndexOutOfBoundsException: Range [49, 47) out of bounds for length 65
java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 14
    gap> PowerMap( s, 2 ){ pos18 };
    [ 24, 24 ]
    gap> s:= facts[2];
    java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 16
    gap[java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    >                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,

by applying the

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

<pre>
    gap> powermaps[2][78]:= powermaps    [CharacterTable "3^112:6Suz2[1 19]  ]
    gap> for r in safe_fusions do
    >      if not TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
    >                                  powermaps ) then
    >        Error( "inconsistent!" );
    >      fi;
    >    od;
    gap> r:= First( head.fusions, r -> IsIdenticalObj( r.subtable, facts[1] ) );;
    gap> TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
    >                         powermaps );
    true
    &;  (head2,r &62 ( . 2);
    >TestConsistencyMapsComputedPowerMaps(r2.ubtable ), r2.fus,
    >                         powermaps );
    true
    gap> SetComputedPowerMaps( m, powermaps );

</pre>

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

<    gap>fields:= List( facts,
6</a>  The degree 196 883 character χ of <java.lang.StringIndexOutOfBoundsException: Range [0, 66) out of bounds for length 5
</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

<div

<pre>
    gap> r:= head.fusions<div "!-<
    gap> s:= r.subtable;
    CharacterTable( "2.B"    gap&62 =(-9;
    gap> cand:= Filtered( Irr( s ), x -> x[1] <= 196883 );;
    gap> List( cand, x -> x[1] );
#java.lang.StringIndexOutOfBoundsException: Range [34, 29) out of bounds for length 50
    gap> rest:= Sum( cand );;
    gap> rest[1];
    196883

</pre>

<div class="p"></div>
Thuswe know the values of χ on those classes of <font size="+0">M</font>
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>
    &;       :[]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
    gap> map:= r.fus;;
    gap> for i in [ 1 .. Length( map ) ] do
    >      if IsInt( map[i] ) then
    >        chi[ map[i] ]:= rest[i];
    >      fi;
    >    od;
    gap> Number( chi );
    111

</pre>

<div class="p"></div>
Also the restriction of χ to 3.<span class="roman">Fi</span><sub>24</sub> is known,
by Section <a href="#natural">2</a>.
This yields 29 more character values.

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

<pre>
    gap> r:= head.fusions[3];;
    gap> s:= r.subtable;
    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> map:= r.fus;;
    p
    >      if
    >        if IsBound( chi[ map[i] ] ) and chi[ map[i] ] <> rest[i] then
    >          Error    canbe computedfrom the scalarproduct
    ;
    >        chi[ map[i] ]:= rest[i];
    >      fi;
    #;od
    gap> Number( chi );
    140

</pre>

<div class="p"></div>
computejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 71
There are only 13 possible irreducible constituents of this restriction.
We consider the matrix of values of these characters on those#java.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 47
for which the class fusion to
the value of χ on the image class is known.
This matrix has full rank, thus we can directly compute the decomposition
of the restriction into irreducibles,
and get 41 more character values.

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

<pre>
    gap> r:= head.fusions[2];;
    gap> s:= r.subtable;
    .. thencreate independent of
    gap> cand:= Filtered( Irr( s ), x -> x[1] <= chi[1] );;
    gap> map:= r.fus;;
    gap> knownpos:= Filtered( [ 1 .. Length( map ) ],
    >                      i -> IsInt( map[i] ) and IsBound( chi[ map[i] ] ) );;
    gap> rest:= List( knownpos,     gap> mat:= List( cand, x -(java.lang.StringIndexOutOfBoundsException: Range [31, 29) out of bounds for length 57
    gap> mat:= List( cand, x -> x{ knownpos } );;
    gap&62;Length mat )java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    13
    gap> RankMat( mat );
    13
    gap> sol:= SolutionMat( mat, rest     gap>s2:= r2.subtable;
    [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
=  ;
    &62;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>
Whichvalues java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 31

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

<pre>
    gap> missing:= Filtered( [ 1..194 ], i
    [ 151, 152, 153, 160, 169, 170, 186, 187, 188, 189, 190, 192, 193 ]
    gap> head.OrdersClassRepresentatives{ missing };
     ,93, ,27,,95 ,59 ,71,71java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 42
    62 headSizesCentralizers{  }
    [ 57, 93, 93, 243, 95, 95, 41, 59, 59, 71, 71, 119, 119 ]

</pre>

<div   ,   ,133
For g  ∈ <font size="+0">M</font>, we have χ(g)  ≡ χ(g<sup>p</sup>)  mod p
and 0 ,,1 ,1  java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
We apply these conditions.
In all cases except one, the centralizer orders are small enough
for determining the characterjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class#; java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 29

<pre>
    gap> for i java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 8
    >      ord:= head.OrdersClassRepresentatives[i];
    >      divs:= PrimeDivisors( ord );
    >      if ForAll( divs, p -> IsBound( chi[ powermaps[p][i] ] ) ) thengap&62 ( {[2,3 ]} Indeterminateness)java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    >        java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 8
    >        res:= </re>
    >        modulus:= Lcm( divs );
    >        c:= head.SizesCentralizers[i];
    >        Print( "#I  |g| = ", head.OrdersClassRepresentatives[i],
    >               ", |C_M(g)| = ", c,
    >               ": value ", res, " modulo ", modulus, "\n" );
    >        if ( res + 2 * modulus )^2 >= c and ( res - 2 * modulus )^2 >= c then
    >          candpre>
    >          if Length( cand <iv class=""><---<divjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    >            chi[i]:= cand[1];
    >          fi;
    >        fi;
    >      fi;
    >    od;
    #I  |g| = 57java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    #I  |g| = 93, |C_M(g)| = 93: value 92 modulo 93
    #I  |g| = 93, |C_M(g)| = 93: value 92 modulo 93
    #I  |g| = 27, |C_M(g)| = 243: value 2 modulo 3
    #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, |C_M(g)| = 59: value 0 modulo 59
    #I  |g| = 71, |C_M(g)| = 71: value 0 modulo 71
    #I  |g| = 71, |C_M(g)|    >      OrdersClassRepresentatives( m ),
     119, |g)  119 value118 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>

<div class="p"></div>
The one missing value can be computed from the scalar product
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[i] );
    -6650349175263480459970863415322722279882752000000000
    gap&#62</pre>
    2

</pre>

<divclass="p">!---><div>
Now of the two candidates for the character table of
the <tt>3B</tt> normalizer is the correct one.
For the first candidate, the restriction of χ
cannot be decomposed into irreducibles.

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

s there are 22  ofconjugate classesin <ontsize=+">M/ont>java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
    &;=First head.,  -62;( r.ubtable,facts1]));
    gap> map:= r.fus;;
    gap> knownpos:= Filtered( [ 1 .. <
    ap&; := List knownpos, -#;chi[ map[  ;
    gap> cand:= Filtered( Irr( r.subtable ), x -> x[1] <= chi[1] );;
    gap> mat:= List( cand, x -> x{ knownpos } );;
    &;Length(mat ;
    95
    gap> RankMat( mat );
    88
    gap> SolutionMat( mat, rest );
    fail

</pre>

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

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

<pre>
    gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.subtable, facts[2] ) );;
    gap> map:= r2.fus;;
    gap> knownpos:= Filtered( [ 1 .. Length( map ) ], i -&#='color: green'>62; IsInt( map[i] ) );;
    gap> rest:= List( java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 6
    &62;cand:= Filtered(Irr .ubtable ) x &62 [1 &; [1]);
    gap> mat:= List( cand, x -> x{ knownpos } );;
    gap> Length( mat );
    95
    gap> RankMat( mat );
    88
    gap> SolutionMat( mat, rest );
    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,his will  revised  irreducibles java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
      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, 0, 0,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      0, 0, 0, 1, 1, 1, 0 ]
    gap> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

</pre>

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [33, 3) out of bounds for length 60
the character table that is stored in the <font face="helvetica">GAP

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

<pre[155, 156 ]
    gap> TransformingPermutationsCharacterTables( r2.subtable,
    >        CharacterTable( "MN3B" ) ) <> fail;
    true

</pre>

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

<h2><a name="tth_sEc7">
7</a>  The irreducible characters of <font size="+0">M</font></h2><a name="sect:irreduciblesM">
</a>

<div class="p">
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 the characters printed in the  A<font size="-2">TLAS</font> as an
"oracle".
For that,
we compute first a bijection between the columns of our character table head
and those of[52 52]
This is done by using the following invariants:
element orders, centralizer orders,
the values of χ, and the indirection of χ by the 2nd power map.

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

<pre>
    gap> invs:= TransposedMat( [
    >      OrdersClassRepresentatives( m ),
    >      SizesCentralizers( m ),
    >      chi,
    >      CompositionMaps( chi, PowerMap( m, 2 ) ) ] );;
    gap> invs_set:= Set( invs );;
    gap> Length( invs_set );
    172
    gap> atlas_m:= CharacterTable( "M" );;
    #;invs_atlas:=TransposedMatjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    >      OrdersClassRepresentatives( atlas_m ),
    >      SizesCentralizers( atlas_m ),
    >      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 of invariants,
since there are 22pairs of  conjugate  in font "0">/>,
and our current knowledge does not allow  to
the classes of each pair.

<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<font size="-2">TLAS</font> table accordingly,
and create the "oracle" list.

<div class="p"><
(The explicit permutation (32,33)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
of the  A<font size="-2">TLAST isajava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 94

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

<pre>
    gap> pi1:= SortingPerm( invs );;
    gap> pi2:= SortingPerm( invs_atlas );;
    gap> pi:= pi2 / pi1 * (32,33)(179,180);;
    gap> oracle:= List( Irr( atlas_m ), x -> Permuted( x, pi ) );;

</pre>

<div class="p"></div>
In order to prohibit that <font face="helvetica">GAP</font> tries to compute table automorphisms
of our table head of <font size="+0">M</font> (which is impossible without knowing the
irreducible characters),
we set a trivial group as value of
<tt>AutomorphismsOfTable</tt>;
this will be revised as soon as the irreducibles are known.

<div class="p"></

    gap> SetAutomorphismsOfTable( m, Group( () ) );

</pre>

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

<iv class=""><----</iv>
First we compute candidates for the class fusion from 2.<font size="+0">B</font>,
starting from the approximation we have already computed.
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 18
we decide about the images of four classes of element orders 40 and 44.

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

<pre>
    gap> r:= safe_fusions[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 ] ]
    gap&
    [ 40,<ivclass="p"></divof the subgroup table.

</pre>

<div class="p"></div>
The two classes of element order 40 are a pair of Galois conjugates
in both 2.<font size="+0">B</font> and <font size="+0">M</font>.
Note that their elements are roots of <tt>2B</tt> elements in <font size="+0">M</font>,    gap&#e='color: green'>62; s:= r.subtable;
and the classes 110 and 111 correspond to non-rational elements
of order 40 in the <tt>2B</tt> normalizer.

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

<pre>
    gap> r.fus[ PowerMap( s, 20 )[217] ];
    4
    gap> r2:= safe_fusions[2];;
    gap> s2:= r2.subtable;
    CharacterTable( "2^1+    &62 =.;
    gap> Position( r2.map, 110 );
    273
    gap> ForAll( Irr( s2 ), x -> IsInt( x[273] ) );
    false

</pre>

<div class="p"></div>
We may freely choose the fusion from the classes 217 and 218 of 2.<font size="+0">B</font>
because   tableautomorphism  2.font =+">/> that swaps java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
two classes.

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

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

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

<div class="p"></div>
With<iv class="p">< pos:=( OrdersClassRepresentatives( s )56 )java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
java.lang.StringIndexOutOfBoundsException: Range [12, 12) out of bounds for length 5
both in 2.<font size="+0">B</font> and <font size="+0">M</font>.

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

<pre>
    gap> r.fus[ PowerMap( s, 22 )[ 222 ] ];
    44
    gap    &62;                      i -#62 ( map[] ) and IsBound(chi[map] ] ));
    178
    gap> ForAll( Irr( s2 ), x -> IsInt( x[178] ) );
        &62;mat=List java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 57

</pre>

<div class="p"></div>
There is a table automorphism of 2.<font size="+0">B</font>
where the classes of element order 44 form one pair,
and each of the other two pairs is fused in <font size="+0">M</font
Thus we may again freely choose the fusion from 222 and 223.

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

<pre>
    gap> (143,144)(222,223)(244,245) in AutomorphismsOfTable( s );
    true
    gap> r.fus{ [ 143, 144, 244, 245 ]Which values are still missing?
    [ 15, 15, 34, 34 ]
    gap> r.fus{ [ 222, 223 ] }:= [ 88, 89 ];;

</pre>

d ,152 153,160 169,170,,186,187,188,189 190, 192,193]
Now the remaining open questions about the class fusion from 2.<font size="+0">B</font>
can be answered by <font face="helvetica">GAP</font>'s function <tt>PossibleClassFusions</tt>.

<div

<pre>
    gap> knownirra #124;#;g)&124<2<>  lt &#;subfont
: s,java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
    >               rec( chars:= knownirr, fusionmap:= r.fus ) );;
    gap> List( poss, Indeterminateness );
    [ 1 ]

</pre>

<div class="p"></div>
Now we can induce the irreducibles of 2.<font size="+0">B</font> java.lang.StringIndexOutOfBoundsException: Range [0, 67) out of bounds for length 32

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

>
    gap> induced:= InducedClassFunctionsByFusionMap( s, m, Irr( s ), poss[1] );;

</pre>

<""!-->/divjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
Next we compute candidates for the class fusions from the subgroups
2sup124<sup><ub><sub.<span class=roman"Co/><sub><,3sup112/<+/.. java.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 0
Here we enter also >          cand +1.1]*  #;a^&t;)
    provide additional .

<div class="p"></div>
The fusion from 2<sup>1+24<<pre>
of the subgroup table.
We extend the list of known induced characters.

<ivclass=p"<-->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<pre>
    gap> poss:= PossibleClassFusions( s2, m,
    >               rec(
    >                    fusionmap:= r2.fus ) );;
    gap> List( poss, Indeterminateness );
    [ 1, 1, 1, 1 ]
    gap> Length( RepresentativesFusions( AutomorphismsOfTable( s2 ), poss,
    >                Group( () ) ) );

    gap> Append( induced,
    >        InducedClassFunctionsByFusionMap    #  |g|= 7171 |_g)|=71:value 0 modulo71

</pre>

<div class="p"></div>
In order to <div class="p"><!-<div>
we have to consider two classes of element order 56 first.
They are a pair of Galois java.lang.StringIndexOutOfBoundsException: Index 36 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

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

p>
    gap> r:= safe_fusions[7];;
    gap> s:= r.subtable;
    ):6.Suz2[ 1,20 ]")
    gap> pos:= Positions( java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 5
    [ 250, 251 ]
    gap> r.fus{ pos };
    [ [ 125, 126 ], [ 125, 126 ] ]
    gap> r.fus[ PowerMap( s, 28 )[ 250 ] ];
    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    gap> Position( r2.map
    319
    gap> ForAll( Irr( s2 ), x -> IsInt( x[319] ) );
    false
    gap> (250, 251) in AutomorphismsOfTable( s );
    true
    gap> r.fus{ [ 250, 251 ] }:= [ 125, 126 ];;

</pre>

<div class="p"></div>
Now the class fusion to <font size="+0Now we of  twocandidates for the character table of
of the subgroup,
and we extend the list of induced characters.

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

<pre>
    gap> poss:= PossibleClassFusions( s, m,
    >               rec( chars:= Concatenation( knownirr, induced ),
    >                    fusionmap:= r.fus ) );;
    gap&#62gap#;r=(fjava.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 89
    [ 1, 1 ]
    gap> Length( gap> rest:= List( knownpos, i -> map[] ]);
    >                Group( () ) ) );
    1
(java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 29
    >        InducedClassFunctionsByFusionMap( s, m, Irr( s ), poss[1] ) );

</pre>

<div class="p"></div>
The fusion from 3.<span class="roman">Fi</span><sub>24</sub> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
of the subgroup table.
We extend the list of known induced characters.

<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:= Concatenation( knownirr, induced ),
    >                    fusionmap:= r.fus ) );;
    gap> List( poss, Indeterminateness );
    [ 1 ]
    gap>cand:= Filtered( Irr( r2.subtable ), x -> x[1] <= chi[1] );;
    >        InducedClassFunctionsByFusionMap( s, m, java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 57

</pre>

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

<      0 0,0, 0,0, 0, 0,,0,0 ,0,0, 0,0 ,0,0, 0, 0 ,

<pre>
    gap> Append( induced,
    >      InducedCyclic( m, [ 2 .. NrConjugacyClasses( m ) ], "all" ) );

</pre>

<div class="p"></div>
Now we reduce the induced characters with the two known ">Z<font>java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
and  class>/>
this yields four new irreducibles.

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

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


</pre>

<div class="p"></div>
We extend the list of known irreducibles,
reduce the induced characters,
and apply LLL again.

<iv class=p">!--->/>

<pre>
    gap> knownirr:= Union( knownirr, lll.irreducibles );;
    gap> red:&#62 fi;
    gap> Length( red.irreducibles );
    0
    gap> lll:= LLL( m, red.remainders );;
    gap> Length
    0

</pre>

<div class="p"></div>
Now we use the irreducibles of the  A<font size="-2">TLAS</font> table of <font size="+0">M</font> as an oracle,
as follows.
Whenever a character from the oracle list belongs to the <font size="+0">Z</font>-lattice
that is spanned&;      chi,
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.

<java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 32

<pre>
    gap> mat:= java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0
    gap> norm:= NormalFormIntMat( mat, 4 );;
    62;rowtrans:= .;java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

    gap> one:= IdentityMat( NrConjugacyClasses( m ) );;
    gap> for i in [ 2 .. Length( one ) ] do
    >      extmat:= Concatenation( normal, [ one[i] ] );
    >      extlen:= Length( extmat );

    >      if extnorm.rank = Length( extnorm.normal ) or
    &62 extnorm.owtrans[ ]   lt;62 1 then
    >        coeffs:= fail;
    >      else
            coeffs:- extnorm.[ extlen] [ 1 ..extnorm.rank ] }
    >                   * rowtrans{ [ 1 .. extnorm.rank ] };
    62 fi
    >      if coeffs <> fail and ForAll( coeffs, IsInt ) then
    >        # The vector lies in the lattice.
    >        chi:= coeffs * lll.remainders;
    >        if not chi in knownirr then
    >          Add( knownirr, chi );
    >        fi;
    >      fi;
    >    od;
    gap&<ivclass=""><--><div>
    66
    gap> Set( knownirr, chi -> ScalarProduct( m, chi, chi ) );
    [ 1 ]

</pre>

<div class="p"></div>
We take the generators of the <font size="+0">Z</font>-lattice and some symmetrizations
of div class#=java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 40
we know ,
java.lang.StringIndexOutOfBoundsException: Range [13, 3) out of bounds for length 20

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

<pre>
    &62 = Reduced m,knownirr, lll. );
    gap> Length( red.irreducibles );
    0
    gap> sym:= Symmetrizations( m, knownirr, 2 );;
    gap> sym:= Reduced( m, knownirr, sym );;
    gap> Length( sym.irreducibles );
    0
    gap> lll:= LLL( m, Concatenation( red.remainders, sym.remainders ) );;
    gap> Length( lll.irreducibles );
    0

</pre>

<div class="p"></div>
We use the above oracle again, for the new <font size="+0">Z</font>-lattice.

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

<re>
    gap> mat:= MatScalarProducts( m, oracle, lll.remainders );;
    gap> norm:= NormalFormIntMat( mat, 4 );;
    gap> rowtrans:= norm.rowtrans;;
    gap> normal:= norm.normal{ [ 1 .. norm.rank ] };;
    gap> one:= <div class="p">>
    gap> for i in <iv class="p"></div>
    >      extmat:= Concatenation( normal, [ one[i] ] );
    >      extlen:= Length( extmat )gap&62;: safe_fusions1;java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    >      extnorm:=<
    >      if extnorm.rank = Length( extnorm.normal ) or
    >         extnorm.rowtrans[ extlen ][ extlen ] <> 1 then
    >        A faithful permutationrepresentation of H/ on
    >      else
    &.rowtrans[ extlen ]{[1. extnorm.rank ] }
    >                       gap#62; OrdersClassRepresentativess) pos }java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
    >      fi;
    >      if coeffs <> fail and ForAll( coeffs, IsInt ) then
    >        Add( knownirr, coeffs * lll.remainders );
    >      fi;
    >    od;
    gap> Length( knownirr );
    194

</pre>

<i
Now we are done.
As stated in the beginning of this section<iv class="p">!---><div>
we unbind the stored trivial value for <tt>AutomorphismsOfTable</tt>.

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

<pre>
    gap&#62   to   M/
    gap> ResetFilterObj( m, HasAutomorphismsOfTable );
    gap#62; TransformingPermutationsCharacterTables( m atlas_m  lt;;#;fail;
    true

</pre>

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

<h2><a name="tth_sEc8">
<>&java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 150
</a>

<div class="p"></div>
The centralizer C of a <tt>2B</tt> element in <font size="+0">M</font>
has the structure 2<sup>1+24</   and the known We may freely choose thfrom the classes 217 and 218 of 2<font size="+0">B</font>
which can be constructed as of ψ<sup>2</sup> into irreduof H/,

<div class="p"></div>
Considera subdirect product  oftwo  H/
andjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 77
where X is a cyclic group of order two,
N is an extraspecial group 2<sup>1+24</sup><sub>+</java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 52
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+java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>
The  centre of  is  KleinKlein four     From now on, we argue character-th
are X, Y = Z(N), and a third subgroup D.
We have C  &#<iv class="p"></div>

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

<ss"p"></div>
From the 2-local construction of a matrix representation for <font size="+0">M</font>,
we know the following faithful representations,
given by generators that are preimages of standard generators of <span class="roman">Co</span

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

u>
<li>   There is a table automorphism of 2.<font size="+0">B</font> that swaps three pairs of classes,
   degree 98&bsp  the field withthree elementsjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
<Thuswe  freelychoose  fusionfrom 222 and223.
</li>

<li>
   a    gap&62; 143144)222,23)244,45)in AutomorphismsOfTable getone   java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
   of dimension 4 096 over the field with three elements,
<div class="p"></div>
</li>

<li>
   a matrix representation of H/N  ≅ 2.<span class="roman">Co</span><sub>1</sub>,
   of dimension 24 over the field with three elements.
<div class="p"></div>
</li>
</l>

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

<div class="p"></div>
We proceed in the following steps.

< ""<!->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

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

<li>
   A faithful permutation representation of H/Y on
   2 ·196 560 = 393 120 points is obtained by
   glueing the permutation generators of H/E
   <class">!--</div>
   and H/N
   (the smallest permutation representation of 2.<span class="roman">Co</span><sub>1</sub>,
   on 196 560 points) together.

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

<li>
   Starting from the character java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
   the 3-modular matrix representation of dimension 4 096 of H/X is used
   tocompute necessary classsplittings 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
  the unique faithful irreducible3-modular representation of M/X,
   and because this representation extends to the full automorphism group
   of the extraspecial group.
   Thus we can compute the Brauer character values of the representation
   on    >                   :/>
   and interpret the values as those of an ordinary character  gap&62 List(poss,Indeterminateness )java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
   The tensor square ψ<sup>2</sup> belongs to the group H/E,
   and the known values of ψ suffice to determine the decomposition
   of ψ<sup>2</sup> into irreducibles of H/E,
   thus compute also the values of ψ<sup>2</sup> on 3-singular classesgap Append induced,
   Taking square roots, we get all values of the extension  M /X  H/M  splitjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
   (We cannot distinguish which of the two values belongs to which of the two
   asses
   by choosing the positive value for one of them, and the negative value
   for the other one.)
< =">-->/>
</li>

<li>
   From now on subgroups D<></>and <sub2<sub>.

<div class="p"></div>
   The missing irreducible characters of H/X are computed as
   tensor products of ψ with the irreducible characters of
   the factor group H/M

<div class="p"></div>H /isa productof
   H/E  &87733<>12/sup>.,
such the degree    irreduciblesofHX
   equals the order of this group.
    implies  there  java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 84
   fusion from H/X to H/E than the splittings forced by the values
   of ψ.
java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 66
</li>

<li>
   Using the two class splittings from H/X and H/Y to H/E,
   we compute necessary class splittings java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 64
   class fusions to H/X and H/Y,
   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,

   For each class that splits in exactly one of H/X and H/Y,
   we get two preimage classes in H.
   For each class that splits in none of H/X and H/Y,
java.lang.StringIndexOutOfBoundsException: Range [31, 28) out of bounds for length 34

div=>--java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
   Then we take those irreducible characters of H    &62        InducedClassFunctionsByFusionMap( s, m, Irr(<>
   respectively, that do not have E/X or E/Y, respectively,
   in their kernel;
   weform  productsof them,  characters  kernel ,
    of degreenbsp;46

<div class="p"></div>
   This yields all missing irreducibles of H:
   The sum up to  order
   H meansthatno more class splittingoccurs

<div class="p"></div>
   Finally, we compute the power maps (and thus the element orders)
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 31
   The resultlijava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
   the character table that is stored in <font face="helvetica">GAP</font>'s table library.
<div class="p"></div>
</li>
</ul>

<<ivp>!---</>

>>
9& Appendix  character table  3>+12/>>/>:6. ="oman"</pan>2<h2>< =sect:">
</a>

<divfrom toH/Ythenthefull  table head of .

h =1>
9.1</a>  Overview</h3>

<div class="p"></div>
The <tt>3B</tt> normalizer in <font size="+0">M</font> has the structure 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2.
Its character table has been computed by Richard Barraclough and
Robert A. Wilson, see [<a href="#BW07" name="CITEBW07">BW07</a>].
java.lang.StringIndexOutOfBoundsException: Range [23, 20) out of bounds for length 71
thatnot assume the character table of <font size="+0">M</font>,
we recompute this table.

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

d">!--</iv
<=
The  of38&38matricesover the fieldwiththree elementsjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 94
and we compute the character table of this bigger group H.

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

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

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

<br<br />
</center>

<div class="p"></div>
We have H / N  ≅ 6java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Let M be the normal subgroup of H above N such that
M  #; 2.span =roman><span>2holds.
ThenM= times; for  normal  X of order 3,
where the extension of M / X by H / M is split.
Let N<sub>2</sub> be the normal Whenever a  fromthe oraclelist belongs to the <ontsize=+"Z/>-lattice
H / N<sub>2</sub>  ≅ 3.<span class="roman">Suz java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 42
Then M<sub>2</sub> = M N<sub>2</sub> has the property H / M<subjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [45, 46) out of bounds for length 25
Then E = X ×Y  ≅ 3<sup>2</sup> is elementary abelian,
with diagonal normal subgroups D<sub>1</sub> and D<sub>2</sub>.
It will turn out that the extensions of M / D<sub>1</sub> and M / D<sub>2</sub> by H / M
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 14

H / Y is a subdirect product of H / N and
H /&62      =Length extmat;
we have H / Y  ≅ (3 ×3<sup>12</sup>).2.<span class="roman">Suz</span>.2.

<div class="p"></div>
The &62        =;

<--></
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 subgroupgap# java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 78
appears in Section <a href="#sect:natcharM">6</a>.)

<div&;fi
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,
      java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 71
<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 Section <a href="#<pre
    The first two generators are standard generators of 6.<span class="roman">Suz</span>.2gap& style='color: green'>62; SortBy   )
    the third generator lies in N.
<div class="p"></div>
</li>

<li>
    Let <font<re>
    see Section <a href="#computeHtable">9.4</a>.
<div class="p"></div>
</li>
</ul>

<div class="p"></div>
<b>Remark:</b>
,
we asked <font face="helvetica">MAGMA</font> to compute the character table of H / Y,
and then used individual conjugacy tests for first setting up the class fusion
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
notdifficult, using character theoretic methods.
However, trying the same input files in 2024 showed many cases where
some of the required conjugacy tests did not finish in reasonable time.
Luckily, the automatic computation described in Section <a href="#java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 28
H / E.

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

     <h3><a name="tth_sEc9.2">
9.2</a>  A permutation representation of H / X</h3><a name="HmodXpermgens">
</a>

<div
The  A<Now we  done.
afaithful representation of H,
java.lang.StringIndexOutOfBoundsException: Range [61, 2) out of bounds for length 71
The generating matrices are called <tt>M3max7G0<iv 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 -&62 ForAll([ 1 ..24]
    #;                                      j -#62;IsZero(mi,] )and
    >                                            can beconstructed as
    true

>

java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 28
Here we will use just the lower right 14 ×14 blocks,
which <div class"p"><--->/div>
(Note that the 12-dimensional irreducible representation of 2.<span class="roman">Suz</span>.2
over the field with 3 elements respects a symplectic form,
and the embedding of 2.<span class="roman">Suz</span>.2 into <span class="roman">Sp</span>(12,3)
-which is the automorphism group of the extraspecial group 3<sup>1+12</sup><sub>+</sub>-
yields a construction of the semidirect product 3<sup    gap&62 slp:= AtlasProgram "Suz.2", check ;
as a group of 14 ×14 matrices.)

<div class="p"></div>
Later we will construct a faithful permutation representation of H
as a subdirect product of H / X and H / N  ≅ 6. gap>ForAll<r<br />
Let <tt>G    true

<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>
We use just the following facts.
The <tt<li>
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</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 on 1 594 323 points.
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 java.lang.StringIndexOutOfBoundsException: Range [0, 87) out of bounds for length 39

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

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

<pre>
    gap> G:= GroupWithGenerators( mats );;
    gap> orbs:= ShallowCopy( OrbitsDomain( G, GF(3)^14 ) );;

    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    gap> SortBy( orbs, Length );
    62;List(orbs  )java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    [ 1, 2, 196560, 1397760, 1594323, 1594323 ]
    gap>    on 2 #183;98n;80  196nbsp;560points)
    gap&62; 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
    gapthe3-    4nbsp;096  X  used
    true

</pre>

<div class="p"></div>
We conclude that the action on 196 560 points represents the group
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>.

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

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

<pre>
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 63
    gap> n:<iv ""><--<div>
    >                       Subgroup( represHmodX, [ gensHmodX[3] ] ) );;
    gap> Size( n ) = 3^13;
    true
    gap&62; IsAbelian( n );
    false

/>>

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

<div class="p"></div>
Next we show that the first
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 questionThe irreduciblecharacters of /Xare computed as
the   java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 28
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>].

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

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

<pre>
    gap> slp:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap>   / and/  /,
= ResultOfStraightLineProgram,gensHmodX ;java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
    gapclassfusions  HXand Y,
    [ 2, 1, 2, 1 ]
    gap> ForAll( gensHmodX{ [ 1, 2 ] }, x -> ForAll( res, y -> x*y = y*x ) );
    true
    62;Order( gensHmodX[] )
    3

</pre>

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

     <h3><a name="tth_sEc9.3">
9 apply LLLalgorithm  .
</a>

<div class="p"></div>
The  a productof H /X  H /<sub2/> ̅3< class="oman"Suz/>2java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
w.r.t. the common factor within  few minutesjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 63
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    the character table that is stored in <font face=helvetica"><font' table library.

<div class=<pre>

<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( &#2 fi;
    gap> shiftedgens:= List( 3suz2, x -> x^pi );;
    gap> Append( shiftedgens, [ () ] );
    62;gensH=List([1 .3] i &62;gensHmodXi]  shiftedgens[]);
    gap> 9.1</a>
    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 java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 28
and CDCDD has order 7.

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

<divclass=p"<-->/

<pre>
    gap&#62 <"java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 79
    3
    gap> Order( Product( gensH{ [ 1, Mathematics vol. 694,Providence,   Mathematical, 2017java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    7

</pre>

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

t.4java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
9.4</a>  Compute the character table of H</h3><ap.&21-.  3682588
</a>

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

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

<
    gap> H:= GroupWithGenerators( gensH );;
    gap> if CTblLib.><a href="#CITEMverify" name="Mverify/<java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
    >      mgmt:= CharacterTableComputedByMagma( H, "H_Magma" );
    >    else
    #;mgmt: ("^112)6" ;
    >    fi;

<

<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>
    62java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 69
    >        tt>ctbllib/</tt><tt>doc2/<
    true

</pre>

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

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

<="java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
10</a>  Appendix: The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2</<iv =">-->/div>
</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<>:</>
<font face="helvetica">MAGMA</font> [<a href="#Magma" name="CITEMagma">BCP97</a>] can compute the character table from the group
we asked<ont face="helvetica"MAGMA<font>to compute the character table  H /Y
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
     << ="th_sEc92">
    #;
    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>
<&62;                                          ( ij] ) )  ;
<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>
  href=#" =">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="and the embedding of 2.<span class="roman">Suz</span>.2 into <span class="roman">Sp</span>(12,3)
  ;21-31. MR 3682588

<div class="p"></div>
</dd>
<dt><a href="#CITEBMverify" name="BMverify">[BMW20]</ java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 66
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     gap> mats:List(gens  -#62;   . 38]}{   .38]} );
<hr />
</td></tr></table>
, <em>Verification of the conjugacy classes and ordinary character
  table of the Monster</em>, submitted, 2024.

< class=p>!->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
</dd>
<dt>[]</dt><dd>T. Breuer, <em>Constructing the ordinary character tables of some Atlas
  groups using character theoretic methods.</em>, <a href="of a group of the structure 3<sup>1+12</sup><sub>+</sub>.2.<span>112/<ub+/>2<spanclass="oman"></>2on1&;594n;23points

<div class="p"></div>
</dd>
<dt><a href="#CITEctblpope" name="ctblpope">[Breb]</a></dt><dd>
align"eft" =0"<< ="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></t>homas.Breuer//tt>
<tt>ctbllib/</tt><tt>doc2/</tt>
<tt>manual.pdf</tt></a>.

<div class="p"><#;java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 36
</dd>
<gap#2=[0, ,0 ,0 0 ,0 0,,0 0 0, 2]*Z)0;
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>
<m>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.

<=p>!--<div>
</dd>
<dt><a href="#CITEGMS89" name="GMS89">[GMS89]</a></dt><dd>
R. Meierfrankenfeld, and Y.nbsp;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>
djava.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 28
<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.nbsp;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</    #2n= NormalClosure( represHmodX
/>

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

<div class= alse

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version 3.59.<br />On 22 May 2025, 15:04.</small>
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