<!-- %W ctblpope.xml GAP 4 package CTblLib Thomas Breuer -->
<Chapter Label="chap:ctblpope">
<Heading>Permutation Characters in &GAP;</Heading>
Date: April 17th, 1999
<P/>
This is a loose collection of examples of computations with
permutation characters and possible permutation characters in
the &GAP; system <Cite Key="GAP"/>.
We mainly use the &GAP; implementation of the algorithms to compute
possible permutation characters that are described
in <Cite Key="BP98copy"/>,
and information from the Atlas of Finite Groups <Cite Key="CCN85"/>.
A <E>possible permutation character</E> of a finite group <M>G</M>
is a character satisfying the conditions listed in Section
<Q>Possible Permutation Characters</Q> of the &GAP; Reference Manual.
<P/>
<List>
<Item>
Sections <Ref Sect="sect:U35sub"/> and <Ref Sect="sect:O82sub"/>
were added in October 2001.
</Item>
<Item>
Section <Ref Subsect="subsect:monsterperm1"/>
was added in June 2009.
</Item>
<Item>
Section <Ref Subsect="subsect:monsterperm2"/>
was added in September 2009.
</Item>
<Item>
Section <Ref Subsect="subsect:monsterperm3"/>
was added in October 2009.
</Item>
<Item>
Section <Ref Subsect="subsect:monsterperm4"/>
was added in November 2009.
</Item>
<Item>
Section <Ref Sect="sect:comp_B"/>
was added in June 2012.
</Item>
<Item>
Section <Ref Sect="sect:comp_2B"/>
was added in October 2017.
</Item>
<Item>
Section <Ref Sect="sect:comp_pi_piprime"/>
was added in December 2021.
</Item>
</List>
<P/>
<!-- %T missing: --> <!-- %T more examples for the combin. and ineq. algorithm, --> <!-- %T the use of tables of marks, --> <!-- %T and the examples in ~/Saxl/M12.2, ~/Saxl/Suz !! --> <!-- %T show an example where the modular criteria are guaranteed by starting --> <!-- %T from the matrix of projective indecomposables! -->
In the following,
the &GAP; Character Table Library <Cite Key="CTblLib"/>
will be used frequently.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:comp_M24">
<Heading>Some Computations with <M>M_{24}</M></Heading>
We start with the sporadic simple Mathieu group <M>G = M_{24}</M>
in its natural action on <M>24</M> points.
<P/>
<Example><![CDATA[
gap> g:= MathieuGroup( 24 );;
gap> SetName( g, "m24" );
gap> Size( g ); IsSimple( g ); NrMovedPoints( g );
244823040
true
24
]]></Example>
<P/>
The conjugacy classes that are computed for a group can be ordered
differently in different &GAP; sessions.
In order to make the output shown in the following examples stable,
we first sort the conjugacy classes of <M>G</M> for our purposes.
<P/>
<Example><![CDATA[
gap> ccl:= AttributeValueNotSet( ConjugacyClasses, g );;
gap> HasConjugacyClasses( g );
false
gap> invariants:= List( ccl, c -> [ Order( Representative( c ) ),
> Size( c ), Size( ConjugacyClass( g, Representative( c )^2 ) ) ] );;
gap> SortParallel( invariants, ccl );
gap> SetConjugacyClasses( g, ccl );
]]></Example>
<P/>
The permutation character <C>pi</C> of <M>G</M> corresponding to the action on
the moved points is constructed.
This action is <M>5</M>-transitive.
(We have set the <Ref Attr="Identifier" BookName="ctbllib"/> value of the
character table because otherwise some default identifier would be chosen,
which depends on the &GAP; session.)
<P/>
<C>pi</C> determines the permutation characters of the <M>G</M>-actions on
related sets,
for example <C>piop</C> on the set of ordered and <C>piup</C> on the set of
unordered pairs of points.
Clearly the action on unordered pairs is not transitive, since the pairs
<M>[ i, i ]</M> form an orbit of their own.
There are exactly two <M>G</M>-orbits on the unordered pairs,
hence the <M>G</M>-action on <M>2</M>-sets of points is transitive.
In terms of characters, the permutation character <C>pihom</C> is the difference
of <C>piup</C> and <C>pi</C> .
Note that &GAP; does not know that this difference is in fact a character;
in general this question is not easy to decide without knowing the
irreducible characters of <M>G</M>,
and up to now &GAP; has not computed the irreducibles.
The point stabilizer in the action on <M>2</M>-sets is in fact a maximal
subgroup of <M>G</M>, which is isomorphic to the automorphism group
<M>M_{22}:2</M> of the Mathieu group <M>M_{22}</M>.
Thus this permutation action is primitive.
But we cannot apply
<Ref Func="IsPrimitive" BookName="ref"/>
to the character <C>pihom</C> for getting
this answer because primitivity of characters is defined in a different
way, cf. <Ref Func="IsPrimitiveCharacter" BookName="ref"/>.
<!-- %T It should be noted that for <M>k > 2</M>, --> <!-- %T the <M>k</M>-th symmetrisation of a permutation character --> <!-- %T does in general not decompose into permutation characters --> <!-- %T corresponding to the action on <M>l</M>-sets, for <M>l \leq k</M>. -->
We could also have computed the transitive permutation character of
degree <M>276</M> using the &GAP; Character Table Library instead of
the group <M>G</M>,
since the character tables of <M>G</M> and all its maximal subgroups are
available, together with the class fusions of the maximal subgroups
into <M>G</M>.
Note that the sequence of conjugacy classes in the library table of
<M>G</M> does in general not agree with the succession computed for the
group.
<!-- %T mention class identification program? -->
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:poss_perm_char_M11">
<Heading>All Possible Permutation Characters of <M>M_{11}</M></Heading>
We compute all possible permutation characters of the Mathieu group
<M>M_{11}</M>, using the three different strategies available in &GAP;.
First we try the algorithm that enumerates all candidates via solving
a system of inequalities,
which is described in <Cite Key="BP98copy" Where="Section 3.2"/>.
<!-- %T This algorithm admits a second search strategy, which uses the inequalities --> <!-- %T for a sort of previewing. --> <!-- %T But this does not work correctly, --> <!-- %T already the GAP 3 version did not do what was promised. --> <!-- %T (Apparently, the additional options were used rarely.) -->
Next we try the improved combinatorial approach that is sketched at the
end of Section 3.2 in <Cite Key="BP98copy"/>.
We get the same characters, except that they may be ordered in a different
way; thus we compare the ordered lists.
<P/>
<Example><![CDATA[
gap> degrees:= DivisorsInt( Size( m11 ) );;
gap> perms2:= [];;
gap> for d in degrees do
> Append( perms2, PermChars( m11, d ) );
> od;
gap> Set( perms ) = Set( perms2 );
true
]]></Example>
<P/>
Finally, we try the algorithm that is based on Gaussian elimination
and that is described in <Cite Key="BP98copy" Where="Section 3.3"/>.
<P/>
<Example><![CDATA[
gap> perms3:= [];;
gap> for d in degrees do
> Append( perms3, PermChars( m11, rec( torso:= [ d ] ) ) );
> od;
gap> Set( perms ) = Set( perms3 );
true
]]></Example>
<P/>
&GAP; provides two more functions to test properties of permutation
characters.
The first one yields no new information in our case,
but the second excludes one possible permutation character;
note that <C>TestPerm5</C> needs a <M>p</M>-modular Brauer table,
and the &GAP; character table library contains all Brauer tables
of <M>M_{11}</M>.
&GAP; knows the table of marks of <M>M_{11}</M>,
from which the permutation characters can be extracted.
It turns out that <M>M_{11}</M> has <M>39</M> conjugacy classes of subgroups
but only <M>36</M> different permutation characters,
so three candidates computed above are in fact not permutation characters.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:act_U62_M22">
<Heading>The Action of <M>U_6(2)</M> on the Cosets of <M>M_{22}</M></Heading>
We are interested in the permutation character of <M>U_6(2)</M>
(see <Cite Key="CCN85" Where="p. 115"/>)
that corresponds to the action on the cosets of a <M>M_{22}</M> subgroup
(see <Cite Key="CCN85" Where="p. 39"/>).
The character tables of both the group and the point stabilizer
are available in the &GAP; character table library,
so we can compute class fusion and permutation character directly;
note that if the class fusion is not stored on the table of the subgroup,
in general one will not get a unique fusion but only a list of candidates
for the fusion.
The six fusions induce three different characters,
they are conjugate under the action of the unique subgroup of order <M>3</M>
in the group of table automorphisms of <M>U_6(2)</M>.
The table automorphisms of order <M>3</M> are induced by group automorphisms
of <M>U_6(2)</M> (see <Cite Key="CCN85" Where="p. 120"/>).
As can be seen from the list of maximal subgroups of <M>U_6(2)</M>
in <Cite Key="CCN85" Where="p. 115"/>,
the three induced characters are in fact permutation characters
which belong to the three classes of maximal subgroups of type <M>M_{22}</M>
in <M>U_6(2)</M>, which are permuted by an outer automorphism of order 3.
Now we want to compute the extension of the above permutation character
to the group <M>U_6(2).2</M>,
which corresponds to the action of this group on the cosets of a <M>M_{22}.2</M>
subgroup.
We see that for the embedding of <M>M_{22}.2</M> into <M>U_6(2).2</M>,
the class fusion is unique,
so we get a unique extension of one of the above permutation characters.
This implies that exactly one class of maximal subgroups of type <M>M_{22}</M>
extends to <M>M_{22}.2</M> in a given group <M>U_6(2).2</M>.
Now we show an alternative way to compute the characters dealt with
in the previous example.
This works also if the character table of the point stabilizer is not
available.
In this situation we can compute all those characters that have certain
properties of permutation characters.
Of course this may take much longer than the above computations,
which needed only a few seconds.
(The following calculations may need several hours,
depending on the computer used.)
For the next step, that is, the computation of the extension of the
permutation character to <M>U_6(2).2</M>, we may use the above information,
since the values on the inner classes are prescribed.
The question which of the three candidates for <M>U_6(2)</M> extends to
<M>U_6(2).2</M> depends on the choice of the class fusion of <M>U_6(2)</M> into
<M>U_6(2).2</M>.
With respect to the class fusion that is stored on the &GAP; library table,
the third candidate extends,
as can be seen from the fact that this one is invariant under the
permutation of conjugacy classes of <M>U_6(2)</M> that is induced by the
action of the chosen supergroup <M>U_6(2).2</M>.
The group <M>O_8^+(3)</M> (see <Cite Key="CCN85" Where="p. 140"/>)
contains a subgroup of type <M>2^{{3+6}}.L_3(2)</M>,
which extends to a maximal subgroup <M>U</M> in <M>O_8^+(3).3</M>.
For the computation of the permutation character,
we cannot use explicit induction since the table of <M>U</M> is not available
in the &GAP; table library.
Since <M>U \cap O_8^+(3)</M> is contained in a <M>O_8^+(2)</M> subgroup
of <M>O_8^+(3)</M>, we can try to find the permutation character of <M>O_8^+(2)</M>
corresponding to the action on the cosets of <M>U \cap O_8^+(3)</M>,
and then induce this character to <M>O_8^+(3)</M>.
This kind of computations becomes more difficult with increasing degree,
so we try to reduce the problem further.
In fact, the <M>2^{{3+6}}.L_3(2)</M> group is contained in a <M>2^6:A_8</M> subgroup
of <M>O_8^+(2)</M>, in which the index is only <M>15</M>;
the unique possible permutation character of this degree can be read off
immediately.
Induction to <M>O_8^+(3)</M> through the chain of subgroups is possible
provided the class fusions are available.
There are <M>24</M> possible fusions from <M>O_8^+(2)</M> into <M>O_8^+(3)</M>,
which are all equivalent w.r.t. table automorphisms of <M>O_8^+(3)</M>.
If we later want to consider the extension of the permutation character
in question to <M>O_8^+(3).3</M> then we have to choose a fusion of an
<M>O_8^+(2)</M> subgroup that does <E>not</E> extend to <M>O_8^+(2).3</M>.
But if for example our question is just whether the resulting permutation
character is multiplicity-free then this can be decided already from the
permutation character of <M>O_8^+(3)</M>.
<!-- %T Alternatively, if the table of marks of the group is available --> <!-- %T then one can extract the permutation characters from it. --> <!-- %T The table of marks of <M>O_8^+(2)</M> is in fact available, --> <!-- %T but it requires some time to read the data into &GAP;, --> <!-- %T since the file is about <M>25</M> MB large. -->
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:action_O73d2_27S7">
<Heading>The Action of <M>O_7(3).2</M> on the Cosets of <M>2^7.S_7</M></Heading>
We want to know whether the permutation character of <M>O_7(3).2</M>
(see <Cite Key="CCN85" Where="p. 108"/>)
on the cosets of its maximal subgroup <M>U</M>
of type <M>2^7.S_7</M> is multiplicity-free.
As in the previous examples, first we try to compute the permutation
character of the simple group <M>O_7(3)</M>.
It turns out that the direct computation of all candidates from the
degree is very time consuming.
But we can use for example the additional information provided by the fact
that <M>U</M> contains an <M>A_7</M> subgroup.
We compute the possible class fusions.
We cannot decide easily which fusion is the right one,
but already the fact that no other fusions are possible
gives us some information about impossible constituents of the
permutation character we want to compute.
Every order <M>3</M> element of <M>U</M> lies in an <M>A_7</M> subgroup of <M>U</M>,
so among the classes of element order <M>3</M>, at most the classes <C>3B</C>, <C>3C</C>,
and <C>3F</C> can have nonzero permutation character values.
The excluded classes of element order <M>6</M> are the square roots of the
excluded order <M>3</M> elements,
likewise the given classes of element orders <M>9</M>, <M>12</M>, and <M>18</M> are
excluded.
The character value on <C>20A</C> must be zero because <M>U</M> does not contain
elements of this order.
So we enter the additional information about these zeros.
We see that this character is already multiplicity free,
so this holds also for its extension to <M>O_7(3).2</M>,
and we need not compute this extension.
(Of course we could compute it in the same way as in the examples above.)
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:action_O8p3d2a_27A8">
<Heading>The Action of <M>O_8^+(3).2_1</M> on the Cosets of <M>2^7.A_8</M></Heading>
We are interested in the permutation character of <M>O_8^+(3).2_1</M>
that corresponds to the action on the cosets of a subgroup of type
<M>2^7.A_8</M>.
The intersection of the point stabilizer with the simple group <M>O_8^+(3)</M>
is of type <M>2^6.A_8</M>.
First we compute the class fusion of these groups,
modulo problems with ambiguities due to table automorphisms.
The ambiguities due to Galois automorphisms disappear when we are
looking for the permutation characters induced by the fusions.
<P/>
<Example><![CDATA[
gap> ind:= List( fus, x -> Induced( sub, o8p3,
> [ TrivialCharacter( sub ) ], x )[1] );;
gap> ind:= Set( ind );;
gap> Length( ind );
6
]]></Example>
<P/>
Now we try to extend the candidates to <M>O_8^+(3).2_1</M>;
the choice of the fusion of <M>O_8^+(3)</M> into <M>O_8^+(3).2_1</M> determines
which of the candidates may extend.
We might be interested in the extension to <M>O_8^+(3).(2^2)_{122}</M>.
It is clear that this cannot be multiplicity free
because of the multiplicity <C>9450aaa</C> in the character
induced from <M>O_8^+(3).2_2</M>.
We could put the extensions to the index two subgroups together,
but it is simpler (and not expensive) to run the same program as above.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:action_S44d4_5225">
<Heading>The Action of <M>S_4(4).4</M> on the Cosets of <M>5^2.[2^5]</M></Heading>
We want to know whether the permutation character corresponding to the
action of <M>S_4(4).4</M> (see <Cite Key="CCN85" Where="p. 44"/>)
on the cosets of its
maximal subgroup of type <M>5^2:[2^5]</M> is multiplicity free.
The library names of subgroups for which the class fusions are stored
are listed as value of the attribute
<Ref Func="NamesOfFusionSources" BookName="ref"/>,
and for groups whose isomorphism type is not determined by the name
this is the recommended way to find out whether the table of the subgroup
is contained in the &GAP; library and known to belong to this group.
(It might be that a table with such a name is contained in the library
but belongs to another group,
and it may also be that the table of the group is contained in the
library --with any name-- but it is not known that this group is
isomorphic to a subgroup of <M>S_4(4).4</M>.)
So there are three candidates.
None of them is multiplicity free,
so we need not decide which of the candidates actually belongs
to the group <M>5^2:[2^5]</M> we have in mind.
(If we would be interested which candidate is the right one,
we could for example look at the intersection with <M>S_4(4)</M>,
and hope for a contradiction to the fact that the group must lie
in a <M>(A_5 \times A_5):2</M> subgroup.)
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:action_Co1_inv_centralizers">
<Heading>The Action of <M>Co_1</M> on the Cosets of Involution Centralizers</Heading>
We compute the permutation characters of the sporadic simple Conway group
<M>Co_1</M> (see <Cite Key="CCN85" Where="p. 180"/>) corresponding to
the actions on the cosets of involution centralizers.
Equivalently, we are interested in the action of <M>Co_1</M> on conjugacy
classes of involutions.
These characters can be computed as follows.
First we take the table of <M>Co_1</M>.
The centralizer of each <C>2A</C> element is a maximal subgroup of <M>Co_1</M>.
This group is also contained in the table library.
So we can compute the permutation character by explicit induction,
and the decomposition in irreducibles is computed with the command
<Ref Func="PermCharInfo" BookName="ref"/>.
The centralizer of a <C>2B</C> element is not maximal.
First we compute which maximal subgroup can contain it.
The character tables of all maximal subgroups of <M>Co_1</M> are contained
in the &GAP;'s table library,
so we may take these tables and look at the group orders.
<P/>
<Example><![CDATA[
gap> centorder:= SizesCentralizers( t )[3];;
gap> maxes:= List( Maxes( t ), CharacterTable );;
gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 );
[ CharacterTable( "(A4xG2(4)):2" ) ]
gap> u:= cand[1];;
gap> index:= Size( u ) / centorder;
3
]]></Example>
<P/>
So there is a unique class of maximal subgroups containing the centralizer
of a <C>2B</C> element, as a subgroup of index <M>3</M>.
We compute the unique permutation character of degree <M>3</M> of this group,
and induce this character to <M>G</M>.
<!-- %T In fact we compute all those characters that have certain properties --> <!-- %T of permutation characters. --> <!-- %T In our situation, this is sufficient. --> <!-- %T --> <!-- %T For a very small degree, the algorithm that needs the <C>torso</C> component --> <!-- %T of the options record is a good choice. --> <!-- %T If one wants to use the combinatorial algorithm for a small degree and --> <!-- %T a not very small number of conjugacy classes, one should suppress the --> <!-- %T computation of bounds by setting the <C>bounds</C> component to <C>false</C>. -->
Finally, we try the same for the centralizer of a <C>2C</C> element.
<P/>
<Example><![CDATA[
gap> centorder:= SizesCentralizers( t )[4];;
gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 );
[ CharacterTable( "Co2" ), CharacterTable( "2^11:M24" ) ]
]]></Example>
<P/>
The group order excludes all except two classes of maximal subgroups.
But the <C>2C</C> centralizer cannot lie in <M>Co_2</M> because the involution
centralizers in <M>Co_2</M> are too small.
We compute the multiplicity free possible permutation characters of
<M>G_2(3)</M> (see <Cite Key="CCN85" Where="p. 60"/>).
For each divisor <M>d</M> of the group order,
we compute all those possible permutation
characters of degree <M>d</M> of <M>G</M> for which each irreducible constituent
occurs with multiplicity at most <M>1</M>;
this is done by prescribing the <C>maxmult</C> component of the second argument
of
<Ref Func="PermChars" BookName="ref"/>
to be the list with <M>1</M> at each position.
For finding out which of these candidates are really permutation
characters, we could inspect them piece by piece, using the information
in <Cite Key="CCN85"/>.
For example, the candidates of degrees <M>351</M>, <M>364</M>, and <M>378</M> are
induced from the trivial characters of maximal subgroups of <M>G</M>,
whereas the candidates of degree <M>546</M> are not permutation characters.
<P/>
Since the table of marks of <M>G</M> is available in &GAP;,
we can extract all permutation characters from the table of marks,
and then filter out the multiplicity free ones.
We compute the primitive permutation characters of degree <M>11\,200</M> of
<M>O_8^+(2)</M> and <M>O_8^+(2).2</M>
(see <Cite Key="CCN85" Where="p. 85"/>).
The character table of the maximal subgroup of type <M>3^4:2^3.S_4</M> in
<M>O_8^+(2)</M> is not available in the &GAP; table library.
But the group extends to a wreath product of <M>S_3</M> and <M>S_4</M> in the
group <M>O_8^+(2).2</M>, and the table of this wreath product can be
constructed easily.
The permutation character <C>pi</C> of <M>O_8^+(2).2</M> can thus be computed by
explicit induction, and the character of <M>O_8^+(2)</M> is obtained by
restriction of <C>pi</C>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:proof_nonexistence">
<Heading>A Proof of Nonexistence of a Certain Subgroup</Heading>
We prove that the sporadic simple Mathieu group <M>G = M_{22}</M>
(see <Cite Key="CCN85" Where="p. 39"/>)
has no subgroup of index <M>56</M>.
In <Cite Key="Isa76"/>, remark after Theorem 5.18, this is stated as an example
of the case that a character may be a possible permutation character but
not a permutation character.
Let us consider the possible permutation character of degree <M>56</M> of
<M>G</M>.
Suppose that <C>pi</C> is a permutation character of <M>G</M>.
Since <M>G</M> is <M>2</M>-transitive on the <M>56</M> cosets of the point stabilizer <M>S</M>,
this stabilizer is transitive on <M>55</M> points,
and thus <M>G</M> has a subgroup <M>U</M> of index <M>56 \cdot 55 = 3080</M>.
We compute the possible permutation character of this degree.
<M>U</M> is contained in <M>S</M>, so only those candidates must be considered
that vanish on all classes where <C>pi</C> vanishes.
Furthermore, the index of <M>U</M> in <M>S</M> is odd, so the Sylow <M>2</M> subgroups
of <M>U</M> and <M>S</M> are isomorphic;
<M>S</M> contains elements of order <M>8</M>, hence also <M>U</M> does.
For getting an overview of the distribution of the elements of <M>U</M> to the
conjugacy classes of <M>G</M>, we use the output of
<Ref Func="PermCharInfo" BookName="ref"/>.
We have four candidates.
For each the above list shows first the character values,
then the cardinality of the intersection of <M>U</M> with the classes,
and then lower bounds for the lengths of <M>U</M>-conjugacy classes of these
elements.
Only those classes of <M>G</M> are shown that contain elements of <M>U</M>
for at least one of the characters.
<P/>
If the first two candidates are permutation characters corresponding to
<M>U</M> then <M>U</M> contains exactly <M>8</M> elements of order <M>3</M>
and thus <M>U</M> has a normal Sylow <M>3</M> subgroup <M>P</M>.
But the order of <M>N_G(P)</M> is bounded by <M>72</M>,
which can be shown as follows.
The only elements in <M>G</M> with centralizer order divisible by <M>9</M>
are of order <M>1</M> or <M>3</M>, so <M>P</M> is self-centralizing in <M>G</M>.
The factor <M>N_G(P)/C_G(P)</M> is isomorphic with a subgroup of
Aut<M>(G) \cong GL(2,3)</M> which has order divisible
by <M>16</M>, hence the order of <M>N_G(P)</M> divides <M>144</M>.
Now note that <M>[ G : N_G(P) ] \equiv 1 \pmod{3}</M> by Sylow's Theorem,
and <M>|G|/144 = 3\,080 \equiv -1 \pmod{3}</M>.
Thus the first two candidates are not permutation characters.
<P/>
If the last two candidates are permutation characters corresponding to
<M>U</M> then <M>U</M> has self-normalizing Sylow subgroups.
This is because the index of a Sylow <M>2</M> normalizer
in <M>G</M> is odd and divides <M>9</M>,
and if it is smaller than <M>9</M> then <M>U</M> contains
at most <M>3 \cdot 15 + 1</M> elements of <M>2</M> power order;
the index of a Sylow <M>3</M> normalizer
in <M>G</M> is congruent to <M>1</M> modulo <M>3</M> and divides <M>16</M>,
and if it is smaller than <M>16</M> then <M>U</M> contains
at most <M>4 \cdot 8</M> elements of order <M>3</M>.
<P/>
But since <M>U</M> is solvable and not a <M>p</M>-group,
not all its Sylow subgroups can be self-normalizing;
note that <M>U</M> has a proper normal subgroup <M>N</M> containing
a Sylow <M>p</M> subgroup <M>P</M> of <M>U</M> for a prime divisor <M>p</M> of <M>|U|</M>,
and <M>U = N \cdot N_U(P)</M> holds by the Frattini argument
(see <Cite Key="Hup67" Where="Satz I.7.8"/>).
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:character_Lyons">
<Heading>A Permutation Character of the Lyons group</Heading>
Let <M>G</M> be a maximal subgroup with structure <M>3^{{2+4}}:2A_5.D_8</M>
in the sporadic simple Lyons group <M>Ly</M>.
We want to compute the permutation character <M>1_G^{Ly}</M>.
(This construction has been explained
in <Cite Key="BP98copy" Where="Section 4.2"/>,
without showing explicit &GAP; code.)
<P/>
In the representation of <M>Ly</M> as automorphism group of the rank <M>5</M>
graph <C>B</C> with <M>9\,606\,125</M> points
(see <Cite Key="CCN85" Where="p. 174"/>),
<M>G</M> is the stabilizer of an edge.
A group <M>S</M> with structure <M>3.McL.2</M> is the point stabilizer.
So the two point stabilizer <M>U = S \cap G</M> is a subgroup of index <M>2</M>
in <M>G</M>.
The index of <M>U</M> in <M>S</M> is <M>15\,400</M>, and according to the list of
maximal subgroups of <M>McL.2</M>
(see <Cite Key="CCN85" Where="p. 100"/>),
the group <M>U</M> is isomorphic to the preimage in <M>3.McL.2</M> of a subgroup <M>H</M>
of <M>McL.2</M> with structure <M>3_+^{{1+4}}:4S_5</M>.
<P/>
Using the improved combinatorial method described
in <Cite Key="BP98copy" Where="Section 3.2"/>,
all possible permutation characters of degree <M>15\,400</M> for the group <M>McL</M>
are computed.
(The method of <Cite Key="BP98copy" Where="Section 3.3"/>
is slower but also needs only a few seconds.)
We get two characters, corresponding to the two classes of maximal
subgroups of index <M>15\,400</M> in <M>McL</M>.
The permutation character <M>\pi = 1_{{H \cap McL}}^{McL}</M> is the one with
nonzero value on the class <C>10A</C>, since the subgroup of structure
<M>2S_5</M> in <M>H \cap McL</M> contains elements of order <M>10</M>.
The character <M>1_H^{McL.2}</M> is an extension of <M>\pi</M>,
so we can use the method of <Cite Key="BP98copy" Where="Section 3.3"/>
to compute all
possible permutation characters for the group <M>McL.2</M> that have
the values of <M>\pi</M> on the classes of <M>McL</M>.
We find that the extension of <M>\pi</M> to a permutation character of <M>McL.2</M>
is unique.
Regarded as a character of <M>3.McL.2</M>, this character is equal to <M>1_U^S</M>.
The fusion of conjugacy classes of <M>S</M> in <M>Ly</M> can be computed from
the character tables of <M>S</M> and <M>Ly</M> given in <Cite Key="CCN85"/>,
it is unique up to Galois automorphisms of the table of <M>Ly</M>.
so we can prescribe the values of <M>1_G^{Ly}</M> on all classes of odd element order.
For elements <M>g</M> of even order we have the weaker condition
<M>U\cap Cl_{Ly}(g) \subseteq G \cap Cl_{Ly}(g)</M>
and thus <M>1_G^{Ly}(g) \geq 1/2 \cdot 1_U^{Ly}(g)</M>,
which gives lower bounds for the value of <M>1_G^{Ly}</M> on the
remaining classes.
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