# The Projective Character Tables of a Solvable Group26:6×2

Authors: Abraham Love Prins
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

The Chevalley–Dickson simple groupG24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.


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## Body

## 1. Introduction

The Chevalley–Dickson simple groupG24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has exactly eight conjugacy classes of maximal subgroups [1]. One of these classes of maximal subgroups of index 1365 in G24 is a 2-local subgroup of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Also, it can be mentioned that 24+6:A5×3 is a parabolic subgroup of G24.The group24+6:A5×3 is the normalizer NG242A4 [1] of an elementary abelian 2-group 2A4 of order 16 in G24, where the generators of 2A4 are 4 commuting involutions found in the class of involutions 2A of G24. Hence, we can construct 24+6:A5×3 as an extension group of N=24 by G=26:A5×3. Using a permutation representation of degree 416 of 24+6:A5×3 found in [2], we can easily verify with the help of the computer algebra system GAP [3] that 24+6:A5×3 exists as a nonsplit extension group G¯=24·26:A5×3. In fact, from the character table of 24+6:A5×3 uploaded in the GAP library, we can deduce that 26 is also normal in 24+6:A5×3. Therefore, with the aid of GAP, it can be easily shown that the group 24+6:A5×3 is also isomorphic to a nonsplit extension of the form 26·24:A5×3.SinceNG242A4≅G¯, the action of G=26:A5×3 on N=24 splits N into two orbits of lengths 1 and 15. By Brauer’s theorem in [4], G also acts on IrrN with two orbits of lengths 1 and 15 and the corresponding inertia factor subgroups are of the forms H1=G and H2=26:6×2. If one wants to construct the ordinary character table of G¯ by the technique of Fischer–Clifford matrices [5], the irreducible ordinary characters IrrH1 of H1 and either the ordinary irreducible characters IrrH2 or a set of irreducible projective characters IrrProjH2,α−1 of H2 with associated nontrivial factor set α−1 are needed. Readers are referred to [6] on a survey of Fischer–Clifford theory.From the uploaded character table in the GAP library, we found thatG¯ has 42 conjugacy classes. Hence, by Fischer–Clifford theory, IrrG¯ = IrrG+IrrProjH2,α−1 = 42. Therefore, IrrProjH2,α−1=16, where IrrG=26. Since IrrH2=30, certainly a set of irreducible projective characters with nontrivial factor set α−1 of H2 is required in the construction of IrrG¯ via Fischer–Clifford theory.Using computations in GAP, the Schur multiplierMH2≅24 of H2 is identified as a group isomorphic to the elementary abelian 2-group 24. Therefore, the group H2 can have up to 15 sets of distinct irreducible projective characters IrrProjH2,αi−1, i=2,3,4,…,16, such that α2∼1. In this paper, the author presents a method (based on the work done in [7, 8]) to compute all the distinct irreducible projective character tables of H2. The author will also show how to choose the appropriate set IrrProjH2,α−1 for H2 if one wants to construct the set IrrG¯ using the technique of Fischer–Clifford matrices.A proof is given of a result, which states that the number of irreducible projective charactersIrrProjG,α−1 of a finite group G associated with some factor set α−1 is always less or equal to the number of the ordinary irreducible characters IrrG of G. Based on the abovementioned proof, GAP codes are developed to find the number IrrProjH2,αi−1 of irreducible projective characters of H2 associated with each factor set αi−1 in a cohomology class αi of MH2. Interested readers are referred to [9–13] for definitions on concepts in ordinary and projective character theory. Computations are carried out with the aid of the computer algebra systems MAGMA [14] and GAP [3], and the notation of ATLAS [1] is mostly used.


## 2. Preliminary Results on Projective Characters

In this section, a brief overview of relevant projective character theory pertaining to our study is given.G will always denote a finite group. Also, a proof of a proposition is given which states that the number of irreducible projective characters of a finite group G associated with some factor set α is always less or equal to the number of the ordinary irreducible characters of G.Definition 1.
A functionα: G×G⟶ℂ∗ is called a factor set of G if αxy,zαx,y=αx,yzαy,z for all x,y,z∈G.
The set of all equivalence classes of factor sets ofG forms a finite abelian group, called the Schur multiplier, and is denoted by MG.Definition 2.
A projective representation of a groupG of degree n over the complex numbers is a map P:G⟶GLn,ℂ, such that(i)
P1=In(ii)
Givenx,y∈G, there exists αx,y∈ℂ∗ such that PxPy=αx,yPxy
The mapα is called the factor set associated with P.
LetP be a projective representation of G with factor set α. Define κg=TracePg for all g∈G. Then, κ is called a projective character of G. We say that κ is irreducible if P is, and κ has a factor set α, where α is the factor set of P.
LetIrrProjG,α denote the set of irreducible projective characters of G associated with the factor set α. An element x∈G is said to be α-regular if αx,g=αg,x for all g∈CGx. It is well known that g∈G is α-regular if and only if κg≠0 for some κ∈IrrProjG,α or equivalently that g is α-irregular if and only if κg=0 for all κ∈IrrProjG,α. The number of irreducible projective characters with factor set α equals the number of α-regular classes of a group G. Projective characters also satisfy the usual orthogonality relations and have analogues to ordinary characters.
As we will see later in Section4, Definition 3 and Remark 1 will play an important role in the computation of the irreducible projective characters of H2.Definition 3.
A groupR is a representation group for G if there exists a homomorphism π from R onto G such that (i) A=kerπ≅MG and (ii) A≤ZR∩R′.Remark 1.
A covering groupC for G will normally be a quotient of R by a subgroup B of A. If A/B has order n, we sometimes refer to the covering group as a n-fold cover of G. Projective representations of G are found in the representation group R for all the equivalence classes of factor sets in MG. However, in an n-fold cover C of G, only the n equivalence classes which C covers will be represented [7].
The following proposition (see [15, 16]) is useful to determine the number of irreducible projective characters of a group G associated with a certain factor set α. It tells us also under which condition IrrProjG,α is strictly less than IrrG. In Section 4, GAP codes (based on Proposition 1) will be used to compute the number IrrProjG,αi−1 of all irreducible projective characters of H2 found in a set IrrProjG,αi−1.Proposition 1.
LetR=MG. G be a representation group of a finite group G, where MG denotes the Schur multiplier of G. Then, the number of irreducible characters IrrR of R which lie over a linear character θ∈IrrMG is less or equal to IrrG.Proof.
The number of irreducible characters IrrR of R which lies over a linear character θ∈IrrMG is given by ∑χ∈IrrR<χ↓MG,θ>/χ1. It is known that the quantity ∑χ∈IrrRχx/χ1≥0 for each x∈MG, and it is nonzero if x is a commutator in R. For any θ∈IrrMG, we have(1)∑x∈MG∑χ∈IrrRχxθx−1χ1≤∑x∈MG∑χ∈IrrRχxχ1=MGgR/MG,where gR/MG is the number of conjugacy classes of R/MG≅G. The last equality follows because the irreducible characters of R with MG in their kernels are precisely those which contain the trivial character on the restriction to MG. Hence, 1/MG∑x∈MG∑χ∈IrrRχxθx−1/χ1 = ∑χ∈IrrR1/MG∑x∈MGχxθx−1/χ1 = ∑χ∈IrrR<χ↓MG,θ>/χ1≤gR/MG = IrrG. Furthermore, if there is a nonidentity element x∈MG∖kerθ which is a commutator in R, then the inequality becomes strict.


## 3. Action of26:A5×3 on 24 and Irr24

As explained in the introductory section, the split extension24+6:A5×3 can be constructed as a nonsplit extension group G¯=24·26:A5×3 of an elementary abelian group N=24 by G=26:A5×3. Since G¯ is the normalizer of N in G24, the action of G¯ on N gives rise to two orbits of lengths 1 and 15. It follows that the action of G on N will also result in two orbits of lengths 1 and 15 with corresponding point stabilizers P1=G and P2=26:6×2. Then, by Brauer’s theorem [4], G also has two orbits of lengths 1 and 15 on IrrN with corresponding inertia factor subgroups H1=G and H2=26:6×2. See Table 1 for a summary of the action (which is self-dual) of 26:A5×3 on 24 and Irr24, respectively.Table 1
Action ofG on N and IrrN.
Action ofG on N
Action ofG on IrrN
Number and size of orbitsQi
O
1
=
1
O
2
=
15
O
1
=
1
O
2
=
15
Structure of stabilizersPi and Hi
P
1
=
2
6
:
A
5
×
3
P
2
=
2
6
:
6
×
2
H
1
=
2
6
:
A
5
×
3
H
2
=
2
6
:
6
×
2
Size of stabilizersPi and Hi
P
1
=
11520
P
2
=
768
H
1
=
11520
H
2
=
768
Number of conjugacy classesg of Pi and Hi
g
P
1
=
26
g
P
2
=
30
g
H
1
=
26
g
H
2
=
30


## 4. Projective Character Tables of Inertia FactorH2

In this section, all the sets IrrProjH2,αi−1 of irreducible projective characters of H2 with associated factor sets αi−1 will be computed. Appropriate GAP codes will be used to assist in the computational aspects of determining the sets IrrProjH2,αi−1. Readers are referred to [7, 8] for a background on the computational techniques being used in this section.Since the Schur multiplier ofMH2≅24 is isomorphic to the elementary abelian group 24 of order 16, we obtain that MH2 contains 15 cohomology classes αi of order 2 and the trivial class 1. Hence, there exist 15 sets of projective characters IrrProjH2,αi−1 lying above the nontrivial factor sets αi−1∈αi, i=2,3,4,…,16, such that αi2∼1. Note that the order H2=28.3 of H2 is the product of powers of two primes, and hence, by the Burnside’s paqb theorem [17], the group H2 is solvable. Since H2 is of relative small order and it is solvable, the following GAP codes [18] (based on Proposition 1) give the number of irreducible projective characters of H2 contained in each set IrrProjH2,αi−1:gap> h := H2gap> f := EpimorphismSchurCover(h)gap> f := InverseGeneralMapping(IsomorphismPcGroup(Source(f))) ∗ fgap> z := Kernel(f)gap> x := Source(f)gap> List(Irr(z), lambda −>Number(Irr(x), chi −>not IsZero(ScalarProduct(RestrictedClassFunction(chi,z), lambda))))30,18,26,14,16,4,16,4,16,4,16,4,16,4,16,4From the output of the above GAP codes, the number of irreducible projective charactersIrrProjH2,αi−1 of H2 contained in each set IrrProjH2,αi−1 is 30, 26, 18, 16, 16, 16, 16, 16, 16, 14, 4, 4, 4, 4, 4, and 4, respectively. The set containing the 30 projective characters is associated with the trivial factor set α1−1=1 of MG, and hence, they are the ordinary irreducible characters IrrH2 of H2. Note in the above GAP codes that “Sourcef” is the full representation group R=MG.H2 of H2 whereas “Kernelf” denotes the Schur multiplier MG.From Remark 1, the irreducible projective characters ofH2 with a given factor set αi−1 can be computed without using the full representation group R=MH2.H2 of H2. Since all the nontrivial factor sets αi−1 of IrrProjH2,αi−1 are of order two, the aim is to find 15 double covers Ci of H2 which contain the desired 15 sets IrrProjH2,αi−1 of projective characters of H2. The nontrivial maximal subgroups Mi of MH2 are computed within GAP, and we found there are exactly 15 of them of order 8 and shape 23. The 15 factor groups Ci=R/Mi, i=2,3,4,…,16, are the double covers 2i⋅H2 which contain the 15 sets IrrProjH2,αi−1 of desired irreducible projective characters of H2 with factor sets αi−1 such that αi2∼1. The following GAP code was used to “restrict” R to H2 to obtain the sets IrrProjH2,αi−1, i=2,3,4,…,16:gap> t:= CharacterTable(“H2”)gap>2t := CharacterTable(“Ci”)gap>F := GetFusionMap(2t, t)gap>map := ProjectionMap(F)gap>projchars := List(2t, x −>{xmap})From the setsIrrProjH2,αi−1 which contain 16 irreducible projective characters each, we obtain exactly 3 distinct irreducible projective character tables. Also, the sets IrrProjH2,αi−1 which contain 4 irreducible projective characters each, give rise to 3 distinct sets of irreducible projective character tables. Hence, we obtain 9 distinct sets IrrProjH2,αi−1 of irreducible projective characters for H2, and they are listed in Tables 2–10. The ordinary irreducible characters IrrH2 of H2 are found in Table 11. Note that the irreducible projective characters with associated factor set αi−1 have values of zero on the αi−1-irregular classes of G. Hence, the number of irreducible projective characters equals the number of αi−1-regular classes of G in each of the below tables. The above results are summarized in the following theorem:Table 2
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
4
0
4
0
0
0
0
0
0
0
1
1
B
B
0
0
D
D
4
0
0
0
1
1
B
B
0
0
D
D
χ
2
4
0
4
0
0
0
0
0
0
0
1
1
−B
−B
0
0
−D
−D
4
0
0
0
1
1
−B
−B
0
0
−D
−D
χ
3
4
0
4
0
0
0
0
0
0
0
A
/A
B
B
0
0
E
−/E
4
0
0
0
A
/A
B
B
0
0
E
−/E
χ
4
4
0
4
0
0
0
0
0
0
0
/A
A
B
B
0
0
−/E
E
4
0
0
0
/A
A
B
B
0
0
−/E
E
χ
5
4
0
4
0
0
0
0
0
0
0
A
/A
−B
−B
0
0
−E
/E
4
0
0
0
A
/A
−B
−B
0
0
−E
/E
χ
6
4
0
4
0
0
0
0
0
0
0
/A
A
−B
−B
0
0
/E
−E
4
0
0
0
/A
A
−B
−B
0
0
/E
−E
χ
7
4
0
4
0
0
0
0
0
0
0
1
1
−B
−B
0
0
−D
−D
−4
0
0
0
−1
−1
B
B
0
0
D
D
χ
8
4
0
4
0
0
0
0
0
0
0
1
1
B
B
0
0
D
D
−4
0
0
0
−1
−1
−B
−B
0
0
−D
−D
χ
9
4
0
4
0
0
0
0
0
0
0
A
/A
−B
−B
0
0
−E
/E
−4
0
0
0
−A
−/A
B
B
0
0
E
−/E
χ
10
4
0
4
0
0
0
0
0
0
0
/A
A
−B
−B
0
0
/E
−E
−4
0
0
0
−/A
−A
B
B
0
0
−/E
E
χ
11
4
0
4
0
0
0
0
0
0
0
A
/A
B
B
0
0
E
−/E
−4
0
0
0
−A
−/A
−B
−B
0
0
−E
/E
χ
12
4
0
4
0
0
0
0
0
0
0
/A
A
B
B
0
0
−/E
E
−4
0
0
0
−/A
−A
−B
−B
0
0
/E
−E
χ
13
12
0
−4
0
0
0
0
0
0
0
0
0
C
−B
0
0
0
0
0
−4
0
0
0
0
−C
B
0
0
0
0
χ
14
12
0
−4
0
0
0
0
0
0
0
0
0
−C
B
0
0
0
0
0
−4
0
0
0
0
C
−B
0
0
0
0
χ
15
12
0
−4
0
0
0
0
0
0
0
0
0
C
−B
0
0
0
0
0
4
0
0
0
0
C
−B
0
0
0
0
χ
16
12
0
−4
0
0
0
0
0
0
0
0
0
−C
B
0
0
0
0
0
4
0
0
0
0
−C
B
0
0
0
0
Note.A = −1−3i/2; B = −2i; C = −6i; D = i; E = E1211.Table 3
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
4
4
0
0
0
0
0
0
0
0
1
1
2
2
0
0
−1
−1
−2
−2
0
0
1
1
−4
0
0
0
−1
−1
χ
2
4
4
0
0
0
0
0
0
0
0
1
1
2
2
0
0
−1
−1
2
2
0
0
−1
−1
4
0
0
0
1
1
χ
3
4
4
0
0
0
0
0
0
0
0
1
1
−2
−2
0
0
1
1
2
2
0
0
−1
−1
−4
0
0
0
−1
−1
χ
4
4
4
0
0
0
0
0
0
0
0
1
1
−2
−2
0
0
1
1
−2
−2
0
0
1
1
4
0
0
0
1
1
χ
5
4
4
0
0
0
0
0
0
0
0
A
/A
2
2
0
0
−A
−/A
−2
−2
0
0
A
/A
−4
0
0
0
−A
−/A
χ
6
4
4
0
0
0
0
0
0
0
0
/A
A
2
2
0
0
−/A
−A
−2
−2
0
0
/A
A
−4
0
0
0
−/A
−A
χ
7
4
4
0
0
0
0
0
0
0
0
A
/A
2
2
0
0
−A
−/A
2
2
0
0
−A
−/A
4
0
0
0
A
/A
χ
8
4
4
0
0
0
0
0
0
0
0
/A
A
2
2
0
0
−/A
−A
2
2
0
0
−/A
−A
4
0
0
0
/A
A
χ
9
4
4
0
0
0
0
0
0
0
0
A
/A
−2
−2
0
0
A
/A
2
2
0
0
−A
−/A
−4
0
0
0
−A
−/A
χ
10
4
4
0
0
0
0
0
0
0
0
/A
A
−2
−2
0
0
/A
A
2
2
0
0
−/A
−A
−4
0
0
0
−/A
−A
χ
11
4
4
0
0
0
0
0
0
0
0
A
/A
−2
−2
0
0
A
/A
−2
−2
0
0
A
/A
4
0
0
0
A
/A
χ
12
4
4
0
0
0
0
0
0
0
0
/A
A
−2
−2
0
0
/A
A
−2
−2
0
0
/A
A
4
0
0
0
/A
A
χ
13
12
−4
0
0
0
0
0
0
0
0
0
0
−6
2
0
0
0
0
−6
2
0
0
0
0
0
4
0
0
0
0
χ
14
12
−4
0
0
0
0
0
0
0
0
0
0
−6
2
0
0
0
0
6
−2
0
0
0
0
0
−4
0
0
0
0
χ
15
12
−4
0
0
0
0
0
0
0
0
0
0
6
−2
0
0
0
0
−6
2
0
0
0
0
0
−4
0
0
0
0
χ
16
12
−4
0
0
0
0
0
0
0
0
0
0
6
−2
0
0
0
0
6
−2
0
0
0
0
0
4
0
0
0
0
Note.A = −1−3i/2.Table 4
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
4
0
0
0
4
0
0
0
0
0
1
1
B
0
0
0
C
C
2
2
0
0
−1
−1
E
E
0
0
−C
−C
χ
2
4
0
0
0
4
0
0
0
0
0
1
1
−B
0
0
0
−C
−C
2
2
0
0
−1
−1
−E
−E
0
0
C
C
χ
3
4
0
0
0
4
0
0
0
0
0
A
/A
B
0
0
0
D
−/D
2
2
0
0
−A
−/A
E
E
0
0
−D
/D
χ
4
4
0
0
0
4
0
0
0
0
0
/A
A
B
0
0
0
−/D
D
2
2
0
0
−/A
−A
E
E
0
0
/D
−D
χ
5
4
0
0
0
4
0
0
0
0
0
A
/A
−B
0
0
0
−D
/D
2
2
0
0
−A
−/A
−E
−E
0
0
D
−/D
χ
6
4
0
0
0
4
0
0
0
0
0
/A
A
−B
0
0
0
/D
−D
2
2
0
0
−/A
−A
−E
−E
0
0
−/D
D
χ
7
4
0
0
0
4
0
0
0
0
0
1
1
B
0
0
0
C
C
−2
−2
0
0
1
1
−E
−E
0
0
C
C
χ
8
4
0
0
0
4
0
0
0
0
0
1
1
−B
0
0
0
−C
−C
−2
−2
0
0
1
1
E
E
0
0
−C
−C
χ
9
4
0
0
0
4
0
0
0
0
0
A
/A
B
0
0
0
D
−/D
−2
−2
0
0
A
/A
−E
−E
0
0
D
−/D
χ
10
4
0
0
0
4
0
0
0
0
0
/A
A
B
0
0
0
−/D
D
−2
−2
0
0
/A
A
−E
−E
0
0
−/D
D
χ
11
4
0
0
0
4
0
0
0
0
0
A
/A
−B
0
0
0
−D
/D
−2
−2
0
0
A
/A
E
E
0
0
−D
/D
χ
12
4
0
0
0
4
0
0
0
0
0
/A
A
−B
0
0
0
/D
−D
−2
−2
0
0
/A
A
E
E
0
0
/D
−D
χ
13
12
0
0
0
−4
0
0
0
0
0
0
0
0
−B
0
0
0
0
−6
2
0
0
0
0
F
−E
0
0
0
0
χ
14
12
0
0
0
−4
0
0
0
0
0
0
0
0
B
0
0
0
0
−6
2
0
0
0
0
−F
E
0
0
0
0
χ
15
12
0
0
0
−4
0
0
0
0
0
0
0
0
−B
0
0
0
0
6
−2
0
0
0
0
−F
E
0
0
0
0
χ
16
12
0
0
0
−4
0
0
0
0
0
0
0
0
B
0
0
0
0
6
−2
0
0
0
0
F
−E
0
0
0
0
Note.A = −1−3i/2; B = 4i; C = i; D = E1211; E = 2i; F = 6i.Table 5
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
8
0
8
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
2
8
0
8
0
0
0
0
0
0
0
A
/A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
3
8
0
8
0
0
0
0
0
0
0
/A
A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
4
24
0
-8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A = −1 − i3.Table 6
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
8
8
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
2
8
8
0
0
0
0
0
0
0
0
A
/A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
3
8
8
0
0
0
0
0
0
0
0
/A
A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
4
24
-8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A = −1 − i3.Table 7
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
8
0
0
0
8
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
2
8
0
0
0
8
0
0
0
0
0
A
/A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
3
8
0
0
0
8
0
0
0
0
0
/A
A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
4
24
0
0
0
-8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A = −1 − i3.Table 8
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
4
4
4
4
4
4
0
0
0
0
−2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
2
4
4
4
4
4
4
0
0
0
0
A
/A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
3
4
4
4
4
4
4
0
0
0
0
/A
A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
4
6
−2
−2
−2
−2
6
0
0
0
0
0
0
0
0
−2
−2
0
0
0
0
B
−B
0
0
0
0
B
B
0
0
χ
5
6
−2
−2
−2
−2
6
0
0
0
0
0
0
0
0
−2
−2
0
0
0
0
−B
B
0
0
0
0
−B
−B
0
0
χ
6
6
−2
−2
−2
−2
6
0
0
0
0
0
0
0
0
2
2
0
0
0
0
B
−B
0
0
0
0
−B
−B
0
0
χ
7
6
−2
−2
−2
−2
6
0
0
0
0
0
0
0
0
2
2
0
0
0
0
−B
B
0
0
0
0
B
B
0
0
χ
8
6
−2
−2
6
−2
−2
0
0
0
0
0
0
0
0
−2
2
0
0
0
0
B
B
0
0
0
0
−B
B
0
0
χ
9
6
−2
−2
6
−2
−2
0
0
0
0
0
0
0
0
−2
2
0
0
0
0
−B
−B
0
0
0
0
B
−B
0
0
χ
10
6
−2
−2
6
−2
−2
0
0
0
0
0
0
0
0
2
−2
0
0
0
0
B
B
0
0
0
0
B
−B
0
0
χ
11
6
−2
−2
6
−2
−2
0
0
0
0
0
0
0
0
2
−2
0
0
0
0
−B
−B
0
0
0
0
−B
B
0
0
χ
12
12
−4
12
−4
−4
−4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
13
12
12
−4
−4
−4
−4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
14
12
−4
−4
−4
12
−4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A =1 + i3; B = −2i.Table 9
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
2
2
2
2
2
2
2
2
2
2
2
A
/A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
3
2
2
2
2
2
2
2
2
2
2
/A
A
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
4
6
6
6
6
6
6
−2
−2
−2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
5
6
−2
6
−2
−2
−2
−2
2
−2
2
0
0
0
0
0
0
0
0
0
0
0
−4
0
0
0
0
0
0
0
0
χ
6
6
−2
6
−2
−2
−2
−2
2
−2
2
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
χ
7
6
−2
6
−2
−2
−2
2
−2
2
−2
0
0
0
0
0
0
0
0
0
0
−4
0
0
0
0
0
0
0
0
0
χ
8
6
−2
6
−2
−2
−2
2
−2
2
−2
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
χ
9
6
6
−2
−2
−2
−2
−2
−2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−4
0
0
χ
10
6
6
−2
−2
−2
−2
−2
−2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
χ
11
6
6
−2
−2
−2
−2
2
2
−2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−4
0
0
0
χ
12
6
6
−2
−2
−2
−2
2
2
−2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
χ
13
6
−2
−2
−2
6
−2
−2
2
2
−2
0
0
0
0
0
B
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
14
6
−2
−2
−2
6
−2
−2
2
2
−2
0
0
0
0
0
−B
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
15
6
−2
−2
−2
6
−2
2
−2
−2
2
0
0
0
0
B
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
16
6
−2
−2
−2
6
−2
2
−2
−2
2
0
0
0
0
−B
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
17
12
−4
−4
12
−4
−4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
18
12
−4
−4
−4
−4
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A = −1 − i3; B = −4i.Table 10
Projective character table ofH2 with factor set αi−1.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
2
2
2
2
2
2
0
0
0
0
−1
−1
−2
−2
0
0
1
1
2
2
0
0
−1
−1
−2
−2
0
0
1
1
χ
2
2
2
2
2
2
2
0
0
0
0
−1
−1
−2
−2
0
0
1
1
−2
−2
0
0
1
1
2
2
0
0
−1
−1
χ
3
2
2
2
2
2
2
0
0
0
0
−1
−1
2
2
0
0
−1
−1
−2
−2
0
0
1
1
−2
−2
0
0
1
1
χ
4
2
2
2
2
2
2
0
0
0
0
−1
−1
2
2
0
0
−1
−1
2
2
0
0
−1
−1
2
2
0
0
−1
−1
χ
5
2
2
2
2
2
2
0
0
0
0
A
/A
−2
−2
0
0
−A
−/A
2
2
0
0
A
/A
−2
−2
0
0
−A
−/A
χ
6
2
2
2
2
2
2
0
0
0
0
/A
A
−2
−2
0
0
−/A
−A
2
2
0
0
/A
A
−2
−2
0
0
−/A
−A
χ
7
2
2
2
2
2
2
0
0
0
0
A
/A
−2
−2
0
0
−A
−/A
−2
−2
0
0
−A
−/A
2
2
0
0
A
/A
χ
8
2
2
2
2
2
2
0
0
0
0
/A
A
−2
−2
0
0
−/A
−A
−2
−2
0
0
−/A
−A
2
2
0
0
/A
A
χ
9
2
2
2
2
2
2
0
0
0
0
A
/A
2
2
0
0
A
/A
−2
−2
0
0
−A
−/A
−2
−2
0
0
−A
−/A
χ
10
2
2
2
2
2
2
0
0
0
0
/A
A
2
2
0
0
/A
A
−2
−2
0
0
−/A
−A
−2
−2
0
0
−/A
−A
χ
11
2
2
2
2
2
2
0
0
0
0
A
/A
2
2
0
0
A
/A
2
2
0
0
A
/A
2
2
0
0
A
/A
χ
12
2
2
2
2
2
2
0
0
0
0
/A
A
2
2
0
0
/A
A
2
2
0
0
/A
A
2
2
0
0
/A
A
χ
13
6
−2
−2
−2
6
−2
0
0
0
0
0
0
−6
2
0
0
0
0
0
0
B
−B
0
0
0
0
−B
B
0
0
χ
14
6
−2
−2
−2
6
−2
0
0
0
0
0
0
−6
2
0
0
0
0
0
0
−B
B
0
0
0
0
B
−B
0
0
χ
15
6
−2
−2
−2
6
−2
0
0
0
0
0
0
6
−2
0
0
0
0
0
0
B
−B
0
0
0
0
B
−B
0
0
χ
16
6
−2
−2
−2
6
−2
0
0
0
0
0
0
6
−2
0
0
0
0
0
0
−B
B
0
0
0
0
−B
B
0
0
χ
17
6
6
−2
−2
−2
−2
0
0
0
0
0
0
0
0
B
B
0
0
0
0
B
B
0
0
6
−2
0
0
0
0
χ
18
6
6
−2
−2
−2
−2
0
0
0
0
0
0
0
0
−B
−B
0
0
0
0
−B
−B
0
0
6
−2
0
0
0
0
χ
19
6
6
−2
−2
−2
−2
0
0
0
0
0
0
0
0
B
B
0
0
0
0
−B
−B
0
0
−6
2
0
0
0
0
χ
20
6
6
−2
−2
−2
−2
0
0
0
0
0
0
0
0
−B
−B
0
0
0
0
B
B
0
0
−6
2
0
0
0
0
χ
21
6
−2
6
−2
−2
−2
0
0
0
0
0
0
0
0
B
−B
0
0
−6
2
0
0
0
0
0
0
−B
−B
0
0
χ
22
6
−2
6
−2
−2
−2
0
0
0
0
0
0
0
0
−B
B
0
0
−6
2
0
0
0
0
0
0
B
B
0
0
χ
23
6
−2
6
−2
−2
−2
0
0
0
0
0
0
0
0
B
−B
0
0
6
−2
0
0
0
0
0
0
B
B
0
0
χ
24
6
−2
6
−2
−2
−2
0
0
0
0
0
0
0
0
−B
B
0
0
6
−2
0
0
0
0
0
0
−B
−B
0
0
χ
25
12
−4
−4
12
−4
−4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
26
12
−4
−4
−4
−4
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A = 1+3i/2; B = −2i.Table 11
Ordinary character table ofH2.
h
H
2
1a
2a
2b
2c
2d
2e
2f
2g
2h
2i
3a
3b
2j
2k
4a
4b
6a
6b
2l
2m
4c
4d
6c
6d
2n
2o
4e
4f
6e
6f
C
H
2
h
768
256
256
256
256
256
64
64
64
64
12
12
192
64
32
32
12
12
192
64
32
32
12
12
192
64
32
32
12
12
χ
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
χ
2
1
1
1
1
1
1
1
1
1
1
1
1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
1
1
1
1
1
1
χ
3
1
1
1
1
1
1
1
1
1
1
1
1
−1
−1
−1
−1
−1
−1
1
1
1
1
1
1
−1
−1
−1
−1
−1
−1
χ
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1
χ
5
1
1
1
1
1
1
1
1
1
1
A
/A
−1
−1
−1
−1
−A
−/A
−1
−1
−1
−1
−A
−/A
1
1
1
1
A
/A
χ
6
1
1
1
1
1
1
1
1
1
1
/A
A
−1
−1
−1
−1
−/A
−A
−1
−1
−1
−1
−/A
−A
1
1
1
1
/A
A
χ
7
1
1
1
1
1
1
1
1
1
1
A
/A
−1
−1
−1
−1
−A
−/A
1
1
1
1
A
/A
−1
−1
−1
−1
−A
−/A
χ
8
1
1
1
1
1
1
1
1
1
1
/A
A
−1
−1
−1
−1
−/A
−A
1
1
1
1
/A
A
−1
−1
−1
−1
−/A
−A
χ
9
1
1
1
1
1
1
1
1
1
1
A
/A
1
1
1
1
A
/A
−1
−1
−1
−1
−A
−/A
−1
−1
−1
−1
−A
−/A
χ
10
1
1
1
1
1
1
1
1
1
1
/A
A
1
1
1
1
/A
A
−1
−1
−1
−1
−/A
−A
−1
−1
−1
−1
−/A
−A
χ
11
1
1
1
1
1
1
1
1
1
1
A
/A
1
1
1
1
A
/A
1
1
1
1
A
/A
1
1
1
1
A
/A
χ
12
1
1
1
1
1
1
1
1
1
1
/A
A
1
1
1
1
/A
A
1
1
1
1
/A
A
1
1
1
1
/A
A
χ
13
3
3
3
3
3
3
−1
−1
−1
−1
0
0
−3
−3
1
1
0
0
−3
−3
1
1
0
0
3
3
−1
−1
0
0
χ
14
3
3
3
3
3
3
−1
−1
−1
−1
0
0
−3
−3
1
1
0
0
3
3
−1
−1
0
0
−3
−3
1
1
0
0
χ
15
3
3
3
3
3
3
−1
−1
−1
−1
0
0
3
3
−1
−1
0
0
−3
−3
1
1
0
0
−3
−3
1
1
0
0
χ
16
3
3
3
3
3
3
−1
−1
−1
−1
0
0
3
3
−1
−1
0
0
3
3
−1
−1
0
0
3
3
−1
−1
0
0
χ
17
6
−2
6
−2
−2
−2
−2
2
−2
2
0
0
0
0
0
0
0
0
−6
2
2
−2
0
0
0
0
0
0
0
0
χ
18
6
−2
6
−2
−2
−2
−2
2
−2
2
0
0
0
0
0
0
0
0
6
−2
−2
2
0
0
0
0
0
0
0
0
χ
19
6
−2
6
−2
−2
−2
2
−2
2
−2
0
0
0
0
0
0
0
0
−6
2
−2
2
0
0
0
0
0
0
0
0
χ
20
6
−2
6
−2
−2
−2
2
−2
2
−2
0
0
0
0
0
0
0
0
6
−2
2
−2
0
0
0
0
0
0
0
0
χ
21
6
6
−2
−2
−2
−2
−2
−2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−6
2
2
−2
0
0
χ
22
6
6
−2
−2
−2
−2
−2
−2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
−2
−2
2
0
0
χ
23
6
6
−2
−2
−2
−2
2
2
−2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−6
2
−2
2
0
0
χ
24
6
6
−2
−2
−2
−2
2
2
−2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
−2
2
−2
0
0
χ
25
6
−2
−2
−2
6
−2
−2
2
2
−2
0
0
−6
2
2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
26
6
−2
−2
−2
6
−2
−2
2
2
−2
0
0
6
−2
−2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
27
6
−2
−2
−2
6
−2
2
−2
−2
2
0
0
−6
2
−2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
28
6
−2
−2
−2
6
−2
2
−2
−2
2
0
0
6
−2
2
−2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
29
12
−4
−4
12
−4
−4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
χ
30
12
−4
−4
−4
−4
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Note.A = −1−3i/2.Theorem 1.
The inertia factor subgroupH2 of the action of G on IrrN has exactly 9 distinct sets of IrrProjH2,αi−1 with associated factor sets αi−1 such that αi2∼1.As mentioned in the introductory section, a setIrrProjH2,αi−1 of 16 irreducible projective characters of H2 with associated factor set αi−1 is required in the construction of IrrG¯, using the technique of Fischer–Clifford matrices. But a choice is to be made amongst the 3 distinct projective characters tables of H2 which contain 16 irreducible projective characters each. The desired set IrrProjH2,αi−1 of 16 irreducible projective characters can be found in the inertia group H¯2=24⋅H2 of G¯ on IrrN.Since we have a known permutation representation ofG¯, the group H¯2 is generated within GAP. The normal subgroups Ei of order 8 of H¯2 are computed. There are three such groups, E1,E2, and E3, which are elementary abelian. The 3 quotient groups Di=H¯2/Ei are double covers for H2 with each containing 46 ordinary irreducible characters. 30 characters from each of the sets IrrDi are the liftings of the ordinary irreducible characters of H2 to Di, while the remaining 16 ordinary characters of each Di represent the set IrrProjH2,αi−1 of irreducible projective characters of H2, which is of interest to us. Using the above GAP code as earlier, the desired set IrrProjH2,αi−1 of projective characters of H2 which is needed in the construction of the ordinary character table of G¯ using Fischer–Clifford theory is identified as Table 3.


## 5. Fischer–Clifford Matrices and Conjugacy Classes ofG¯

For the readers who are interested in the construction of the ordinary character table ofG¯=24·26:A5×3 using the technique of Fischer–Clifford matrices (see, for example, [6, 9, 19–23]), the Fischer–Clifford matrices (Table 12) and the conjugacy classes (Table 13) of G¯ are supplied. The conjugacy classes are arranged in a format obtained by the technique of coset analysis (see, for example, [6, 9, 21, 24]). The set of irreducible characters of G¯ will be partitioned into 2 blocks Δ1=χj∣1≤j≤26 and Δ2=χj∣27≤j≤42 corresponding to the inertia factor groups H1 and H2, respectively, where χj∈IrrG¯.Table 12
Fischer–Clifford matrices ofG¯.
M
g
M
g
M
1
A
=
1
1
15
−
1
M
2
A
=
1
M
2
B = 1
M
2
C
=
1
1
1
−
1
M
2
D
=
1
1
1
1
1
−
1
1
−
1
−
1
1
1
−
1
1
1
−
1
−
1
M
2
E
=
1
1
1
1
1
−
1
1
−
1
−
1
1
1
−
1
1
1
−
1
−
1
M
3
A
=
1
1
15
−
1
M
3
B
=
1
1
15
−
1
M
3
C
=
1
M
3
D
=
1
M
3
E
=
1
M
4
A
=
1
M
4
B
=
1
M
5
A
=
1
M
5
B
=
1
M
6
A
=
1
1
1
1
1
−
1
1
−
1
1
−
1
−
1
1
1
1
−
1
−
1
M
6
B
=
1
1
1
1
1
−
1
1
−
1
1
−
1
−
1
1
1
1
−
1
−
1
M
6
C
=
1
M
6
D
=
1
M
6
E
=
1
M
10
A
=
1
M
10
B
=
1
M
15
A
=
1
M
15
B
=
1
M
15
C
=
1
M
15
D
=
1Table 13
Classes ofG¯.
g
G
k
f
j
x
G
¯
C
G
¯
x
1
A
16
f
1
=
1
1
A
184320
f
2
=
15
2
A
12288
2
A
16
f
1
=
16
2
B
3840
2
B
16
f
1
=
16
2
C
768
2
C
16
f
1
=
8
4
A
512
f
2
=
8
4
B
512
2
D
4
f
1
=
1
2
D
768
f
2
=
1
4
C
768
f
3
=
1
4
D
768
f
4
=
1
4
E
768
2
E
4
f
1
=
1
2
E
256
f
2
=
1
4
F
256
f
3
=
1
4
G
256
f
4
=
1
4
H
256
3
A
16
f
1
=
1
3
A
2880
f
2
=
15
6
A
192
3
B
16
f
1
=
1
3
B
2880
f
2
=
15
6
B
192
3
C
1
f
1
=
1
3
C
36
3
D
1
f
1
=
1
3
D
36
3
E
1
f
1
=
1
3
E
36
4
A
4
f
1
=
4
8
A
32
4
B
4
f
1
=
4
8
B
32
5
A
1
f
1
=
1
5
A
60
5
B
1
f
1
=
1
5
B
60
6
A
4
f
1
=
1
6
C
48
f
2
=
1
12
A
48
f
3
=
1
12
B
48
f
4
=
1
12
C
48
6
B
4
f
1
=
1
6
D
48
f
2
=
1
12
D
48
f
3
=
1
12
E
48
f
4
=
1
12
F
48
6
C
1
f
1
=
1
6
E
12
6
D
1
f
1
=
1
6
F
12
6
E
1
f
1
=
1
6
G
12
10A
1
f
1
=
1
10A
20
10B
1
f
1
=
1
10B
20
15A
1
f
1
=
1
15A
15
15B
1
f
1
=
1
15B
15
15C
1
f
1
=
1
15C
15
15D
1
f
1
=
1
15D
15



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*Source: 8684742-2019-11-27.xml*