Multiplicity-Free Permutation Characters in GAP, part 2

Thomas Breuer

Lehrstuhl D für Mathematik RWTH, 52056 Aachen, Germany

July 21th, 2003

Abstract

We complete the classification of the multiplicity-free permutation actions of nearly simple groups that involve a sporadic simple group, which had been started in [BL96] and [LM].

Contents

1 Introduction 2

2 The Approach 3 2.1 Computing Possible Permutation Characters . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Verifying the Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Isoclinic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Tests for GAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 The Groups 8 3.1 G = 2.M_{12} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 G = 2.M_{12}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 G = 2.M_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 G = 2.M_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 G = 3.M_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.6 G = 3.M_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.7 G = 4.M_{22} and G = 12.M_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.8 G = 4.M_{22}.2 and G = 12.M_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.9 G = 6.M_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.10 G = 6.M_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.11 G = 2.J_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.12 G = 2.J_{2}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.13 G = 2.HS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.14 G = 2.HS.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.15 G = 3.J_{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.16 G = 3.J_{3}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.17 G = 3.McL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.18 G = 3.McL.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.19 G = 2.Ru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.20 G = 2.Suz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.21 G = 2.Suz.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.22 G = 3.Suz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.23 G = 3.Suz.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.24 G = 6.Suz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.25 G = 6.Suz.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.26 G = 3.ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.27 G = 3.ON.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.28 G = 2.F i_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.29 G = 2.F i_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.30 G = 3.F i_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.31 G = 3.F i_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.32 G = 6.F i_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.33 G = 6.F i_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.34 G = 2.Co_{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.35 G = 3.F_{3+} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.36 G = 3.F_{3+}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.37 G = 2.B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Appendix: Explicit Computations with Groups 33 4.1 2^{4} : A_{6} type subgroups in 2.M_{22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 2^{4} : S_{5} type subgroups in M_{22}.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1 Introduction

In [BL96], the multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups were classified. Based on this list, the multiplicity-free permutation characters of the central extensions of the sporadic simple groups were classified in [LM]. The purpose of this writeup is to show how the multiplicity-free permutation characters of the automorphic extensions of the central extensions of the sporadic simple groups can be computed, to verify the calculations in [LM] (and to correct an error, see Section 3.32), and to provide a testfile for the GAP functions and the database. The database has been extended in the sense that also most of the character tables of the multiplicityfree permutation modules of the sporadic simple groups and their automorphic and central extensions have been computed, see [Höh01, Mül03] for details.

2 The Approach

Suppose that a group G contains a normal subgroup N. If π is a faithful multiplicity-free permutation character of G then π = 1_{U}^{G} for a subgroup U of G that intersects N trivially, so π contains a constituent 1_{UN}^{G} of degree π(1)/|N|, which can be viewed as a multiplicity-free permutation character of the factor group G/N. Moreover, no constituent of the difference π − 1_{UN}^{G} has N in its kernel.

So if we know all multiplicity-free permutation characters of the factor group G/N then we can compute all candidates for multiplicity-free permutation characters of G by “filling up” each such character \overline{π} with a linear combination of characters not containing N in their kernels, of total degree (|N| − 1) · π(1), and such that the sum is a possible permutation character of G. For this situation, GAP provides a special variant of the function PermChars. In a second step, the candidates are inspected whether the required point stabilizers (and if yes, how many conjugacy classes of them) exist. Finally, the permutation characters are verified by explicit induction from the character tables of the point stabilizers.

The multiplicity-free permutation actions of the sporadic simple groups and their automorphism groups are known by [BL96], so this approach is suitable for these groups.

For central extensions of sporadic simple groups, the multiplicity-free permutation characters have been classified in [LM]; this note describes a slightly different approach, so we will give an independent confirmation of their results.

First we load the Character Table Library [Bre04b] of the GAP system [GAP04], read the file with GAP functions for computing multiplicity-free permutation characters, and the known data for the sporadic simple groups and their automorphism groups.

    gap> LoadPackage( "ctbllib" );
    true
    gap> RereadPackage( "ctbllib", "tst/multfree.g" );
    true
    gap> RereadPackage( "ctbllib", "tst/multfree.dat" );
    true

In order to compare the results computed below with the database that is already available, we load also this database.

    gap> RereadPackage( "ctbllib", "tst/multfre2.dat" );
    true

2.1 Computing Possible Permutation Characters

Then we define the GAP functions that are needed in the following.

FaithfulCandidates takes the character table tbl of a group G and the name factname of a factor group F of G for which the multiplicity-free permutation characters are known, and returns a list of lists, the entry at the i-th position being the list of possible permutation characters of G that are multiplicity-free and such that the sum of all constituents that are characters of F is the i-th multiplicity-free permutation character of F. As a side-effect, if the i-th entry is nonempty then information is printed about the structure of the point-stabilizer in F and the number of candidates found.

    gap> FaithfulCandidates:= function( tbl, factname )
    >    local factinfo, factchars, facttbl, fus, sizeN, faith, i;
    >
    >    # Fetch the data for the factor group.
    >    factinfo:= MultFreePermChars( factname );

>    factchars:= List( factinfo, x -> x.character );
>    facttbl:= UnderlyingCharacterTable( factchars[1] );
>    fus:= GetFusionMap( tbl, facttbl );
>    sizeN:= Size( tbl ) / Size( facttbl );
>
>    # Compute faithful possible permutation characters.
>    faith:= List( factchars, pi -> PermChars( tbl,
>                      rec( torso:= [ sizeN * pi[1] ],
>                           normalsubgroup:= ClassPositionsOfKernel( fus ),
>                           nonfaithful:= pi{ fus } ) ) );
>
>    # Take only the multiplicity-free ones.
>    faith:= List( faith, x -> Filtered( x, pi -> ForAll( Irr( tbl ),
>                      chi -> ScalarProduct( tbl, pi, chi ) < 2 ) ) );
>
>    # Print info about the candidates.
>    for i in [ 1 .. Length( faith ) ] do
>      if not IsEmpty( faith[i] ) then
>        Print( i, ":  subgroup ", factinfo[i].subgroup,
>               ", degree ", faith[i][1][1],
>               " (", Length( faith[i] ), " cand.)\n" );
>      fi;
>    od;
>
>    # Return the candidates.
>    return faith;
>    end;;

2.2 Verifying the Candidates

In the verification step, we check which of the given candidates of G are induced from a given subgroup S. For that, we use the following function. Its arguments are the character table s of S, the character tables tbl2 and tbl of G and its derived subgroup G′ of index 2 (if G is perfect then 0 must be entered for tbl2), the list candidates of characters of G, and one of the strings "all", "extending", which means that we consider either all possible class fusions of s into tbl2 or only those whose image does not lie in G′. Note that the table of the derived subgroup of G is needed because we want to express the decomposition of the permutation characters relative to G′.

The idea is that we know that n different permutation characters arise from subgroups isomorphic with S (with the additional property that the image of the embedding of S into G is not contained in G′ if the last argument is "extending"), and that candidates is a set of possible permutation characters, of length n. If the possible fusions between the character tables s and tbl2 lead to exactly to the given n permutation characters then we have proved that they are in fact the permutation characters of G in question. In this case, VerifyCandidates prints information about the decomposition of the permutation characters. If none of candidates arises from the possible embeddings of S into G then the function prints that S does not occur. In all other cases, the function signals an error (so this will not happen in the calls to this function below).

gap> VerifyCandidates:= function( s, tbl, tbl2, candidates, admissible )
>    local fus, der, pi;
>
>    if tbl2 = 0 then
>      tbl2:= tbl;
>    fi;

>
>    # Compute the possible class fusions, and induce the trivial character.
>    fus:= PossibleClassFusions( s, tbl2 );
>    if admissible = "extending" then
>      der:= Set( GetFusionMap( tbl, tbl2 ) );
>      fus:= Filtered( fus, map -> not IsSubset( der, map ) );
>    fi;
>    pi:= Set( List( fus, map -> Induced( s, tbl2,
>            [ TrivialCharacter( s ) ], map )[1] ) );
>
>    # Compare the two lists.
>    if pi = SortedList( candidates ) then
>      Print( "G = ", Identifier( tbl2 ), ":  point stabilizer ",
>             Identifier( s ), ", ranks ",
>             List( pi, x -> Length( ConstituentsOfCharacter(x) ) ), "\n" );
>      if Size( tbl ) = Size( tbl2 ) then
>        Print( PermCharInfo( tbl, pi ).ATLAS, "\n" );
>      else
>        Print( PermCharInfoRelative( tbl, tbl2, pi ).ATLAS, "\n" );
>      fi;
>    elif IsEmpty( Intersection( pi, candidates ) ) then
>      Print( "G = ", Identifier( tbl2 ), ":  no ", Identifier( s ), "\n" );
>    else
>      Error( "problem with verify" );
>    fi;
>    end;;

Since in most cases the character tables of possible point stabilizers are contained in the GAP Character Table Library, the above function provides an easy test. Alternatively, we could compute all faithful possible permutation characters (not only the multiplicity-free ones) of the degree in question; if there are as many different such characters as are known to be induced from point stabilizers and if no other subgroups of this index exist then the characters are indeed permutation characters, and we can compare them with the multiplicity-free characters computed before. In the verification of the candidates, the following situations occur.

Lemma 2.1 Let Φ: Ĝ → G be a group epimorphism, with K = ker(Φ) cyclic of order m, and let H be a subgroup of G such that m is coprime to the order of the commutator factor group of H. If it is known that Φ^{−1}(H) is a direct product of H with K then the preimages under Φ of the G-conjugates of H in Ĝ contain one Ĝ-class of subgroups that are isomorphic with H and intersect trivially with K. (This holds for example if the order of the Schur multiplier of H is coprime to m.)

Lemma 2.2 Let Φ: Ĝ → G be a group epimorphism, with K = ker(Φ) of order 3, such that the derived subgroup G′ of G has index 2 in G and such that K is not central in G. Consider a subgroup H of G with a subgroup H_{0} = H ∩ G′ of index 2 in H, and such that it is known that the preimage Φ^{−1}(H_{0}) is a direct product of H_{0} with K. (This holds for example if the order of the Schur multiplier of H_{0} is coprime to 3.) Then each complement of K in Φ^{−1}(H_{0}) extends in Φ^{−1}(H) to exactly one complement of K that is isomorphic with H.

Lemma 2.3 Let Φ: Ĝ → G be a group epimorphism, with K = ker(Φ) of order 2. Consider a subgroup H of G, with derived subgroup H′ of index 2 in H and such that Φ^{−1}(H′) has the structure 2 × H′.

(i) Suppose that there is an element h ∈ H \ H′ such that the squares of the preimages of h in Ĝ lie in the unique subgroup of index 2 in Φ^{−1}(H′). (This holds for example if the preimages of h are involutions.) Then Φ^{−1}(H) has the type K × H.

(ii) If Φ^{−1}(H) has the type K ×H then this group contains exactly two subgroups that are isomorphic with H.

(iii) Suppose that case (ii) applies and that there is h ∈ H \H′ whose two preimages are not conjugate in Ĝ and such that none of the two subgroups of the type H in Φ^{−1}(H) contains elements in the two conjugacy classes that contain the preimages of h. Then the two subgroups of the type H induce different permutation characters of Ĝ, in particular exactly two conjugacy classes of subgroups of the type H in Ĝ arise from the conjugates of H in G.

With character theoretic methods, we can check a weaker form of Lemma 2.3 (i). Namely, the conditions are clearly satisfied if there is a conjugacy class C of elements in H that is not contained in H′ and such that the class of Ĝ that contains the squares of the preimages of C is not contained in the images of the classes of 2 × H′ that lie outside H′.

The function CheckConditionsForLemma3 tests this, and prints a message if Lemma 2.3 (i) applies becaue of this situation. More precisely, the arguments are (in this order) the character tables of H′, H, G, and Ĝ, and one of the strings "all", "extending"; the last argument means that either all embeddings of H into G are considered or only those which do not lie inside the derived subgroup of G.

The function assumes that s0 is the character table of the derived subgroup of s, and that H′ lifts to a direct product in Ĝ.

gap> CheckConditionsForLemma3:= function( s0, s, fact, tbl, admissible )
>    local s0fuss, sfusfact, der, outerins, outerinfact, preim, squares, dp,
>          dpfustbl, s0indp, other, goodclasses;
>
>    if Size( s ) <> 2 * Size( s0 ) then
>      Error( "<s> must be twice as large as <s0>" );
>    fi;
>
>    s0fuss:= GetFusionMap( s0, s );
>    sfusfact:= PossibleClassFusions( s, fact );
>    if admissible = "extending" then
>      der:= ClassPositionsOfDerivedSubgroup( fact );
>      sfusfact:= Filtered( sfusfact, map -> not IsSubset( der, map ) );
>    fi;
>    outerins:= Difference( [ 1 .. NrConjugacyClasses( s ) ], s0fuss );
>    outerinfact:= Set( List( sfusfact, map -> Set( map{ outerins } ) ) );
>    if Length( outerinfact ) <> 1 then
>      Error( "classes of `", s, "' inside `", fact, "' not determined" );
>    fi;
>
>    preim:= Flat( InverseMap( GetFusionMap( tbl, fact ) ){ outerinfact[1] } );
>    squares:= Set( PowerMap( tbl, 2 ){ preim } );
>    dp:= s0 * CharacterTable( "Cyclic", 2 );
>    dpfustbl:= PossibleClassFusions( dp, tbl );
>    s0indp:= GetFusionMap( s0, dp );
>    other:= Difference( [ 1 .. NrConjugacyClasses( dp ) ], s0indp );
>    goodclasses:= List( dpfustbl, map -> Intersection( squares,
>                            Difference( map{ s0indp }, map{ other } ) ) );
>    if not IsEmpty( Intersection( goodclasses ) ) then
>      Print( Identifier( tbl ), ":  ", Identifier( s ),
>             " lifts to a direct product,\n",
>             "proved by squares in ", Intersection( goodclasses ), ".\n" );
>    elif ForAll( goodclasses, IsEmpty ) then

>      Print( Identifier( tbl ), ":  ", Identifier( s ),
>             " lifts to a nonsplit extension.\n" );
>    else
>      Print( "sorry, no proof of the splitting!\n" );
>    fi;
>    end;;

Let us assume we are in the situation of Lemma 2.3. Then Φ^{−1}(H) is a direct product ⟨z⟩×H, where z is an involution. The derived subgroup of Φ^{−1}(H) is H_{0} ≡ H′, and Φ^{−1}(H) contains two subgroups H_{1}, H_{2} which are isomorphic with H, and such that H_{2} = H_{0} ∪ {hz; h ∈ H_{1} \ H_{0}}. If the embedding of H_{1}, say, into Ĝ has the properties that an element outside H_{0} is mapped into a class C of Ĝ that is different from zC and such that no element of H_{1} lies in zC then zC contains element of H_{2} but C does not. In particular, the permutation characters of the two actions of Ĝ on the cosets of H_{1} and H_{2}, respectively, are necessarily different.

We check this with the following function. Its arguments are one class fusion from the character table of H_{1} to that of Ĝ, the factor fusion from the character table of Ĝ to that of G, and the list of positions of the classes of H_{0} in the character table of H_{1}. The return value is true if there are two different permutation characters, and false if this cannot be proved using the criterion.

gap> NecessarilyDifferentPermChars:= function( fusion, factfus, inner )
>    local outer, inv;
>
>    outer:= Difference( [ 1 .. Length( fusion ) ], inner );
>    fusion:= fusion{ outer };
>    inv:= Filtered( InverseMap( factfus ), IsList );
>    return ForAny( inv, pair -> Length( Intersection( pair, fusion ) ) = 1 );
>    end;;

2.3 Isoclinic Groups

For dealing with the character tables of groups of the type 2.G.2 that are isoclinic to those whose tables are printed in the Atlas ([CCN^{+}85]), it is necessary to store explicitly the factor fusion from 2.G.2 onto G.2 and the subgroup fusion from 2.G into 2.G.2, in order to make the above functions work. Note that these maps coincide for the two isoclinism types.

gap> IsoclinicTable:= function( tbl, tbl2, facttbl )
>    local subfus, factfus;
>
>    subfus:= GetFusionMap( tbl, tbl2 );
>    factfus:= GetFusionMap( tbl2, facttbl );
>    tbl2:= CharacterTableIsoclinic( tbl2 );
>    StoreFusion( tbl, subfus, tbl2 );
>    StoreFusion( tbl2, factfus, facttbl );
>    return tbl2;
>    end;;

2.4 Tests for GAP

With the following function, we check whether the characters computed here coincide with the lists computed in [LM].

gap> CompareWithDatabase:= function( name, chars )
>    local info;

>
>    info:= MultFreePermChars( name );
>    info:= List( info, x -> x.character );;
>    if SortedList( info ) <> SortedList( Concatenation( chars ) ) then
>      Error( "contradiction 1 for ", name );
>    fi;
>    end;;

If the character tables of all maximal subgroups of G are known then we could use alternatively the same method (and in fact the same GAP functions) as in the classification in [BL96]. This is shown in the following sections where applicable, using the following function.

gap> CompareWithCandidatesByMaxes:= function( name, faith )
>    local tbl, poss;
>
>    tbl:= CharacterTable( name );
>    if not HasMaxes( tbl ) then
>      Error( "no maxes stored for ", name );
>    fi;
>    poss:= PossiblePermutationCharactersWithBoundedMultiplicity( tbl, 1 );
>    poss:= List( poss.permcand, l -> Filtered( l,
>                 pi -> ClassPositionsOfKernel( pi ) = [ 1 ] ) );
>    if SortedList( Concatenation( poss ) )
>       <> SortedList( Concatenation( faith ) ) then
>      Error( "contradiction 2 for ", name );
>    fi;
>    end;;

3 The Groups

In the following, we use Atlas notation (see [CCN^{+}85]) for the names of the groups. In particular, 2 × G and G × 2 denote the direct product of the group G with a cyclic group of order 2, and G.2 and 2.G denote an upward and downward extension, respectively, of G by a cyclic group of order 2, such that these groups are not direct products.

For groups of the structure 2.G.2 where the character table of G is contained in the Atlas, we use the name 2.G.2 for the isoclinism type whose character table is printed in the Atlas, and (2.G.2)^{∗} for the other isoclinism type.

Most of the computations that are shown in the following use only information from the GAP Character Table Library. The (few) explicit computations with groups are collected in Section 4.

3.1 G = 2.M_{12}

The group 2.M_{12} has ten faithful multiplicity-free permutation actions, with point stabilizers of the types M_{11} (twice), A_{6}.2_{1} (twice), 3^{2}.2.S_{4} (four classes), and 3^{2} : 2.A_{4} (twice).

gap> tbl:= CharacterTable( "2.M12" );;
gap> faith:= FaithfulCandidates( tbl, "M12" );;
1:  subgroup $M_{11}$, degree 24 (1 cand.)
2:  subgroup $M_{11}$, degree 24 (1 cand.)
5:  subgroup $A_6.2_1 \leq A_6.2^2$, degree 264 (1 cand.)
8:  subgroup $A_6.2_1 \leq A_6.2^2$, degree 264 (1 cand.)
11:  subgroup $3^2.2.S_4$, degree 440 (2 cand.)

12:  subgroup $3^2:2.A_4 \leq 3^2.2.S_4$, degree 880 (1 cand.)
13:  subgroup $3^2.2.S_4$, degree 440 (2 cand.)
14:  subgroup $3^2:2.A_4 \leq 3^2.2.S_4$, degree 880 (1 cand.)

There are two classes of M_{11} subgroups in M_{12} as well as in 2.M_{12}, so we apply Lemma 2.1.

gap> VerifyCandidates( CharacterTable( "M11" ), tbl, 0,
>        Concatenation( faith[1], faith[2] ), "all" );
G = 2.M12:  point stabilizer M11, ranks [ 3, 3 ]
[ "1a+11a+12a", "1a+11b+12a" ]

According to the list of maximal subgroups of 2.M_{12}, any A_{6}.2^{2} subgroup in M_{12} lifts to a group of the structure A_{6}.D_{8} in M_{12}, which contains two conjugate subgroups of the type A_{6}.2_{1}; so we get two classes of such subgroups, with the same permutation character.

gap> Maxes( tbl );
[ "2xM11", "2.M12M2", "A6.D8", "2.M12M4", "2.L2(11)", "2x3^2.2.S4",
  "2.M12M7", "2.M12M8", "2.M12M9", "2.M12M10", "2.A4xS3" ]
gap> faith[5] = faith[8];
true
gap> VerifyCandidates( CharacterTable( "A6.2_1" ), tbl, 0, faith[5], "all" );
G = 2.M12:  point stabilizer A6.2_1, ranks [ 7 ]
[ "1a+11ab+12a+54a+55a+120b" ]

The 3^{2}.2.S_{4} type subgroups of M_{12} lift to direct products with the centre of 2.M_{12}, each such group contains two subgroups of the type 3^{2}.2.S_{4} which induce different permutation characters, for example because the involutions in 3^{2}.2.S_{4} \ 3^{2}.2.A_{4} lie in the two preimages of the class 2B of M_{12}.

gap> s:= CharacterTable( "3^2.2.S4" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable( "M12" );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, 0, Concatenation( faith[11], faith[13] ), "all" );
G = 2.M12:  point stabilizer 3^2.2.S4, ranks [ 7, 7, 9, 9 ]
[ "1a+11a+54a+55a+99a+110ab", "1a+11b+54a+55a+99a+110ab",
  "1a+11a+12a+44ab+54a+55a+99a+120b", "1a+11b+12a+44ab+54a+55a+99a+120b" ]

Each 3^{2}.2.S_{4} type group contains a unique subgroup of the type 3^{2}.2.A_{4}, we get two classes of such subgroups, with different permutation characters because already the corresponding characters for M_{12} are different; we verify the candidates by inducing the degree two permutation characters of the 3^{2}.2.S_{4} type groups to 2.M_{12}.

gap> fus:= PossibleClassFusions( s, tbl );;
gap> deg2:= PermChars( s, 2 );
[ Character( CharacterTable( "3^2.2.S4" ), [ 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0
     ] ) ]
gap> pi:= Set( List( fus, map -> Induced( s, tbl, deg2, map )[1] ) );;
gap> pi = SortedList( Concatenation( faith[12], faith[14] ) );
true
gap> PermCharInfo( tbl, pi ).ATLAS;
[ "1a+11a+12a+44ab+45a+54a+55ac+99a+110ab+120ab",
  "1a+11b+12a+44ab+45a+54a+55ab+99a+110ab+120ab" ]
gap> CompareWithDatabase( "2.M12", faith );
gap> CompareWithCandidatesByMaxes( "2.M12", faith );

3.2 G = 2.M_{12}.2

The group 2.M_{12}.2 that is printed in the Atlas has three faithful multiplicity-free permutation actions, with point stabilizers of the types M_{11} and L_{2}(11).2 (twice), respectively.

gap> tbl2:= CharacterTable( "2.M12.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M12.2" );;
1:  subgroup $M_{11}$, degree 48 (1 cand.)
2:  subgroup $L_2(11).2$, degree 288 (2 cand.)

The two classes of subgroups of the type M_{11} in 2.M_{12} are fused in 2.M_{12}.2, so we get one class of these subgroups.

gap> VerifyCandidates( CharacterTable( "M11" ), tbl, tbl2, faith[1], "all" );
G = 2.M12.2:  point stabilizer M11, ranks [ 5 ]
[ "1a^{\\pm}+11ab+12a^{\\pm}" ]

The outer involutions in the maximal subgroups of the type L_{2}(11).2 in M_{12}.2 lift to involutions in 2.M_{12}.2; moreover, those subgroups of the type L_{2}(11).2 that are novelties (so the intersection with M_{12} lies in M_{11} subgroups) contain 2B elements which lift to involutions in 2.M_{12}.2, so the L_{2}(11) subgroup lifts to a group of the type 2 × L_{2}(11), and Lemma 2.3 yields two classes of subgroups. The permutation characters are different, for example because each each of the two candidates contains elements in one of the two preimages of the class 2B. (The function CheckConditionsForLemma3 fails here, because of the two classes of maximal subgroups L_{2}(11).2 in M_{12}.2. One of them contains 2A elements, the other contains 2B elements. Only the latter type of subgroups, whose intersection with M_{12} is not maximal in M_{12}, lifts to subgroups of 2.M_{12}.2 that contain L_{2}(11).2 subgroups.)

gap> s:= CharacterTable( "L2(11).2" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable( "M12.2" );;
gap> factfus:= GetFusionMap( tbl2, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[2], "all" );
G = 2.M12.2:  point stabilizer L2(11).2, ranks [ 7, 7 ]
[ "1a^++11ab+12a^{\\pm}+55a^++66a^++120b^-",
  "1a^++11ab+12a^{\\pm}+55a^++66a^++120b^+" ]
gap> CompareWithDatabase( "2.M12.2", faith );

The group (2.M_{12}.2)^{∗} of the isoclinism type that is not printed in the Atlas has one faithful multiplicity-free permutation action, with point stabilizer of the type M_{11}; as this subgroup lies inside 2.M_{12}, its existence is clear, and the permutation character in both groups of the type 2.M_{12}.2 is the same.

gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M12.2" );;
1:  subgroup $M_{11}$, degree 48 (1 cand.)
gap> CompareWithDatabase( "Isoclinic(2.M12.2)", faith );

Note that in (2.M_{12}.2)^{∗}, the subgroup of the type (2×L_{2}(11)).2 is a nonsplit extension, so the unique index 2 subgroup in this group contains the centre of 2.M_{12}.2, in particular there is no subgroup of the type L_{2}(11).2.

gap> s:= CharacterTable( "L2(11).2" );;
gap> PossibleClassFusions( s, tbl2 );
[  ]

3.3 G = 2.M_{22}

The group 2.M_{22} has four faithful multiplicity-free permutation actions, with point stabilizers of the types 2^{4} : A_{5}, A_{7} (twice), and 2^{3} : L_{3}(2), by Lemma 2.1.

gap> tbl:= CharacterTable( "2.M22" );;
gap> faith:= FaithfulCandidates( tbl, "M22" );;
3:  subgroup $2^4:A_5 \leq 2^4:A_6$, degree 924 (1 cand.)
4:  subgroup $A_7$, degree 352 (1 cand.)
5:  subgroup $A_7$, degree 352 (1 cand.)
7:  subgroup $2^3:L_3(2)$, degree 660 (1 cand.)

Note that one class of subgroups of the type 2^{4} : A_{5} in the maximal subgroup of the type 2^{4} : A_{6} as well as the A_{7} and 2^{3} : L_{3}(2) subgroups lift to direct products in 2.M_{22}. A proof for 2^{4} : A_{5} using explicit computations with the group can be found in Subsection 4.1.

gap> Maxes( tbl );
[ "2.L3(4)", "2.M22M2", "2xA7", "2xA7", "2.M22M5", "2x2^3:L3(2)",
  "(2xA6).2_3", "2xL2(11)" ]
gap> s:= CharacterTable( "P1/G1/L1/V1/ext2" );;
gap> VerifyCandidates( s, tbl, 0, faith[3], "all" );
G = 2.M22:  point stabilizer P1/G1/L1/V1/ext2, ranks [ 8 ]
[ "1a+21a+55a+126ab+154a+210b+231a" ]
gap> faith[4] = faith[5];
true
gap> VerifyCandidates( CharacterTable( "A7" ), tbl, 0, faith[4], "all" );
G = 2.M22:  point stabilizer A7, ranks [ 5 ]
[ "1a+21a+56a+120a+154a" ]
gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[7], "all" );
G = 2.M22:  point stabilizer 2^3:sl(3,2), ranks [ 7 ]
[ "1a+21a+55a+99a+120a+154a+210b" ]
gap> CompareWithDatabase( "2.M22", faith );
gap> CompareWithCandidatesByMaxes( "2.M22", faith );

3.4 G = 2.M_{22}.2

The group 2.M_{22}.2 that is printed in the Atlas has eight faithful multiplicity-free permutation actions, with point stabilizers of the types 2^{4} : S_{5} (twice), A_{7}, 2^{3} : L_{3}(2) × 2 (twice), 2^{3} : L_{3}(2), and L_{2}(11).2 (twice).

gap> tbl2:= CharacterTable( "2.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
6:  subgroup $2^4:S_5 \leq 2^4:S_6$, degree 924 (2 cand.)
7:  subgroup $A_7$, degree 704 (1 cand.)
11:  subgroup $2^3:L_3(2) \times 2$, degree 660 (2 cand.)
12:  subgroup $2^3:L_3(2) \leq 2^3:L_3(2) \times 2$, degree 1320 (2 cand.)
16:  subgroup $L_2(11).2$, degree 1344 (2 cand.)

The character table of the 2^{4} : S_{5} type subgroup is contained in the GAP Character Table Library, with identifier w(d5) (cf. Subsection 4.2).

gap> s:= CharacterTable( "w(d5)" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable( "M22.2" );;
gap> factfus:= GetFusionMap( tbl2, facttbl );;

gap> ForAll( PossibleClassFusions( s, tbl2 ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" );
G = 2.M22.2:  point stabilizer w(d5), ranks [ 7, 7 ]
[ "1a^++21a^++55a^++126ab+154a^++210b^-+231a^-",
  "1a^++21a^++55a^++126ab+154a^++210b^++231a^-" ]

The two classes of the type A_{7} subgroups in 2.M_{22} are fused in 2.M_{22}.2.

gap> VerifyCandidates( CharacterTable( "A7" ), tbl, tbl2, faith[7], "all" );
G = 2.M22.2:  point stabilizer A7, ranks [ 10 ]
[ "1a^{\\pm}+21a^{\\pm}+56a^{\\pm}+120a^{\\pm}+154a^{\\pm}" ]

The preimages of the 2^{3} : L_{3}(2) × 2 type subgroups of M_{22}.2 in 2.M_{22}.2 are direct products, by the discussion of 2.M_{22} and Lemma 2.3 (i). So Lemma 2.3 (iii) yields two classes, with different permutation characters.

gap> s:= CharacterTable( "2x2^3:L3(2)" );;
gap> s0:= CharacterTable( "2^3:sl(3,2)" );;
gap> s0fuss:= PossibleClassFusions( s0, s );;
gap> StoreFusion( s0, s0fuss[1], s );
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" );
2.M22.2:  2x2^3:L3(2) lifts to a direct product,
proved by squares in [ 1, 5, 14, 16 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" );
G = 2.M22.2:  point stabilizer 2x2^3:L3(2), ranks [ 7, 7 ]
[ "1a^++21a^++55a^++99a^++120a^-+154a^++210b^-",
  "1a^++21a^++55a^++99a^++120a^++154a^++210b^+" ]

There is one class of subgroups of the type 2^{3} : L_{3}(2) in 2.M_{22}. One of the two candidates of degree 1 320 is excluded because it does not arise from a possible class fusion.

gap> s:= CharacterTable( "M22M6" );;
gap> fus:= PossibleClassFusions( s, tbl );;
gap> pi1320:= Set( List( fus, x -> Induced( s, tbl2,
>                                  [ TrivialCharacter( s ) ], x )[1] ) );;
gap> IsSubset( faith[12], pi1320 );
true
gap> faith[12]:= pi1320;;
gap> VerifyCandidates( s, tbl, tbl2, faith[12], "all" );
G = 2.M22.2:  point stabilizer 2^3:sl(3,2), ranks [ 14 ]
[ "1a^{\\pm}+21a^{\\pm}+55a^{\\pm}+99a^{\\pm}+120a^{\\pm}+154a^{\\pm}+210b^{\\\
pm}" ]

The preimages of the L_{2}(11).2 type subgroups of M_{22}.2 in 2.M_{22}.2 are direct products by Lemma 2.3 (i), so Lemma 2.3 (iii) yields two classes, with different permutation characters.

gap> s:= CharacterTable( "L2(11).2" );;
gap> s0:= CharacterTable( "L2(11)" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );

2.M22.2:  L2(11).2 lifts to a direct product,
proved by squares in [ 1, 4, 10, 13 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" );
G = 2.M22.2:  point stabilizer L2(11).2, ranks [ 10, 10 ]
[ "1a^++21a^-+55a^++56a^{\\pm}+120a^-+154a^++210a^-+231a^-+440a^+",
  "1a^++21a^-+55a^++56a^{\\pm}+120a^++154a^++210a^-+231a^-+440a^-" ]
gap> CompareWithDatabase( "2.M22.2", faith );

The group (2.M_{22}.2)^{∗} of the isoclinism type that is not printed in the Atlas has two faithful multiplicity-free permutation actions, with point stabilizers of the types A_{7} and 2^{3} : L_{3}(2).

gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
7:  subgroup $A_7$, degree 704 (1 cand.)
12:  subgroup $2^3:L_3(2) \leq 2^3:L_3(2) \times 2$, degree 1320 (2 cand.)
gap> faith[12]:= Filtered( faith[12], chi -> chi in pi1320 );;
gap> CompareWithDatabase( "Isoclinic(2.M22.2)", faith );

The two classes of subgroups lie inside 2.M_{22}, so their existence has been discussed already above.

3.5 G = 3.M_{22}

The group 3.M_{22} has four faithful multiplicity-free permutation actions, with point stabilizers of the types 2^{4} : A_{5}, 2^{4} : S_{5}, 2^{3} : L_{3}(2), and L_{2}(11).

gap> tbl:= CharacterTable( "3.M22" );;
gap> faith:= FaithfulCandidates( tbl, "M22" );;
3:  subgroup $2^4:A_5 \leq 2^4:A_6$, degree 1386 (1 cand.)
6:  subgroup $2^4:S_5$, degree 693 (1 cand.)
7:  subgroup $2^3:L_3(2)$, degree 990 (1 cand.)
9:  subgroup $L_2(11)$, degree 2016 (1 cand.)

The existence of one class of each of these types follows from Lemma 2.1.

gap> VerifyCandidates( CharacterTable( "P1/G1/L1/V1/ext2" ), tbl, 0, faith[3], "all" );
G = 3.M22:  point stabilizer P1/G1/L1/V1/ext2, ranks [ 13 ]
[ "1a+21abc+55a+105abcd+154a+231abc" ]
gap> VerifyCandidates( CharacterTable( "M22M5" ), tbl, 0, faith[6], "all" );
G = 3.M22:  point stabilizer 2^4:s5, ranks [ 10 ]
[ "1a+21abc+55a+105abcd+154a" ]
gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[7], "all" );
G = 3.M22:  point stabilizer 2^3:sl(3,2), ranks [ 13 ]
[ "1a+21abc+55a+99abc+105abcd+154a" ]
gap> VerifyCandidates( CharacterTable( "M22M8" ), tbl, 0, faith[9], "all" );
G = 3.M22:  point stabilizer L2(11), ranks [ 16 ]
[ "1a+21abc+55a+105abcd+154a+210abc+231abc" ]
gap> CompareWithDatabase( "3.M22", faith );
gap> CompareWithCandidatesByMaxes( "3.M22", faith );

The group 3.M_{22}.2 has five faithful multiplicity-free permutation actions, with point stabilizers of the types 2^{4} : S_{5}, 2^{5} : S_{5}, 2^{4} : (A_{5} × 2), 2^{3} : L_{3}(2) × 2, and L_{2}(11).2.

gap> tbl2:= CharacterTable( "3.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
6:  subgroup $2^4:S_5 \leq 2^4:S_6$, degree 1386 (1 cand.)
8:  subgroup $2^5:S_5$, degree 693 (1 cand.)
10:  subgroup $2^4:(A_5 \times 2) \leq 2^5:S_5$, degree 1386 (1 cand.)
11:  subgroup $2^3:L_3(2) \times 2$, degree 990 (1 cand.)
16:  subgroup $L_2(11).2$, degree 2016 (1 cand.)

Subgroups of these types exist by Lemma 2.2. The verification is straightforward in all cases except that of 2^{4} : (A_{5} × 2).

gap> VerifyCandidates( CharacterTable( "w(d5)" ), tbl, tbl2, faith[6], "all" );
G = 3.M22.2:  point stabilizer w(d5), ranks [ 9 ]
[ "1a^++21a^+bc+55a^++105adbc+154a^++231a^-bc" ]
gap> VerifyCandidates( CharacterTable( "M22.2M4" ), tbl, tbl2, faith[8], "all" );
G = 3.M22.2:  point stabilizer M22.2M4, ranks [ 7 ]
[ "1a^++21a^+bc+55a^++105adbc+154a^+" ]
gap> VerifyCandidates( CharacterTable( "2x2^3:L3(2)" ), tbl, tbl2, faith[11], "all" );
G = 3.M22.2:  point stabilizer 2x2^3:L3(2), ranks [ 9 ]
[ "1a^++21a^+bc+55a^++99a^+bc+105adbc+154a^+" ]
gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" );
G = 3.M22.2:  point stabilizer L2(11).2, ranks [ 11 ]
[ "1a^++21a^-bc+55a^++105adbc+154a^++210a^-bc+231a^-bc" ]

In the remaining case, we note that the 2^{4} : (A_{5} × 2) type subgroup has index 2 in the maximal subgroup of the type 2^{5} : S_{5}, whose character table is available via the identifier M22.2M4. It is sufficient to show that exactly one of the three index 2 subgroups in this group induces a multiplicityfree permutation character of 3.M_{22}.2, and this can be done by inducing the degree 2 permutation characters of 2^{5} : S_{5} to 3.M_{22}.2.

gap> s:= CharacterTable( "M22.2M4" );;
gap> lin:= LinearCharacters( s );
[ Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
      1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 ] ),
  Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, -1, -1, -1,
      -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1 ] ),
  Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, -1, -1, -1,
      -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1 ] ) ]
gap> perms:= List( lin{ [ 2 .. 4 ] }, chi -> chi + lin[1] );;
gap> sfustbl2:= PossibleClassFusions( s, tbl2 );;
gap> Length( sfustbl2 );
2
gap> ind1:= Induced( s, tbl2, perms, sfustbl2[1] );;
gap> ind2:= Induced( s, tbl2, perms, sfustbl2[2] );;
gap> PermCharInfo( tbl2, ind1 ).ATLAS;
[ "1ab+21ab+42aa+55ab+154ab+210ccdd", "1a+21ab+42a+55a+154a+210bcd+462a",
  "1a+21aa+42a+55a+154a+210acd+462a" ]
gap> PermCharInfo( tbl2, ind2 ).ATLAS;

[ "1a+21aa+42a+55a+154a+210acd+462a", "1a+21ab+42a+55a+154a+210bcd+462a",
  "1ab+21ab+42aa+55ab+154ab+210ccdd" ]
gap> ind1[2] = ind2[2];
true
gap> [ ind1[2] ] = faith[10];
true
gap> CompareWithDatabase( "3.M22.2", faith );

3.7 G = 4.M_{22} and G = 12.M_{22}

The group 4.M_{22} and hence also the group 12.M_{22} has no faithful multiplicity-free permutation action.

gap> tbl:= CharacterTable( "4.M22" );;
gap> faith:= FaithfulCandidates( tbl, "2.M22" );;
gap> CompareWithDatabase( "4.M22", faith );
gap> CompareWithCandidatesByMaxes( "4.M22", faith );

3.8 G = 4.M_{22}.2 and G = 12.M_{22}.2

The two isoclinism types of groups of the type 4.M_{22}.2 and hence also all groups of the type 12.M_{22}.2 have no faithful multiplicity-free permutation actions.

gap> tbl2:= CharacterTable( "4.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
gap> CompareWithDatabase( "4.M22.2", faith );
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
gap> CompareWithDatabase( "Isoclinic(4.M22.2)", faith );

3.9 G = 6.M_{22}

The group 6.M_{22} has two faithful multiplicity-free permutation actions, with point stabilizers of the types 2^{4} : A_{5} and 2^{3} : L_{3}(2).

gap> tbl:= CharacterTable( "6.M22" );;
gap> faith:= FaithfulCandidates( tbl, "3.M22" );;
1:  subgroup $2^4:A_5 \rightarrow (M_{22},3)$, degree 2772 (1 cand.)
3:  subgroup $2^3:L_3(2) \rightarrow (M_{22},7)$, degree 1980 (1 cand.)

The existence of one class of each of these subgroups follows from the treatment of 2.M_{22} and 3.M_{22}.

gap> VerifyCandidates( CharacterTable( "P1/G1/L1/V1/ext2" ), tbl, 0, faith[1], "all" );
G = 6.M22:  point stabilizer P1/G1/L1/V1/ext2, ranks [ 22 ]
[ "1a+21abc+55a+105abcd+126abcdef+154a+210bef+231abc" ]
gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[3], "all" );
G = 6.M22:  point stabilizer 2^3:sl(3,2), ranks [ 17 ]
[ "1a+21abc+55a+99abc+105abcd+120a+154a+210b+330de" ]
gap> CompareWithDatabase( "6.M22", faith );
gap> CompareWithCandidatesByMaxes( "6.M22", faith );

The group 6.M_{22}.2 that is printed in the Atlas has six faithful multiplicity-free permutation actions, with point stabilizers of the types 2^{4} : S_{5} (twice), 2^{3} : L_{3}(2) × 2 (twice), and L_{2}(11).2 (twice).

gap> tbl2:= CharacterTable( "6.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
6:  subgroup $2^4:S_5 \leq 2^4:S_6$, degree 2772 (2 cand.)
11:  subgroup $2^3:L_3(2) \times 2$, degree 1980 (2 cand.)
16:  subgroup $L_2(11).2$, degree 4032 (2 cand.)

We know that 2.M_{22}.2 contains two classes of subgroups isomorphic with each of the required point stabilizers, so we apply Lemma 2.2.

gap> s:= CharacterTable( "w(d5)" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" );
G = 6.M22.2:  point stabilizer w(d5), ranks [ 14, 14 ]
[ "1a^++21a^+bc+55a^++105adbc+126abcfde+154a^++210b^-ef+231a^-bc",
  "1a^++21a^+bc+55a^++105adbc+126abcfde+154a^++210b^+ef+231a^-bc" ]

(Since 6.M_{22} contains subgroups of the type 2^{3} : L_{3}(2) × 2 in which we are not interested, we must use "extending" as the last argument of VerifyCandidates for this case.)

gap> s:= CharacterTable( "2x2^3:L3(2)" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" );
G = 6.M22.2:  point stabilizer 2x2^3:L3(2), ranks [ 12, 12 ]
[ "1a^++21a^+bc+55a^++99a^+bc+105adbc+120a^-+154a^++210b^-+330de",
  "1a^++21a^+bc+55a^++99a^+bc+105adbc+120a^++154a^++210b^++330de" ]
gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" );
G = 6.M22.2:  point stabilizer L2(11).2, ranks [ 20, 20 ]
[ "1a^++21a^-bc+55a^++56a^{\\pm}+66abcd+105adbc+120a^-bc+154a^++210a^-cdgjhi+2\
31a^-bc+440a^+",
  "1a^++21a^-bc+55a^++56a^{\\pm}+66abcd+105adbc+120a^+bc+154a^++210a^-cdgjhi+2\
31a^-bc+440a^-" ]
gap> CompareWithDatabase( "6.M22.2", faith );

The group (6.M_{22}.2)^{∗} of the isoclinism type that is not printed in the Atlas has no faithful multiplicityfree permutation action.

gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
gap> CompareWithDatabase( "Isoclinic(6.M22.2)", faith );

3.11 G = 2.J_{2}

The group 2.J_{2} has one faithful multiplicity-free permutation action, with point stabilizer of the type U_{3}(3), by Lemma 2.1.

gap> tbl:= CharacterTable( "2.J2" );;
gap> faith:= FaithfulCandidates( tbl, "J2" );;
1:  subgroup $U_3(3)$, degree 200 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U3(3)" ), tbl, 0, faith[1], "all" );
G = 2.J2:  point stabilizer U3(3), ranks [ 5 ]
[ "1a+36a+50ab+63a" ]
gap> CompareWithDatabase( "2.J2", faith );
gap> CompareWithCandidatesByMaxes( "2.J2", faith );

3.12 G = 2.J_{2}.2

The group 2.J_{2}.2 that is printed in the Atlas has no faithful multiplicity-free permutation action.

gap> tbl2:= CharacterTable( "2.J2.2" );;
gap> faith:= FaithfulCandidates( tbl2, "J2.2" );;
gap> CompareWithDatabase( "2.J2.2", faith );

The group (2.J_{2}.2)^{∗} of the isoclinism type that is not printed in the Atlas has four faithful multiplicityfree permutation actions, with point stabilizers of the types U_{3}(3).2 (twice) and 3.A_{6}.2_{3} (twice).

gap> facttbl:= CharacterTable( "J2.2" );;
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "J2.2" );;
1:  subgroup $U_3(3).2$, degree 200 (1 cand.)
5:  subgroup $3.A_6.2_3 \leq 3.A_6.2^2$, degree 1120 (1 cand.)

The existence of two classes of each of these subgroups follows from Lemma 2.3. (Note that the Schur multiplier of U_{3}(3) is trivial and 6.A_{6} does not admit an automorphic extension that has a factor group A_{6}.2_{3}.)

gap> s0:= CharacterTable( "U3(3)" );;
gap> s:= CharacterTable( "U3(3).2" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
Isoclinic(2.J2.2):  U3(3).2 lifts to a direct product,
proved by squares in [ 1, 3, 8, 16 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[1], "all" );
G = Isoclinic(2.J2.2):  point stabilizer U3(3).2, ranks [ 4 ]
[ "1a^++36a^++50ab+63a^+" ]
gap> s0:= CharacterTable( "3.A6" );;
gap> s:= CharacterTable( "3.A6.2_3" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
Isoclinic(2.J2.2):  3.A6.2_3 lifts to a direct product,
proved by squares in [ 3, 10, 16, 25 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[5], "all" );
G = Isoclinic(2.J2.2):  point stabilizer 3.A6.2_3, ranks [ 12 ]
[ "1a^++14c^{\\pm}+21ab+50ab+63a^{\\pm}+90a^++126a^++175a^-+216a^{\\pm}" ]
gap> faith[1]:= faith[1]{ [ 1, 1 ] };;
gap> faith[5]:= faith[5]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "Isoclinic(2.J2.2)", faith );

3.13 G = 2.HS

The group 2.HS has five faithful multiplicity-free permutation actions, with point stabilizers of the types U_{3}(5) (twice), A_{8}, and M_{11} (twice).

gap> tbl:= CharacterTable( "2.HS" );;
gap> faith:= FaithfulCandidates( tbl, "HS" );;
3:  subgroup $U_3(5) \leq U_3(5).2$, degree 704 (1 cand.)
5:  subgroup $U_3(5) \leq U_3(5).2$, degree 704 (1 cand.)
8:  subgroup $A_8 \leq A_8.2$, degree 4400 (1 cand.)
10:  subgroup $M_{11}$, degree 11200 (1 cand.)
11:  subgroup $M_{11}$, degree 11200 (1 cand.)

Lemma 2.1 applies in all cases; note that 2.HS does not admit an embedding of 2.A_{8}.

gap> VerifyCandidates( CharacterTable( "U3(5)" ), tbl, 0,
>       Concatenation( faith[3], faith[5] ), "all" );
G = 2.HS:  point stabilizer U3(5), ranks [ 6, 6 ]
[ "1a+22a+154c+175a+176ab", "1a+22a+154b+175a+176ab" ]
gap> PossibleClassFusions( CharacterTable( "2.A8" ), tbl );
[  ]
gap> VerifyCandidates( CharacterTable( "A8" ), tbl, 0, faith[8], "all" );
G = 2.HS:  point stabilizer A8, ranks [ 13 ]
[ "1a+22a+77a+154abc+175a+176ab+693a+770a+924ab" ]
gap> VerifyCandidates( CharacterTable( "M11" ), tbl, 0,
>       Concatenation( faith[10], faith[11] ), "all" );
G = 2.HS:  point stabilizer M11, ranks [ 16, 16 ]
[ "1a+22a+56a+77a+154c+175a+176ab+616ab+770a+825a+1056a+1980ab+2520a",
  "1a+22a+56a+77a+154b+175a+176ab+616ab+770a+825a+1056a+1980ab+2520a" ]
gap> CompareWithDatabase( "2.HS", faith );
gap> CompareWithCandidatesByMaxes( "2.HS", faith );

3.14 G = 2.HS.2

The group 2.HS.2 that is printed in the Atlas has four faithful multiplicity-free permutation actions, with point stabilizers of the types A_{8} × 2 (twice) and A_{8}.2 (twice).

gap> tbl2:= CharacterTable("2.HS.2");;
gap> faith:= FaithfulCandidates( tbl2, "HS.2" );;
10:  subgroup $A_8 \times 2 \leq A_8.2 \times 2$, degree 4400 (1 cand.)
11:  subgroup $A_8.2 \leq A_8.2 \times 2$, degree 4400 (1 cand.)

The existence of two classes of subgroups for each candidate follows from Lemma 2.3. (Since there are A_{8} × 2 type subgroups inside 2.HS in which we are not interested, we must use "extending" as the last argument of VerifyCandidates.)

gap> facttbl:= CharacterTable( "HS.2" );;
gap> factfus:= GetFusionMap( tbl2, facttbl );;
gap> s0:= CharacterTable( "A8");;
gap> s:= s0 * CharacterTable( "Cyclic", 2 );
CharacterTable( "A8xC2" )
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.HS.2:  A8xC2 lifts to a direct product,
proved by squares in [ 1, 6, 13, 20, 30 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[10], "extending" );
G = 2.HS.2:  point stabilizer A8xC2, ranks [ 10 ]
[ "1a^++22a^++77a^++154a^+bc+175a^++176ab+693a^++770a^++924ab" ]
gap> s:= CharacterTable( "A8.2" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" );
2.HS.2:  A8.2 lifts to a direct product,
proved by squares in [ 1, 6, 13 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "all" );
G = 2.HS.2:  point stabilizer A8.2, ranks [ 10 ]
[ "1a^++22a^-+77a^++154a^+bc+175a^++176ab+693a^++770a^-+924ab" ]
gap> faith[10]:= faith[10]{ [ 1, 1 ] };;
gap> faith[11]:= faith[11]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "2.HS.2", faith );

The group (2.HS.2)^{∗} of the isoclinism type that is not printed in the Atlas has no faithful multiplicityfree permutation action.

gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "HS.2" );;
gap> CompareWithDatabase( "Isoclinic(2.HS.2)", faith );

3.15 G = 3.J_{3}

The group 3.J_{3} has no faithful multiplicity-free permutation action.

gap> tbl:= CharacterTable( "3.J3" );;
gap> faith:= FaithfulCandidates( tbl, "J3" );;
gap> CompareWithDatabase( "3.J3", faith );

3.16 G = 3.J_{3}.2

The group 3.J_{3}.2 has no faithful multiplicity-free permutation action.

gap> tbl2:= CharacterTable( "3.J3.2" );;
gap> faith:= FaithfulCandidates( tbl2, "J3.2" );;
gap> CompareWithDatabase( "3.J3.2", faith );

3.17 G = 3.McL

The group 3.McL has one faithful multiplicity-free permutation action, with point stabilizer of the type 2.A_{8}, by Lemma 2.1.

gap> tbl:= CharacterTable( "3.McL" );;
gap> faith:= FaithfulCandidates( tbl, "McL" );;
6:  subgroup $2.A_8$, degree 66825 (1 cand.)
gap> VerifyCandidates( CharacterTable( "2.A8" ), tbl, 0, faith[6], "all" );
G = 3.McL:  point stabilizer 2.A8, ranks [ 14 ]
[ "1a+252a+1750a+2772ab+5103abc+5544a+6336ab+8064ab+9625a" ]
gap> CompareWithDatabase( "3.McL", faith );
gap> CompareWithCandidatesByMaxes( "3.McL", faith );

3.18 G = 3.McL.2

The group 3.McL.2 has one faithful multiplicity-free permutation action, with point stabilizer of the type (2.A_{8}.2)^{∗}, by Lemma 2.2.

gap> tbl2:= CharacterTable( "3.McL.2" );;
gap> faith:= FaithfulCandidates( tbl2, "McL.2" );;
9:  subgroup $2.S_8$, degree 66825 (1 cand.)
gap> s:= CharacterTable( "Isoclinic(2.A8.2)" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[9], "all" );
G = 3.McL.2:  point stabilizer Isoclinic(2.A8.2), ranks [ 10 ]
[ "1a^++252a^++1750a^++2772ab+5103a^+bc+5544a^++6336ab+8064ab+9625a^+" ]
gap> CompareWithDatabase( "3.McL.2", faith );

3.19 G = 2.Ru

The group 2.Ru has one faithful multiplicity-free permutation action, with point stabilizer of the type 2F_{4}(2)′, by Lemma 2.1.

gap> tbl:= CharacterTable( "2.Ru" );;
gap> faith:= FaithfulCandidates( tbl, "Ru" );;
2:  subgroup ${^2F_4(2)^{\prime}} \leq {^2F_4(2)^{\prime}}.2$, degree 16240 (
1 cand.)
gap> VerifyCandidates( CharacterTable( "2F4(2)'" ), tbl, 0, faith[2], "all" );
G = 2.Ru:  point stabilizer 2F4(2)', ranks [ 9 ]
[ "1a+28ab+406a+783a+3276a+3654a+4032ab" ]
gap> CompareWithDatabase( "2.Ru", faith );

3.20 G = 2.Suz

The group 2.Suz has one faithful multiplicity-free permutation action, with point stabilizer of the type U_{5}(2), by Lemma 2.1.

gap> tbl:= CharacterTable( "2.Suz" );;
gap> faith:= FaithfulCandidates( tbl, "Suz" );;
4:  subgroup $U_5(2)$, degree 65520 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U5(2)" ), tbl, 0, faith[4], "all" );
G = 2.Suz:  point stabilizer U5(2), ranks [ 10 ]
[ "1a+143a+364abc+5940a+12012a+14300a+16016ab" ]
gap> CompareWithDatabase( "2.Suz", faith );

3.21 G = 2.Suz.2

The group 2.Suz.2 that is printed in the Atlas has four faithful multiplicity-free permutation actions, with point stabilizers of the types U_{5}(2).2 (twice) and 3^{5} : (M_{11} × 2) (twice), respectively.

gap> tbl2:= CharacterTable( "2.Suz.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
8:  subgroup $U_5(2).2$, degree 65520 (1 cand.)
12:  subgroup $3^5:(M_{11} \times 2)$, degree 465920 (1 cand.)

We verify the conditions of Lemma 2.3.

gap> s0:= CharacterTable( "U5(2)" );;
gap> s:= CharacterTable( "U5(2).2" );;
gap> facttbl:= CharacterTable( "Suz.2" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.Suz.2:  U5(2).2 lifts to a direct product,
proved by squares in [ 1, 8, 13, 19, 31, 44 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[8], "all" );
G = 2.Suz.2:  point stabilizer U5(2).2, ranks [ 8 ]
[ "1a^++143a^-+364a^+bc+5940a^++12012a^-+14300a^-+16016ab" ]
gap> s0:= CharacterTable( "SuzM5" );
CharacterTable( "3^5:M11" )
gap> s:= CharacterTable( "Suz.2M6" );
CharacterTable( "3^5:(M11x2)" )
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.Suz.2:  3^5:(M11x2) lifts to a direct product,

proved by squares in [ 1, 4, 8, 10, 19, 22, 26, 39 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[12], "all" );
G = 2.Suz.2:  point stabilizer 3^5:(M11x2), ranks [ 14 ]
[ "1a^++364a^{\\pm}bc+5940a^++12012a^-+14300a^-+15015ab+15795a^++16016ab+54054\
a^++100100a^-b^{\\pm}" ]
gap> faith[8]:= faith[8]{ [ 1, 1 ] };;
gap> faith[12]:= faith[12]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "2.Suz.2", faith );

The group (2.Suz.2)^{∗} of the isoclinism type that is not printed in the Atlas has no faithful multiplicityfree permutation action.

gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
gap> CompareWithDatabase( "Isoclinic(2.Suz.2)", faith );

3.22 G = 3.Suz

The group 3.Suz has four faithful multiplicity-free permutation actions, with point stabilizers of the types G_{2}(4), U_{5}(2), 2_{−}^{1+6}.U_{4}(2), and 2^{4+6} : 3A_{6}, respectively, by Lemma 2.1.

gap> tbl:= CharacterTable( "3.Suz" );;
gap> faith:= FaithfulCandidates( tbl, "Suz" );;
1:  subgroup $G_2(4)$, degree 5346 (1 cand.)
4:  subgroup $U_5(2)$, degree 98280 (1 cand.)
5:  subgroup $2^{1+6}_-.U_4(2)$, degree 405405 (1 cand.)
6:  subgroup $2^{4+6}:3A_6$, degree 1216215 (1 cand.)
gap> Maxes( tbl );
[ "3xG2(4)", "3^2.U4(3).2_3'", "3xU5(2)", "3x2^(1+6)_-.U4(2)", "3^6.M11",
  "3xJ2.2", "3x2^(4+6).3A6", "(A4x3.L3(4)).2", "3x2^(2+8):(A5xS3)",
  "3xM12.2", "3.3^(2+4):2(A4x2^2).2", "(3.A6xA5):2", "(3^(1+2):4xA6).2",
  "3xL3(3).2", "3xL3(3).2", "3xL2(25)", "3.A7" ]
gap> VerifyCandidates( CharacterTable( "G2(4)" ), tbl, 0, faith[1], "all" );
G = 3.Suz:  point stabilizer G2(4), ranks [ 7 ]
[ "1a+66ab+780a+1001a+1716ab" ]
gap> VerifyCandidates( CharacterTable( "U5(2)" ), tbl, 0, faith[4], "all" );
G = 3.Suz:  point stabilizer U5(2), ranks [ 14 ]
[ "1a+78ab+143a+364a+1365ab+4290ab+5940a+12012a+14300a+27027ab" ]
gap> VerifyCandidates( CharacterTable( "SuzM4" ), tbl, 0, faith[5], "all" );
G = 3.Suz:  point stabilizer 2^1+6.u4q2, ranks [ 23 ]
[ "1a+66ab+143a+429ab+780a+1716ab+3432a+5940a+6720ab+14300a+18954abc+25025a+42\
900ab+64350cd+66560a" ]
gap> VerifyCandidates( CharacterTable( "SuzM7" ), tbl, 0, faith[6], "all" );
G = 3.Suz:  point stabilizer 2^4+6:3a6, ranks [ 27 ]
[ "1a+364a+780a+1001a+1365ab+4290ab+5940a+12012a+14300a+15795a+25025a+27027ab+\
42900ab+66560a+75075a+85800ab+88452a+100100a+104247ab+139776ab" ]
gap> CompareWithDatabase( "3.Suz", faith );

3.23 G = 3.Suz.2

The group 3.Suz.2 has four faithful multiplicity-free permutation actions, with point stabilizers of the types G_{2}(4).2, U_{5}(2).2, 2_{−}^{1+6}.U_{4}(2).2, and 2^{4+6} : 3S_{6}, respectively. We know from the treatment of 3.Suz that we can apply Lemma 2.2.

gap> tbl2:= CharacterTable( "3.Suz.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
1:  subgroup $G_2(4).2$, degree 5346 (1 cand.)
8:  subgroup $U_5(2).2$, degree 98280 (1 cand.)
10:  subgroup $2^{1+6}_-.U_4(2).2$, degree 405405 (1 cand.)
13:  subgroup $2^{4+6}:3S_6$, degree 1216215 (1 cand.)
gap> Maxes( CharacterTable( "Suz.2" ) );
[ "Suz", "G2(4).2", "3_2.U4(3).(2^2)_{133}", "U5(2).2", "2^(1+6)_-.U4(2).2",
  "3^5:(M11x2)", "J2.2x2", "2^(4+6):3S6", "(A4xL3(4):2_3):2",
  "2^(2+8):(S5xS3)", "M12.2x2", "3^(2+4):2(S4xD8)", "(A6:2_2xA5).2",
  "(3^2:8xA6).2", "L2(25).2_2", "A7.2" ]
gap> VerifyCandidates( CharacterTable( "G2(4).2" ), tbl, tbl2, faith[1], "all" );
G = 3.Suz.2:  point stabilizer G2(4).2, ranks [ 5 ]
[ "1a^++66ab+780a^++1001a^++1716ab" ]
gap> VerifyCandidates( CharacterTable( "U5(2).2" ), tbl, tbl2, faith[8], "all" );
G = 3.Suz.2:  point stabilizer U5(2).2, ranks [ 10 ]
[ "1a^++78ab+143a^-+364a^++1365ab+4290ab+5940a^++12012a^-+14300a^-+27027ab" ]
gap> VerifyCandidates( CharacterTable( "Suz.2M5" ), tbl, tbl2, faith[10], "all" );
G = 3.Suz.2:  point stabilizer 2^(1+6)_-.U4(2).2, ranks [ 16 ]
[ "1a^++66ab+143a^-+429ab+780a^++1716ab+3432a^++5940a^++6720ab+14300a^-+18954a\
^-bc+25025a^++42900ab+64350cd+66560a^+" ]
gap> VerifyCandidates( CharacterTable( "Suz.2M8" ), tbl, tbl2, faith[13], "all" );
G = 3.Suz.2:  point stabilizer 2^(4+6):3S6, ranks [ 20 ]
[ "1a^++364a^++780a^++1001a^++1365ab+4290ab+5940a^++12012a^-+14300a^-+15795a^+\
+25025a^++27027ab+42900ab+66560a^++75075a^++85800ab+88452a^++100100a^++104247a\
b+139776ab" ]
gap> CompareWithDatabase( "3.Suz.2", faith );

3.24 G = 6.Suz

The group 6.Suz has one faithful multiplicity-free permutation action, with point stabilizer of the type U_{5}(2), by Lemma 2.1.

gap> tbl:= CharacterTable( "6.Suz" );;
gap> faith:= FaithfulCandidates( tbl, "2.Suz" );;
1:  subgroup $U_5(2) \rightarrow (Suz,4)$, degree 196560 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U5(2)" ), tbl, 0, faith[1], "all" );
G = 6.Suz:  point stabilizer U5(2), ranks [ 26 ]
[ "1a+12ab+78ab+143a+364abc+924ab+1365ab+4290ab+4368ab+5940a+12012a+14300a+160\
16ab+27027ab+27456ab" ]
gap> CompareWithDatabase( "6.Suz", faith );

3.25 G = 6.Suz.2

The group 6.Suz.2 that is printed in the Atlas has two faithful multiplicity-free permutation actions, with point stabilizers of the type U_{5}(2).2 (twice). We know from the treatment of 6.Suz that we can apply Lemma 2.2.

gap> tbl2:= CharacterTable( "6.Suz.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
8:  subgroup $U_5(2).2$, degree 196560 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U5(2).2" ), tbl, tbl2, faith[8], "all" );
G = 6.Suz.2:  point stabilizer U5(2).2, ranks [ 16 ]

[ "1a^++12ab+78ab+143a^-+364a^+bc+924ab+1365ab+4290ab+4368ab+5940a^++12012a^-+\
14300a^-+16016ab+27027ab+27456ab" ]
gap> faith[8]:= faith[8]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "6.Suz.2", faith );

It follows from the treatment of 2.Suz.2 that the group (6.Suz.2)^{∗} of the isoclinism type that is not printed in the Atlas does not have a faithful multiplicity-free permutation action.

3.26 G = 3.ON

The group 3.ON has four faithful multiplicity-free permutation actions, with point stabilizers of the types L_{3}(7).2 (twice) and L_{3}(7) (twice). (The Schur multiplier of L_{3}(7).2 is trivial, so the L_{3}(7) type subgroups lift to direct products with the centre of 3.ON, that is, we can apply Lemma 2.1.)

gap> tbl:= CharacterTable( "3.ON" );;
gap> faith:= FaithfulCandidates( tbl, "ON" );;
1:  subgroup $L_3(7).2$, degree 368280 (1 cand.)
2:  subgroup $L_3(7) \leq L_3(7).2$, degree 736560 (1 cand.)
3:  subgroup $L_3(7).2$, degree 368280 (1 cand.)
4:  subgroup $L_3(7) \leq L_3(7).2$, degree 736560 (1 cand.)
gap> VerifyCandidates( CharacterTable( "L3(7).2" ), tbl, 0,
>        Concatenation( faith[1], faith[3] ), "all" );
G = 3.ON:  point stabilizer L3(7).2, ranks [ 11, 11 ]
[ "1a+495ab+10944a+26752a+32395b+52668a+58653bc+63612ab",
  "1a+495cd+10944a+26752a+32395a+52668a+58653bc+63612ab" ]
gap> VerifyCandidates( CharacterTable( "L3(7)" ), tbl, 0,
>        Concatenation( faith[2], faith[4] ), "all" );
G = 3.ON:  point stabilizer L3(7), ranks [ 15, 15 ]
[ "1a+495ab+10944a+26752a+32395b+37696a+52668a+58653bc+63612ab+85064a+122760ab\
",
  "1a+495cd+10944a+26752a+32395a+37696a+52668a+58653bc+63612ab+85064a+122760ab\
" ]
gap> CompareWithDatabase( "3.ON", faith );

3.27 G = 3.ON.2

The group 3.ON.2 has no faithful multiplicity-free permutation action.

gap> tbl2:= CharacterTable( "3.ON.2" );;
gap> faith:= FaithfulCandidates( tbl2, "ON.2" );;
gap> CompareWithDatabase( "3.ON.2", faith );

3.28 G = 2.F i_{22}

The group 2.F i_{22} has seven faithful multiplicity-free permutation actions, with point stabilizers of the types O_{7}(3) (twice), O_{8}^{+}(2) : S_{3} (twice), O_{8}^{+}(2) : 3, and O_{8}^{+}(2) : 2 (twice).

gap> tbl:= CharacterTable( "2.Fi22" );;
gap> faith:= FaithfulCandidates( tbl, "Fi22" );;
2:  subgroup $O_7(3)$, degree 28160 (2 cand.)
3:  subgroup $O_7(3)$, degree 28160 (2 cand.)
4:  subgroup $O_8^+(2).3.2$, degree 123552 (2 cand.)
5:  subgroup $O_8^+(2).3 \leq O_8^+(2).3.2$, degree 247104 (1 cand.)
6:  subgroup $O_8^+(2).2 \leq O_8^+(2).3.2$, degree 370656 (2 cand.)

The two classes of maximal subgroups of the type O_{7}(3) in F i_{22} induce the same permutation character and lift to two classes of the type 2 × O_{7}(3) in 2.F i_{22}. We get the same two candidates for these two classes. One of them belongs to the first class of O_{7}(3) subgroups in 2.F i_{22}, the other candidate belongs to the second class; this can be seen from the fact that the outer automorphism of F i_{22} swaps the two classes of O_{7}(3) subgroups, and the lift of this automorphism to 2.F i_{22} interchanges the candidates –this action can be read off from the embedding of 2.F i_{22} into any group of the type 2.F i_{22}.2.

gap> faith[2] = faith[3];
true
gap> tbl2:= CharacterTable("2.Fi22.2");;
gap> embed:= GetFusionMap( tbl, tbl2 );;
gap> swapped:= Filtered( InverseMap( embed ), IsList );
[ [ 3, 4 ], [ 17, 18 ], [ 25, 26 ], [ 27, 28 ], [ 33, 34 ], [ 36, 37 ],
  [ 42, 43 ], [ 51, 52 ], [ 59, 60 ], [ 63, 65 ], [ 64, 66 ], [ 71, 72 ],
  [ 73, 75 ], [ 74, 76 ], [ 81, 82 ], [ 85, 87 ], [ 86, 88 ], [ 89, 90 ],
  [ 93, 94 ], [ 95, 98 ], [ 96, 97 ], [ 99, 100 ], [ 103, 104 ],
  [ 107, 110 ], [ 108, 109 ], [ 113, 114 ] ]
gap> perm:= Product( List( swapped, pair -> ( pair[1], pair[2] ) ) );;
gap> Permuted( faith[2][1], perm ) = faith[2][2];
true
gap> VerifyCandidates( CharacterTable( "O7(3)" ), tbl, 0, faith[2], "all" );
G = 2.Fi22:  point stabilizer O7(3), ranks [ 5, 5 ]
[ "1a+352a+429a+13650a+13728b", "1a+352a+429a+13650a+13728a" ]
gap> faith[2]:= [ faith[2][1] ];;
gap> faith[3]:= [ faith[3][2] ];;

All involutions in F i_{22} lift to involutions in 2.F i_{22}, so the preimages of the maximal subgroups of the type O_{8}^{+}(2).S_{3} in F i_{22} have the type 2 × O_{8}^{+}(2).S_{3}. We apply Lemma 2.3, using that the two subgroups of the type O_{8}^{+}(2).S_{3} contain involutions outside O_{8}^{+}(2) which lie in the two nonconjugate preimages of the class 2A of F i_{22}; this proves the existence of the two candidates of degree 123 552.

gap> s:= CharacterTable( "O8+(2).S3" );;
gap> s0:= CharacterTable( "O8+(2).3" );;
gap> facttbl:= CharacterTable( "Fi22" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl, "all" );
2.Fi22:  O8+(2).3.2 lifts to a direct product,
proved by squares in [ 1, 8, 10, 12, 20, 23, 30, 46, 55, 61, 91 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( CharacterTable( "O8+(2).S3" ), tbl, 0, faith[4], "all" );
G = 2.Fi22:  point stabilizer O8+(2).3.2, ranks [ 6, 6 ]
[ "1a+3080a+13650a+13728b+45045a+48048c",
  "1a+3080a+13650a+13728a+45045a+48048b" ]

The existence of one class of O_{8}^{+}(2).3 subgroups follows from Lemma 2.1, and the proof for O_{8}^{+}(2).S_{3} also establishes two classes of O_{8}^{+}(2).2 subgroups, with different permutation characters,

gap> VerifyCandidates( CharacterTable( "O8+(2).3" ), tbl, 0, faith[5], "all" );
G = 2.Fi22:  point stabilizer O8+(2).3, ranks [ 11 ]
[ "1a+1001a+3080a+10725a+13650a+13728ab+45045a+48048bc+50050a" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, 0, faith[6], "all" );

G = 2.Fi22:  point stabilizer O8+(2).2, ranks [ 11, 11 ]
[ "1a+352a+429a+3080a+13650a+13728b+45045a+48048ac+75075a+123200a",
  "1a+352a+429a+3080a+13650a+13728a+45045a+48048ab+75075a+123200a" ]
gap> CompareWithDatabase( "2.Fi22", faith );

3.29 G = 2.F i_{22}.2

The group 2.F i_{22}.2 that is printed in the Atlas has seven faithful multiplicity-free permutation actions, with point stabilizers of the types O_{7}(3), O_{8}^{+}(2) : S_{3}, O_{8}^{+}(2) : 3 × 2 (twice), O_{8}^{+}(2) : 2, and 2F_{4}(2) (twice).

gap> tbl2:= CharacterTable( "2.Fi22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
3:  subgroup $O_7(3)$, degree 56320 (1 cand.)
5:  subgroup $O_8^+(2).3.2 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
6:  subgroup $O_8^+(2).3 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
10:  subgroup $O_8^+(2).2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
16:  subgroup ${^2F_4(2)}$, degree 7185024 (1 cand.)

The third, fifth, and tenth multiplicity-free permutation character of F i_{22}.2 are induced from subgroups of the types O_{7}(3), O_{8}^{+}(2).S_{3}, and O_{8}^{+}(2).2 that lie inside F i_{22}, and we have discussed above that these groups lift to direct products in 2.F i_{22}. In fact all such subgroups of 2.F i_{22}.2 lie inside 2.F i_{22}, and the two classes of such subgroups in 2.F i_{22} are fused in 2.F i_{22}.2, hence we get only one class of these subgroups.

gap> VerifyCandidates( CharacterTable( "O7(3)" ), tbl, tbl2, faith[3], "all" );
G = 2.Fi22.2:  point stabilizer O7(3), ranks [ 9 ]
[ "1a^{\\pm}+352a^{\\pm}+429a^{\\pm}+13650a^{\\pm}+13728ab" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).S3" ), tbl, tbl2, faith[5], "all" );
G = 2.Fi22.2:  point stabilizer O8+(2).3.2, ranks [ 10 ]
[ "1a^{\\pm}+3080a^{\\pm}+13650a^{\\pm}+13728ab+45045a^{\\pm}+48048bc" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, tbl2, faith[10], "all" );
G = 2.Fi22.2:  point stabilizer O8+(2).2, ranks [ 20 ]
[ "1a^{\\pm}+352a^{\\pm}+429a^{\\pm}+3080a^{\\pm}+13650a^{\\pm}+13728ab+45045a\
^{\\pm}+48048a^{\\pm}bc+75075a^{\\pm}+123200a^{\\pm}" ]

The sixth multiplicity-free permutation character of F i_{22}.2 is induced from a subgroup of the type O_{8}^{+}(2).3 × 2 that does not lie inside F i_{22}. As we have discussed above, the O_{8}^{+}(2).3 type subgroup of F i_{22} lifts to a subgroup of the type 2 × O_{8}^{+}(2).3 in 2.F i_{22}, and the outer involutions in the subgroup O_{8}^{+}(2).3 × 2 of F i_{22}.2 lift to involutions in 2.F i_{22}.2, so there are two subgroups of the type O_{8}^{+}(2).3 × 2 not containing the centre of 2.F i_{22}.2, which induce the same permutation character. Since also 2.F i_{22} contains subgroups of the type O_{8}^{+}(2).3 × 2, we must use "extending" as the last

argument of VerifyCandidates.
gap> s:= CharacterTable( "O8+(2).3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "extending" );
G = 2.Fi22.2:  point stabilizer O8+(2).3xC2, ranks [ 9 ]
[ "1a^++1001a^-+3080a^++10725a^++13650a^++13728ab+45045a^++48048bc+50050a^+" ]
gap> faith[6]:= faith[6]{ [ 1, 1 ] };;

By Lemma 2.3, the subgroup ^{2}F_{4}(2) of F i_{22}.2 lifts to 2 ×^{2}F_{4}(2) in 2.F i_{22}.2; for that, note that the class 4D of ^{2}F_{4}(2) does not lie inside ^{2}F_{4}(2)′ and the preimages in 2.F i_{22}.2 of the images in F i_{22}.2 square into the subgroup ^{2}F_{4}(2)′ of the direct product 2×^{2}F_{4}(2)′. Since the group 2×^{2}F_{4}(2) contains two subgroups of the type ^{2}F_{4}(2), with normalizer 2×^{2}F_{4}(2), there are two classes of such subgroups, which induce the same permutation character.

gap> facttbl:= CharacterTable( "Fi22.2" );;
gap> s0:= CharacterTable( "2F4(2)'" );;
gap> s:= CharacterTable( "2F4(2)" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.Fi22.2:  2F4(2)'.2 lifts to a direct product,
proved by squares in [ 5, 38, 53 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[16], "all" );
G = 2.Fi22.2:  point stabilizer 2F4(2)'.2, ranks [ 13 ]
[ "1a^++1001a^++1430a^++13650a^++30030a^++133056a^{\\pm}+289575a^-+400400ab+57\
9150a^++675675a^-+1201200a^-+1663200ab" ]
gap> faith[16]:= faith[16]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "2.Fi22.2", faith );

The group (2.F i_{22}.2)^{∗} of the isoclinism type that is not printed in the Atlas has seven faithful multiplicity-free permutation actions, with point stabilizers of the types O_{7}(3), O_{8}^{+}(2) : S_{3} (three times), and O_{8}^{+}(2) : 2 (three times).

gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
3:  subgroup $O_7(3)$, degree 56320 (1 cand.)
5:  subgroup $O_8^+(2).3.2 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
7:  subgroup $O_8^+(2).S_3 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
10:  subgroup $O_8^+(2).2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
11:  subgroup $O_8^+(2).2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)

The characters arising from the third, fifth, and tenth multiplicity-free permutation character of F i_{22}.2 are induced from subgroups of 2.F i_{22}, so these actions have been verified above.

The seventh multiplicity-free permutation character of F i_{22}.2 is induced from an O_{8}^{+}(2) : S_{3} type subgroup that does not lie inside F i_{22}. By Lemma 2.3 (i), this subgroup lifts to a direct product in (2.F i_{22}.2)^{∗}; this yields two actions (because the O_{8}^{+}(2) : S_{3} type subgroups have index 2 in their normalizer), which induce the same permutation character. Note that the involutions in O_{8}^{+}(2) : S_{3} \O_{8}^{+}(2) lie in the class 2F of F i_{22}.2, and these elements lift to involutions in (2.F i_{22}.2)^{∗}. Since also 2.F i_{22} contains subgroups of the type O_{8}^{+}(2) : S_{3}, we must use "extending" as the last argument of

VerifyCandidates.

This argument also proves the existence of two classes of O_{8}^{+}(2) : 2 type subgroups that are not contained in 2.F i_{22}; they arise from the 11th multiplicity-free permutation character of F i_{22}.2.

gap> s0:= CharacterTable( "O8+(2).3" );;
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" );
Isoclinic(2.Fi22.2):  O8+(2).3.2 lifts to a direct product,
proved by squares in [ 1, 7, 9, 11, 18, 21, 26, 39, 47, 52, 73 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[7], "extending" );
G = Isoclinic(2.Fi22.2):  point stabilizer O8+(2).3.2, ranks [ 9 ]
[ "1a^++1001a^++3080a^++10725a^-+13650a^++13728ab+45045a^++48048bc+50050a^-" ]
gap> faith[7]:= faith[7]{ [ 1, 1 ] };;
gap> s:= CharacterTable( "O8+(2).2" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" );

G = Isoclinic(2.Fi22.2):  point stabilizer O8+(2).2, ranks [ 19 ]
[ "1a^++352a^{\\pm}+429a^{\\pm}+1001a^++3080a^++10725a^-+13650a^++13728ab+4504\
5a^++48048a^{\\pm}bc+50050a^-+75075a^{\\pm}+123200a^{\\pm}" ]
gap> faith[11]:= faith[11]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "Isoclinic(2.Fi22.2)", faith );

3.30 G = 3.F i_{22}

The group 3.F i_{22} has six faithful multiplicity-free permutation actions, with point stabilizers of the types O_{8}^{+}(2) : S_{3}, O_{8}^{+}(2) : 3 (twice), O_{8}^{+}(2) : 2, 2^{6} : S_{6}(2), and ^{2}F_{4}(2).

gap> tbl:= CharacterTable( "3.Fi22" );;
gap> faith:= FaithfulCandidates( tbl, "Fi22" );;
4:  subgroup $O_8^+(2).3.2$, degree 185328 (1 cand.)
5:  subgroup $O_8^+(2).3 \leq O_8^+(2).3.2$, degree 370656 (2 cand.)
6:  subgroup $O_8^+(2).2 \leq O_8^+(2).3.2$, degree 555984 (1 cand.)
8:  subgroup $2^6:S_6(2)$, degree 2084940 (1 cand.)
9:  subgroup ${^2F_4(2)^{\prime}}$, degree 10777536 (1 cand.)

The preimages of the maximal subgroups of the type O_{8}^{+}(2).S_{3} in F i_{22} have the type 3 × O_{8}^{+}(2).S_{3}, because the Schur multiplier of O_{8}^{+}(2) has order 4 and the only central extension of S_{3} by a group of order 3 is 3 × S_{3}. Each such preimage contains one subgroup of the type O_{8}^{+}(2).S_{3} with one subgroup of the type O_{8}^{+}(2).3, two conjugate O_{8}^{+}(2).3 subgroups which are not contained in O_{8}^{+}(2).S_{3}, and one class of O_{8}^{+}(2).2 subgroups. The two classes of O_{8}^{+}(2).3 subgroups contain elements of order 3 outside O_{8}^{+}(2) which lie in nonconjugate preimages of the class 3A of F i_{22}, so we get two classes of O_{8}^{+}(2).3 subgroups in 3.F i_{22} which induce different permutation characters.

gap> VerifyCandidates( CharacterTable( "O8+(2).S3" ), tbl, 0, faith[4], "all" );
G = 3.Fi22:  point stabilizer O8+(2).3.2, ranks [ 10 ]
[ "1a+351ab+3080a+13650a+19305ab+42120ab+45045a" ]
gap> s:= CharacterTable( "O8+(2).3" );;
gap> fus:= PossibleClassFusions( s, tbl );;
gap> facttbl:= CharacterTable( "Fi22" );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> outer:= Difference( [ 1 .. NrConjugacyClasses( s ) ],
>                ClassPositionsOfDerivedSubgroup( s ) );;
gap> outerfus:= List( fus, map -> map{ outer } );
[ [ 13, 13, 18, 18, 46, 46, 50, 50, 59, 59, 75, 75, 95, 95, 98, 98, 95, 95,
      116, 116, 142, 142, 148, 148, 157, 157, 160, 160 ],
  [ 14, 15, 18, 18, 47, 48, 51, 52, 59, 59, 76, 77, 96, 97, 99, 100, 96, 97,
      116, 116, 143, 144, 149, 150, 158, 159, 161, 162 ],
  [ 15, 14, 18, 18, 48, 47, 52, 51, 59, 59, 77, 76, 97, 96, 100, 99, 97, 96,
      116, 116, 144, 143, 150, 149, 159, 158, 162, 161 ] ]
gap> preim:= InverseMap( factfus )[5];
[ 13, 14, 15 ]
gap> List( outerfus, x -> List( preim, i -> i in x ) );
[ [ true, false, false ], [ false, true, true ], [ false, true, true ] ]
gap> VerifyCandidates( s, tbl, 0, faith[5], "all" );
G = 3.Fi22:  point stabilizer O8+(2).3, ranks [ 11, 17 ]
[ "1a+1001a+3080a+10725a+13650a+27027ab+45045a+50050a+96525ab",
  "1a+351ab+1001a+3080a+7722ab+10725a+13650a+19305ab+42120ab+45045a+50050a+540\
54ab" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, 0, faith[6], "all" );
G = 3.Fi22:  point stabilizer O8+(2).2, ranks [ 17 ]

[ "1a+351ab+429a+3080a+13650a+19305ab+27027ab+42120ab+45045a+48048a+75075a+965\
25ab" ]

Lemma 2.1 applies to the maximal subgroups of the types 2^{6} : S_{6}(2) and ^{2}F_{4}(2)′ in F i_{22} and their preimages in 3.F i_{22}.

gap> VerifyCandidates( CharacterTable( "2^6:s6f2" ), tbl, 0, faith[8], "all" );
G = 3.Fi22:  point stabilizer 2^6:s6f2, ranks [ 24 ]
[ "1a+351ab+429a+1430a+3080a+13650a+19305ab+27027ab+30030a+42120ab+45045a+7507\
5a+96525ab+123552ab+205920a+320320a+386100ab" ]
gap> VerifyCandidates( CharacterTable( "2F4(2)'" ), tbl, 0, faith[9], "all" );
G = 3.Fi22:  point stabilizer 2F4(2)', ranks [ 25 ]
[ "1a+1001a+1430a+13650a+19305ab+27027ab+30030a+51975ab+289575a+386100ab+40040\
0ab+405405ab+579150a+675675a+1201200a+1351350efgh" ]
gap> CompareWithDatabase( "3.Fi22", faith );

3.31 G = 3.F i_{22}.2

The group 3.F i_{22}.2 has seven faithful multiplicity-free permutation actions, with point stabilizers of the types O_{8}^{+}(2) : S_{3} × 2, O_{8}^{+}(2) : 3 × 2, O_{8}^{+}(2) : S_{3} (twice), O_{8}^{+}(2) : 2 × 2, 2^{7} : S_{6}(2), and ^{2}F_{4}(2).

gap> tbl2:= CharacterTable( "3.Fi22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
4:  subgroup $O_8^+(2).3.2 \times 2$, degree 185328 (1 cand.)
6:  subgroup $O_8^+(2).3 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 370656 (
1 cand.)
7:  subgroup $O_8^+(2).S_3 \leq O_8^+(2).3.2 \times 2$, degree 370656 (
2 cand.)
8:  subgroup $O_8^+(2).2 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 555984 (
1 cand.)
9:  subgroup $O_8^+(2).3 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
14:  subgroup $2^7:S_6(2)$, degree 2084940 (1 cand.)
16:  subgroup ${^2F_4(2)}$, degree 10777536 (1 cand.)

Let H be a subgroup of the type O_{8}^{+}(2) : S_{3} × 2 in F i_{22}.2; it induces the 4th multiplicity-free permutation character of F i_{22}.2. The intersection of H with F i_{22} is of the type O_{8}^{+}(2) : S_{3}; it lifts to a direct product in 3.F i_{22}, which contains one subgroup of the type O_{8}^{+}(2) : S_{3} that is normal in the preimage of H. By Lemma 2.2, we get one class of subgroups of the type O_{8}^{+}(2) : S_{3} × 2 in 3.F i_{22}.2. The same argument yields one class of each of the types O_{8}^{+}(2) : 3 × 2 and O_{8}^{+}(2) : 2 × 2, which arise from the 6th and 8th multiplicity-free permutation character of F i_{22}.2, respectively.

gap> s:= CharacterTable( "O8+(2).S3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[4], "all" );
G = 3.Fi22.2:  point stabilizer O8+(2).3.2xC2, ranks [ 7 ]
[ "1a^++351ab+3080a^++13650a^++19305ab+42120ab+45045a^+" ]
gap> s:= CharacterTable( "O8+(2).3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" );
G = 3.Fi22.2:  point stabilizer O8+(2).3xC2, ranks [ 12 ]
[ "1a^++351ab+1001a^-+3080a^++7722ab+10725a^++13650a^++19305ab+42120ab+45045a^\
++50050a^++54054ab" ]
gap> s:= CharacterTable( "O8+(2).2" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[8], "all" );
G = 3.Fi22.2:  point stabilizer O8+(2).2xC2, ranks [ 12 ]
[ "1a^++351ab+429a^++3080a^++13650a^++19305ab+27027ab+42120ab+45045a^++48048a^\
++75075a^++96525ab" ]

Let H be a subgroup of the type O_{8}^{+}(2) : S_{3} in F i_{22}.2 that is not contained in F i_{22}; it induces the 7-th multiplicity-free permutation character of F i_{22}.2. The intersection of H with F i_{22} is of the type O_{8}^{+}(2) : 3; it lifts to a direct product in 3.F i_{22}, which contains four subgroups of the type O_{8}^{+}(2) : 3, three of them not containing the centre of 3.F i_{22}. By Lemma 2.2, we get three subgroups of the type O_{8}^{+}(2) : S_{3} in 3.F i_{22}.2, two of which are conjugate; they induce two different permutation characters, so we get two classes.

(Since there are O_{8}^{+}(2).S_{3} type subgroups also inside 3.F i_{22}, we must use "extending" as the last

argument of VerifyCandidates.)
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable("Fi22.2");;
gap> sfustbl2:= PossibleClassFusions( s, tbl2,
>        rec( permchar:= faith[7][1] ) );;
gap> ForAll( sfustbl2,
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[7], "extending" );
G = 3.Fi22.2:  point stabilizer O8+(2).3.2, ranks [ 9, 12 ]
[ "1a^++1001a^++3080a^++10725a^-+13650a^++27027ab+45045a^++50050a^-+96525ab",
  "1a^++351ab+1001a^++3080a^++7722ab+10725a^-+13650a^++19305ab+42120ab+45045a^\
++50050a^-+54054ab" ]

The nineth multiplicity-free permutation character of F i_{22}.2 is induced from a subgroup of the type O_{8}^{+}(2).3 that lies inside F i_{22} and is known to lift to s group of the type 3 × O_{8}^{+}(2).3 in 3.F i_{22}. All subgroups of index three in this group either contain the centre of 3.F i_{22} or have the type O_{8}^{+}(2).3, and it turns out that the permutation characters of 3.F i_{22}.2 induced from these subgroups are not multiplicity-free. So the candidate can be excluded.

gap> VerifyCandidates( CharacterTable( "O8+(2).3" ), tbl, tbl2, faith[9], "all" );
G = 3.Fi22.2:  no O8+(2).3
gap> faith[9]:= [];;

Lemma 2.2 guarantees the existence of one class of subgroups of each of the types 2^{7} : S_{6}(2) and 2F_{4}(2).

gap> VerifyCandidates( CharacterTable( "2^7:S6(2)" ), tbl, tbl2, faith[14], "all" );
G = 3.Fi22.2:  point stabilizer 2^7:S6(2), ranks [ 17 ]
[ "1a^++351ab+429a^++1430a^++3080a^++13650a^++19305ab+27027ab+30030a^++42120ab\
+45045a^++75075a^++96525ab+123552ab+205920a^++320320a^++386100ab" ]
gap> VerifyCandidates( CharacterTable( "2F4(2)" ), tbl, tbl2, faith[16], "all" );
G = 3.Fi22.2:  point stabilizer 2F4(2)'.2, ranks [ 17 ]
[ "1a^++1001a^++1430a^++13650a^++19305ab+27027ab+30030a^++51975ab+289575a^-+38\
6100ab+400400ab+405405ab+579150a^++675675a^-+1201200a^-+1351350efgh" ]
gap> CompareWithDatabase( "3.Fi22.2", faith );

3.32 G = 6.F i_{22}

The group 6.F i_{22} has six faithful multiplicity-free permutation actions, with point stabilizers of the types O_{8}^{+}(2) : S_{3} (twice), O_{8}^{+}(2) : 3 (twice), and O_{8}^{+}(2) : 2 (twice).

gap> tbl:= CharacterTable( "6.Fi22" );;
gap> facttbl:= CharacterTable( "3.Fi22" );;
gap> faith:= FaithfulCandidates( tbl, "3.Fi22" );;

1:  subgroup $O_8^+(2):S_3 \rightarrow (Fi_{22},4)$, degree 370656 (2 cand.)
2:  subgroup $O_8^+(2):3 \rightarrow (Fi_{22},5)$, degree 741312 (1 cand.)
3:  subgroup $O_8^+(2):3 \rightarrow (Fi_{22},5)$, degree 741312 (1 cand.)
4:  subgroup $O_8^+(2):2 \rightarrow (Fi_{22},6)$, degree 1111968 (2 cand.)

From the discussion of the cases 2.F i_{22} and 3.F i_{22}, we conclude that the maximal subgroups of the type O_{8}^{+}(2).S_{3} lift to groups of the type 6 × O_{8}^{+}(2).S_{3} in 6.F i_{22}. So Lemma 2.3 (iii) yields two classes of O_{8}^{+}(2) : S_{3} type subgroups, which induce different permutation characters.

gap> s:= CharacterTable( "O8+(2).S3" );;
gap> s0:= CharacterTable( "O8+(2).3" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl, "all" );
6.Fi22:  O8+(2).3.2 lifts to a direct product,
proved by squares in [ 1, 22, 28, 30, 46, 55, 76, 104, 131, 141, 215 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl ),
>        map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, 0, faith[1], "all" );
G = 6.Fi22:  point stabilizer O8+(2).3.2, ranks [ 14, 14 ]
[ "1a+351ab+3080a+13650a+13728b+19305ab+42120ab+45045a+48048c+61776cd",
  "1a+351ab+3080a+13650a+13728a+19305ab+42120ab+45045a+48048b+61776ab" ]

Each subgroup of the type O_{8}^{+}(2) : 3 in 3.F i_{22} lifts to a direct product in 6.F i_{22}, which yields one action; as the two constituents that are permutation characters of 3.F i_{22} are different, we get two different permutation characters induced from O_{8}^{+}(2) : 3.

gap> VerifyCandidates( CharacterTable( "O8+(2).3" ), tbl, 0,
>        Concatenation( faith[2], faith[3] ), "all" );
G = 6.Fi22:  point stabilizer O8+(2).3, ranks [ 17, 25 ]
[ "1a+1001a+3080a+10725a+13650a+13728ab+27027ab+45045a+48048bc+50050a+96525ab+\
123552cd",
  "1a+351ab+1001a+3080a+7722ab+10725a+13650a+13728ab+19305ab+42120ab+45045a+48\
048bc+50050a+54054ab+61776abcd" ]

Each subgroup of the type O_{8}^{+}(2) : 2 in 3.F i_{22} lifts to a direct product in 6.F i_{22}, which yields two actions; the permutation characters are different by the argument used for O_{8}^{+}(2) : S_{3}.

gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, 0, faith[4], "all" );
G = 6.Fi22:  point stabilizer O8+(2).2, ranks [ 25, 25 ]
[ "1a+351ab+352a+429a+3080a+13650a+13728b+19305ab+27027ab+42120ab+45045a+48048\
ac+61776cd+75075a+96525ab+123200a+123552cd",
  "1a+351ab+352a+429a+3080a+13650a+13728a+19305ab+27027ab+42120ab+45045a+48048\
ab+61776ab+75075a+96525ab+123200a+123552cd" ]
gap> CompareWithDatabase( "6.Fi22", faith );

(Note that the rank 17 permutation character above was missing in the first version of [LM].)

3.33 G = 6.F i_{22}.2

The group 6.F i_{22}.2 that is printed in the Atlas has four faithful multiplicity-free permutation actions, with point stabilizers of the types O_{8}^{+}(2) : 3 × 2 (twice) and ^{2}F_{4}(2) (twice).

gap> tbl2:= CharacterTable( "6.Fi22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
6:  subgroup $O_8^+(2).3 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (
1 cand.)
16:  subgroup ${^2F_4(2)}$, degree 21555072 (1 cand.)

Each O_{8}^{+}(2) : 3×2 type subgroup of 3.F i_{22}.2 gives rise to two subgroups of the same type in 6.F i_{22}.2, so we get two classes inducing the same permutation character. (Since there are O_{8}^{+}(2).3 × 2 type subgroups also inside 6.F i_{22}, we must use "extending" as the last argument of VerifyCandidates.)

gap> s:= CharacterTable( "O8+(2).3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "extending" );
G = 6.Fi22.2:  point stabilizer O8+(2).3xC2, ranks [ 16 ]
[ "1a^++351ab+1001a^-+3080a^++7722ab+10725a^++13650a^++13728ab+19305ab+42120ab\
+45045a^++48048bc+50050a^++54054ab+61776adbc" ]

The subgroup of the type 6 ×^{2}F_{4}(2)′ of 6.F i_{22} extends to 6 ×^{2}F_{4}(2) in 6.F i_{22}.2, which contains two subgroups of the type ^{2}F_{4}(2), by Lemma 2.3; so we get two classes of such subgroups, which induce the same permutation character.

gap> VerifyCandidates( CharacterTable( "2F4(2)" ), tbl, tbl2, faith[16], "all" );
G = 6.Fi22.2:  point stabilizer 2F4(2)'.2, ranks [ 22 ]
[ "1a^++1001a^++1430a^++13650a^++19305ab+27027ab+30030a^++51975ab+133056a^{\\p\
m}+289575a^-+386100ab+400400ab+405405ab+579150a^++675675a^-+1201200a^-+1351350\
efgh+1663200ab+1796256adbc" ]
gap> faith[6]:= faith[6]{ [ 1, 1 ] };;
gap> faith[16]:= faith[16]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "6.Fi22.2", faith );

The group (6.F i_{22}.2)^{∗} of the isoclinism type that is not printed in the Atlas has two faithful multiplicity-free permutation actions, with point stabilizers of the type O_{8}^{+}(2) : S_{3} (twice).

gap> facttbl:= CharacterTable( "Fi22.2" );;
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
7:  subgroup $O_8^+(2).S_3 \leq O_8^+(2).3.2 \times 2$, degree 741312 (
2 cand.)

The existence of O_{8}^{+}(2) : S_{3} type subgroups (not contained in 6.F i_{22}) follows from Lemma 2.2 and the existence of these subgroups in (2.F i_{22}.2)^{∗}; we get one class for each of the two classes in (2.F i_{22}.2)^{∗}, with different permutation characters.

gap> s:= CharacterTable( "O8+(2).S3" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[7], "extending" );
G = Isoclinic(6.Fi22.2):  point stabilizer O8+(2).3.2, ranks [ 12, 16 ]
[ "1a^++1001a^++3080a^++10725a^-+13650a^++13728ab+27027ab+45045a^++48048bc+500\
50a^-+96525ab+123552cd",
  "1a^++351ab+1001a^++3080a^++7722ab+10725a^-+13650a^++13728ab+19305ab+42120ab\
+45045a^++48048bc+50050a^-+54054ab+61776adbc" ]
gap> CompareWithDatabase( "Isoclinic(6.Fi22.2)", faith );

3.34 G = 2.Co_{1}

The group 2.Co_{1} has two faithful multiplicity-free permutation actions, with point stabilizers of the types Co_{2} and Co_{3}, respectively, by Lemma 2.1.

gap> tbl:= CharacterTable( "2.Co1" );;
gap> faith:= FaithfulCandidates( tbl, "Co1" );;
1:  subgroup $Co_2$, degree 196560 (1 cand.)
5:  subgroup $Co_3$, degree 16773120 (1 cand.)
gap> VerifyCandidates( CharacterTable( "Co2" ), tbl, 0, faith[1], "all" );
G = 2.Co1:  point stabilizer Co2, ranks [ 7 ]
[ "1a+24a+299a+2576a+17250a+80730a+95680a" ]
gap> VerifyCandidates( CharacterTable( "Co3" ), tbl, 0, faith[5], "all" );
G = 2.Co1:  point stabilizer Co3, ranks [ 12 ]
[ "1a+24a+299a+2576a+17250a+80730a+95680a+376740a+1841840a+2417415a+5494125a+6\
446440a" ]
gap> CompareWithDatabase( "2.Co1", faith );

3.35 G = 3.F_{3+}

The group 3.F_{3+} has two faithful multiplicity-free permutation actions, with point stabilizers of the types F i_{23} and O_{10}^{−}(2), respectively, by Lemma 2.1.

gap> tbl:= CharacterTable( "3.F3+" );;
gap> faith:= FaithfulCandidates( tbl, "F3+" );;
1:  subgroup $Fi_{23}$, degree 920808 (1 cand.)
2:  subgroup $O_{10}^-(2)$, degree 150532080426 (1 cand.)
gap> VerifyCandidates( CharacterTable( "Fi23" ), tbl, 0, faith[1], "all" );
G = 3.F3+:  point stabilizer Fi23, ranks [ 7 ]
[ "1a+783ab+57477a+249458a+306153ab" ]
gap> VerifyCandidates( CharacterTable( "O10-(2)" ), tbl, 0, faith[2], "all" );
G = 3.F3+:  point stabilizer O10-(2), ranks [ 43 ]
[ "1a+783ab+8671a+57477a+64584ab+249458a+306153ab+555611a+1666833a+6724809ab+1\
9034730ab+35873145a+43779879ab+48893768a+79452373a+195019461ab+203843871ab+415\
098112a+1050717096ab+1264015025a+1540153692a+1818548820ab+2346900864a+32086535\
25a+10169903744a+10726070355ab+13904165275a+15016498497ab+17161712568a+2109675\
1104ab" ]
gap> CompareWithDatabase( "3.F3+", faith );

3.36 G = 3.F_{3+}.2

The group 3.F_{3+}.2 has two faithful multiplicity-free permutation actions, with point stabilizers of the types F i_{23} × 2 and O_{10}^{−}(2).2, respectively, by Lemma 2.2.

gap> tbl2:= CharacterTable( "3.F3+.2" );;
gap> faith:= FaithfulCandidates( tbl2, "F3+.2" );;
1:  subgroup $Fi_{23} \times 2$, degree 920808 (1 cand.)
3:  subgroup $O_{10}^-(2).2$, degree 150532080426 (1 cand.)
gap> VerifyCandidates( CharacterTable( "2xFi23" ), tbl, tbl2, faith[1], "all" );
G = 3.F3+.2:  point stabilizer 2xFi23, ranks [ 5 ]
[ "1a^++783ab+57477a^++249458a^++306153ab" ]
gap> VerifyCandidates( CharacterTable( "O10-(2).2" ), tbl, tbl2, faith[3], "all" );
G = 3.F3+.2:  point stabilizer O10-(2).2, ranks [ 30 ]
[ "1a^++783ab+8671a^-+57477a^++64584ab+249458a^++306153ab+555611a^-+1666833a^+\
+6724809ab+19034730ab+35873145a^++43779879ab+48893768a^-+79452373a^++195019461\
ab+203843871ab+415098112a^-+1050717096ab+1264015025a^++1540153692a^++181854882\
0ab+2346900864a^-+3208653525a^++10169903744a^-+10726070355ab+13904165275a^++15\
016498497ab+17161712568a^++21096751104ab" ]
gap> CompareWithDatabase( "3.F3+.2", faith );

3.37 G = 2.B

The group 2.B has one faithful multiplicity-free permutation action, with point stabilizer of the type F i_{23}, by Lemma 2.1.

gap> tbl:= CharacterTable( "2.B" );;
gap> faith:= FaithfulCandidates( tbl, "B" );;
4:  subgroup $Fi_{23}$, degree 2031941058560000 (1 cand.)
gap> VerifyCandidates( CharacterTable( "Fi23" ), tbl, 0, faith[4], "all" );
G = 2.B:  point stabilizer Fi23, ranks [ 34 ]
[ "1a+4371a+96255a+96256a+9458750a+10506240a+63532485a+347643114a+356054375a+4\
10132480a+4221380670a+4275362520a+8844386304a+9287037474a+13508418144a+3665765\
3760a+108348770530a+309720864375a+635966233056a+864538761216a+1095935366250a+4\
322693806080a+6145833622500a+6619124890560a+10177847623680a+12927978301875a+38\
348970335820a+60780833777664a+89626740328125a+110949141022720a+211069033500000\
a+284415522641250b+364635285437500a+828829551513600a" ]
gap> CompareWithDatabase( "2.B", faith );

4 Appendix: Explicit Computations with Groups

Only in the proofs for the groups involving M_{22}, explicit computations with the groups were necessary. These computations are collected in this appendix.

4.1 2^{4} : A_{6} type subgroups in 2.M_{22}

We show that the preimage in 2.M_{22} of each maximal subgroup of the type 2^{4} : A_{6} in M_{22} contains one class of subgroups of the type 2 × 2^{4} : A_{5}. For that, we first note that there are two classes of subgroups of the type 2^{4} : A_{5} inside 2^{4} : A_{6}, and that the A_{5} subgroups lift to groups of the type 2 × A_{5} because 2.M_{22} does not admit an embedding of 2.A_{6}.

gap> tbl:= CharacterTable( "2.M22" );;
gap> PossibleClassFusions( CharacterTable( "2.A6" ), tbl );
[  ]

Now we fetch a permutation representation of 2.M_{22} on 352 points, from the Atlas of Group Representations (see [Wil]), via the GAP package AtlasRep (see [Bre04a]), and compute generators for the second class of maximal subgroups, via the straight line program for M_{22}.

gap> LoadPackage( "atlasrep" );
true
gap> gens:= OneAtlasGeneratingSet( "2.M22", NrMovedPoints, 352 );;
gap> slp:= AtlasStraightLineProgram( "M22", "maxes", 2 );;
gap> sgens:= ResultOfStraightLineProgram( slp.program, gens.generators );;
gap> s:= Group( sgens );;  Size( s );
11520
gap> 2^5 * 360;
11520

The subgroup acts intransitively on the 352 points. We switch to the representation on 192 points, and compute the normal subgroup N of order 2^{5}.

gap> orbs:= Orbits( s, MovedPoints( s ) );;
gap> List( orbs, Length );
[ 160, 192 ]
gap> s:= Action( s, orbs[2] );;
gap> Size( s );
11520
gap> syl2:= SylowSubgroup( s, 2 );;
gap> repeat
>   x:= Random( syl2 );
>   n:= NormalClosure( s, SubgroupNC( s, [ x ] ) );
> until Size( n ) = 32;

The point stabilizer S in this group has type A_{5}, and generates together with N one of the desired subgroups of the type 2^{5} : A_{5}. However, S does not normalize a subgroup of order 2^{4}, and so there is no subgroup of the type 2^{4} : A_{5}.

gap> stab:= Stabilizer( s, 192 );;
gap> sub:= ClosureGroup( n, stab );;
gap> Size( sub );
1920
gap> Set( List( Elements( n ),
>         x -> Size( NormalClosure( sub, SubgroupNC( sub, [ x ] ) ) ) ) );
[ 1, 2, 32 ]

A representative of the other class of A_{5} type subgroups can be found by taking an element x of order three that is not conjugate to one in S, and to choose an element y of order five such that the product is an involution.

gap> syl3:= SylowSubgroup( s, 3 );;
gap> repeat three:= Random( stab ); until Order( three ) = 3;
gap> repeat other:= Random( syl3 );
>        until Order( other ) = 3 and not IsConjugate( s, three, other );
gap> syl5:= SylowSubgroup( s, 5 );;
gap> repeat y:= Random( syl5 )^Random( s ); until Order( other*y ) = 2;
gap> a5:= Group( other, y );;
gap> IsConjugate( s, a5, stab );
false
gap> sub:= ClosureGroup( n, a5 );;
gap> Size( sub );
1920
gap> Set( List( Elements( n ),
>         x -> Size( NormalClosure( sub, SubgroupNC( sub, [ x ] ) ) ) ) );
[ 1, 2, 16, 32 ]

This proves the existence of one class of the desired subgroups. Finally, we show that the character table of these groups is indeed the one we used in Section 3.3.

gap> g:= First( Elements( n ),
>       x -> Size( NormalClosure( sub, SubgroupNC( sub, [ x ] ) ) ) = 16 );;
gap> compl:= ClosureGroup( a5, g );;
gap> Size( compl );
960
gap> tbl:= CharacterTable( compl );;
gap> IsRecord( TransformingPermutationsCharacterTables( tbl,
>        CharacterTable( "P1/G1/L1/V1/ext2" ) ) );
true

4.2 2^{4} : S_{5} type subgroups in M_{22}.2

A maximal subgroup of the type 2^{4} : S_{6} in M_{22}.2 is perhaps easiest found as the point stabilizer in the degree 77 permutation representation. In order to find its index 6 subgroups, the degree 22 permutation representation of M_{22}.2 is more suitable because the restriction to the 2^{4} : S_{6} type subgroup has orbits of the lengths 6 and 16, where the action of the orbit of length 6 is the natural permutation action of S_{6}.

So we choose the sum of the two representations, of total degree 99. For convenience, we find this representation as the point stabilizer in the degree 100 representation of HS.2, which is contained in the Atlas of Group Representations (see [Wil]).

gap> gens:= OneAtlasGeneratingSet( "HS.2", NrMovedPoints, 100 );;
gap> stab:= Stabilizer( Group( gens.generators ), 100 );;
gap> orbs:= Orbits( stab, MovedPoints( stab ) );;
gap> List( orbs, Length );
[ 77, 22 ]
gap> pnt:= First( orbs, x -> Length( x ) = 77 )[1];;
gap> m:= Stabilizer( stab, pnt );;
gap> Size( m );
11520

Now we find two nonconjugate subgroups of the type 2^{4} : S_{5} as the stabilizer of a point and of a total in S_{6}, respectively (cf. [CCN^{+}85, p. 4]).

gap> orbs:= Orbits( m, MovedPoints( m ) );;
gap> List( orbs, Length );
[ 60, 16, 6, 16 ]
gap> six:= First( orbs, x -> Length( x ) = 6 );;
gap> p:= ( six[1], six[2] )( six[3], six[4] )( six[5], six[6] );;
gap> conj:= ( six[2], six[4], six[5], six[6], six[3] );;
gap> total:= List( [ 0 .. 4 ], i -> p^( conj^i ) );;
gap> stab1:= Stabilizer( m, six[1] );;
gap> stab2:= Stabilizer( m, Set( total ), OnSets );;
gap> IsConjugate( m, stab1, stab2 );
false

We identify the character tables of the two groups in the GAP Character Table Library.

gap> s1:= CharacterTable( stab1 );;
gap> s2:= CharacterTable( stab2 );;
gap> NrConjugacyClasses( s1 );  NrConjugacyClasses( s2 );
12
18
gap> lib1:= CharacterTable( "2^4:s5" );;
gap> IsRecord( TransformingPermutationsCharacterTables( lib1, s1 ) );
true
gap> lib2:= CharacterTable( "w(d5)" );;
gap> IsRecord( TransformingPermutationsCharacterTables( lib2, s2 ) );
true

The first subgroup does not lead to multiplicity-free permutation characters of 2.M_{22}.2. Note that there are two classes of subgroups of this type in M_{22}.2, one of them is contained in M_{22} and the other is not. The action on the cosets of the former is multiplicity-free, but it does not lift to a multiplicity-free candidate of 2.M_{22}.2; and the action on the cosets of the latter is not multiplicityfree.

gap> tbl:= CharacterTable( "M22" );;
gap> tbl2:= CharacterTable( "M22.2" );;
gap> s1fustbl2:= PossibleClassFusions( s1, tbl2 );
[ [ 1, 2, 2, 4, 4, 2, 5, 3, 7, 5, 10, 6 ],
  [ 1, 2, 12, 15, 4, 2, 5, 3, 16, 15, 17, 6 ] ]
gap> pi:= List( s1fustbl2, map -> Induced( s1, tbl2,
>             [ TrivialCharacter( s1 ) ], map )[1] );
[ Character( CharacterTable( "M22.2" ), [ 462, 46, 12, 6, 6, 2, 4, 0, 0, 2,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( "M22.2" ), [ 462, 30, 12, 2, 2, 2, 0, 0, 0, 0,
      0, 56, 0, 0, 12, 2, 2, 0, 0, 0, 0 ] ) ]
gap> PermCharInfoRelative( tbl, tbl2, pi ).ATLAS;
[ "1a^{\\pm}+21a^{\\pm}+55a^{\\pm}+154a^{\\pm}",
  "1a^++21(a^+)^{2}+55a^++154a^++210a^+" ]

So only the second type of 2^{4} : S_{5} type subgroups can lift to the multiplicity-free candidate in question, and this situation is dealt with in Section 3.4.

gap> s2fustbl2:= PossibleClassFusions( s2, tbl2 );;
gap> pi:= List( s2fustbl2, map -> Induced( s2, tbl2,
>             [ TrivialCharacter( s2 ) ], map )[1] );
[ Character( CharacterTable( "M22.2" ), [ 462, 30, 3, 2, 2, 2, 3, 0, 0, 0, 0,
      28, 20, 4, 8, 1, 2, 0, 1, 0, 0 ] ) ]
gap> PermCharInfoRelative( tbl, tbl2, pi ).ATLAS;
[ "1a^++21a^++55a^++154a^++231a^-" ]

References

[BL96] Thomas Breuer and Klaus Lux, The multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups, Comm. Alg. 24 (1996), no. 7, 2293– 2316.

[Bre04a] Thomas Breuer, Manual for the GAP 4 Package AtlasRep, Version 1.2, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 2004.

[Bre04b] Thomas Breuer, Manual for the GAP Character Table Library, Version 1.1, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 2004.

[CCN^{+}85] J[ohn] H. Conway, R[obert] T. Curtis, S[imon] P. Norton, R[ichard] A. Parker, and R[obert] A. Wilson, Atlas of finite groups, Oxford University Press, 1985.

[GAP04] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4, 2004,

      http://www.gap-system.org.

[Höh01] Ines Höhler, Vielfachheitsfreie Permutationsdarstellungen und die Invarianten zugehöriger Graphen, Examensarbeit, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 2001.

[LM] S. A. Linton and Z. E. Mpono, Multiplicity-free permutation characters of covering groups of sporadic simple groups.

[Mül03] Jürgen Müller, On endomorphism rings and character tables, Habilitationsschrift, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 2003.

[Wil]  Robert  A.  Wilson,  ATLAS  of  Finite  Group  Representations,

      http://www.mat.bham.ac.uk/atlas/.