# Fischer-Clifford Matrices and Character Table of the Maximal Subgroup(29:(L3(4)):2 of U6(2):2

Authors: Abraham Love Prins; Ramotjaki Lucky Monaledi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

The automorphism groupU6(2):2 of the unitary group U6(2)≅Fi21 has a maximal subgroup G¯ of the form (29:(L3(4)):2 of order 20643840. In this paper, Fischer-Clifford theory is applied to the split extension group G¯ to construct its character table. Also, class fusion from G¯ into the parent group U6(2):2 is determined.


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## Body

## 1. Introduction

The unitary simple groupU=U6(2) has outer automorphisms of orders 2, 3, and 6 and hence automorphism groups of the forms U1=U6(2):2, U2=U6(2):3, and U3=U6(2):S3 exist for U (see the ATLAS [1]). The reader is referred to [2] for more information about the construction of matrix representations for the covering and automorphism groups of U6(2). Recently in [3], a 3-local identification is given for the group PSU6≅U6(2) and its automorphism groups PSU6(2):3, PSU6(2):2, and PSU6(2):S3. Also, we found in the ATLAS that one of the 16 maximal subgroups of U is a split extension group A of the type 29:L3(4) of index 891 and has order 10321920. The group A has automorphism groups of the form A1=(29:L3(4)):2, A2=(29:L3(4)):3, and A3=(29:L3(4)):S3 which sit maximally inside the groups U1, U2, and U3, respectively.The character table ofA is stored in the GAP Library [4], whereas the character table of A1 and A2 are not yet uploaded in GAP. In this paper, the unknown Fischer-Clifford matrices [5] of A1 and its associated character table will be constructed. Readers are referred to [6] on a survey of Fischer-Clifford theory. It is interesting to note that our group A1 is the stabilizer StabU1(v1) of a singular vector v1∈V20(2), where V20(2) is the irreducible module of dimension 20 over the field GF(2) for U6(2):2 (see [3]). The method of coset analysis as discussed in [7–9] will be used in the computation of the conjugacy classes of elements of the group A1. The Fischer-Clifford matrices and character table of A3 were determined in [10].


## 2. Group29:(L3(4):2)

In this section, using a suitable permutation representation ofU6(2):2, we identify our group A1=(29:L3(4)):2 as a split extension 29 by L3(4):2 with the aid of GAP [4] and MAGMA [11]. Then with the help of MAGMA we represent L3(4):2 as a matrix group G of degree 9 over the Galois field GF(2). Since G acts absolutely irreducibly on its natural module 29, a split extension S of the form 29:(L3(4):2) exists. Then we create S as a subgroup of SL10(2) and show with the help of MAGMA that A1 is indeed an isomorphic copy of S.We constructU1=U6(2):2 within GAP, using its smallest permutation representation of degree 672 found in the online ATLAS of Finite Group Representations [12]. Next, we use the GAP commands “MS≔ ConjugacyClassesMaximalSubgroups (U1)”, “A1≔MS[4]”, and “Size(A1)” to represent A1=(29:(L3(4)):2 as a permutation group on 672 points and then use this permutation representation to construct A1 within MAGMA. Using the sequence of MAGMA commands “a,b≔ChiefSeries(A1)”, “N≔ a[3]”, “NormalSubgroups(A1)”, “IsNormal(A1,a[3])”, “IsElementaryAbelian(N)”, “C≔ Complements(A1,N)”, “Order(C[1])”, “C[1] meet N”, and “IsIsomorphic(C[1], L3(4):2)”, we verified that A1=(29:L3(4)):2≅29:(L3(4):2).HavingA1 as a permutation group on 672 points, we use the MAGMA commands “M≔GModule(A1,N)” and "‘M:Maximal” to represent L3(4):2 as a matrix group G of degree 9 over the Galois field GF(2). Thus we obtain the matrix group G=g1,g2 having 14 conjugacy classes of elements, where o(g1)=4 and o(g2)=14. The generators g1 and g2 of G are as follows:(1)g1=100001110010101000110100000100111001001111111011110001010010000101011100010010101g2=100110011110001111000001110010101110101011000111110111011101111101100101010101100.The MAGMA command “IsAbsolutelyIrreducible(G)” tells us that the action of the matrix groupG on its natural module 29 is absolutely irreducible. Thus a split extension of the type 29:(L3(4):2) does exist. Hence we can construct 29:(L3(4):2) as a subgroup S of SL10(2) such that S=s1,s2,s3 and L3(4):2=s1,s2, where o(s1)=4, o(s2)=14, and o(s3)=2. The generators of the matrix group S of degree 10 over GF(2) are as follows:(2)s1=1000011100010101000011010000001001110010001111111001111000100100100000101011100001001010100000000001s2=1001100110110001111000000111000101011100101011000011111011100111011110101100101001010110000000000001s3=1000000000010000000000100000000001000000000010000000000100000000001000000000010000000000101000000001.Since we can representS and A1 as a matrix and permutation group, respectively, we use the MAGMA command “IsIsomorphic (S,A1)’’ to confirm that S≅A1. Hence we can regard A1=(29:L3(4)):2 as the split extension S=29:(L3(4):2).


## 3. The Conjugacy Classes ofG¯=29:(L3(4):2)

In this section, the conjugacy classes ofG¯ are computed using the technique of coset analysis and readers are encouraged to consult [8, 9] for a sound theoretical background on this technique.Throughout the remainder of this chapter, letG¯=29:(L3(4):2) be a split extension of N=29 by L3(4):2, where N is the vector space V9(2) of dimension 9 over GF(2) on which the linear group G acts. Since G=g1,g2 is represented as a matrix group, we used the MAGMA commands “O≔ Orbits(G)”, “#O”, “#O[1]”, “#O[2]”,#O[3]”, and “#O[4]” to compute the orbit lengths of the action of G on N. We obtain 4 orbits of lengths 1, 21, 210, and 280 and using the MAGMA commands “P1≔ Stabilizer(G,O[1,1])”, “P2≔ Stabilizer(G,O[2,1])”, “P3≔ Stabilizer(G,O[3,1])”, and “P4≔ Stabilizer(G,O[4,1])”, we are able to compute the corresponding point stabilizers Pi, i=1,2,3,4, which are subgroups of G. With the aid of MAGMA and also checking the indices of the maximal subgroups of G in the ATLAS, the structures of the stabilizers Pi are identified as P1=L3(4):2, P2=24:S5, P3=24:(2×S3), and P4=32:(Q8.2), where P2 and P4 are maximal subgroups of P1. We should note here that the group L3(4):2 has two nonconjugate isomorphic maximal subgroups L1=P2 and L2, having the same structure 24:S5. The stabilizer P3 sits maximally in L2. Alternatively, we can use [13] to identify the structures of the groups Pi. Since the action of G on N does not fix any nontrivial subspace of 29, we have that 29 is an irreducible module for G. We can readily verify this fact by using the MAGMA command “IsIrreducible(G)”.Letχ(L3(4):2∣29) be the permutation character of L3(4):2 on the classes of 29. Then, from methods that were developed by Mpono [14], we obtain that χ(L3(4):2∣29) = IP1P1 +IP2P1+IP3P1+IP4P1 = 4×1a+4×20a+2×35a+45a+45b+2×64a+2×70a, where IP1P1, IP2P1, IP3P1, and IP4P1 are the identity characters of the point stabilizers Pi, i=1,2,3,4, induced to G. Note that the identity characters IPiP1 are identified with the permutation characters χ(L3(4):2∣Pi) of L3(4):2 acting on the classes of the point stabilizers Pi. We found that IP1P1 = χ(L3(4):3∣P1)=1a, IP2P1 = χ(L3(4):2∣P2)=1a+20a, IP3P1 = χ(L3(4):2∣P3)=1a+2×20a+35a+64a+70a, and IP4P1 = χ(L3(4):3∣P4)=1a+20a+35a+45a+45b+64a+70a. The permutation characters χ(L3(4):2∣Pi) are written in terms of the ordinary irreducible characters of G. Since we have the generators g1 and g2 for G, we compute the character tables of G and the Pi’s directly in MAGMA and use these tables together with the fusion maps of the stabilizers into G, to compute χ(L3(4):2∣Pi) and χ(L3(4):2∣29). The values of χ(L3(4):2∣29) on the different classes of G determine the number k of fixed points of each g∈G in 29. The values of k are listed in Table 1.Table 1
The values ofχ(L3(4):2∣29) on the different classes of L3(4):2.
[ h ] L 3 ( 4 ) : 2 1 A 2 A 2 B 3 A 4 A 4 B 4 C 5 A 6 A 7 A 7 B 8 A 14 A 14 B χ ( L 3 ( 4 ) : 3 ∣ P 1 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 χ ( L 3 ( 4 ) : 3 ∣ P 2 ) 21 7 5 3 1 1 3 1 1 0 0 1 0 0 χ ( L 3 ( 4 ) : 3 ∣ P 3 ) 210 28 18 3 2 2 4 0 1 0 0 0 0 0 χ ( L 3 ( 4 ) : 3 ∣ P 4 ) 280 28 8 1 4 4 0 0 1 0 0 2 0 0  k 512 64 32 8 8 8 8 2 4 1 1 4 1 1The values ofk enabled us to determine the number fj of orbits Qi’s, 1≤i≤k, which have fused together under the action of CG(g), for each class representative g∈G, to form one orbit △f. Mpono in [14] used the technique of coset analysis to develop Programmes A and B in CAYLEY [15] for the computation of the conjucacy classes of a split extension G¯=N:G, where N is an elementary abelian p-group for a prime p on which a linear group G acts. Ali [16] adapted Programmes A and B to be used in MAGMA. Programme A computes the values of the fj′s, whereas Programme B determines the order of the elements for each conjugacy class [x] in G¯. We obtain that G¯ has exactly 49 conjugacy classes. The parameters dj and w are defined in [14]. The centralizer order of each class [x] of G¯ is computed using the formula CG¯x=(k/f)CGg. All the information involving the conjugacy classes of G¯ are listed in Table 2.Table 2
The conjugacy classes of elements ofG¯=29:(L3(4):2).
[g]G k f j d j w [ x ] G ¯ x G ¯ C G ¯ x 1 A 512 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 1 A 1 20643840 f 2 = 21 ( 0,0 , 0,0 , 1,1 , 0,1 , 1 ) ( 0,0 , 0,0 , 1,1 , 0,1 , 1 ) 2 A 21 983040 f 3 = 210 ( 1,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 1,0 , 0,0 , 0,0 , 0,0 , 0 ) 2 B 210 98304 f 4 = 280 ( 1,0 , 1,0 , 1,1 , 1,0 , 1 ) ( 1,0 , 1,0 , 1,1 , 1,0 , 1 ) 2 C 280 73728  2 A 64 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 2 D 960 21504 f 2 = 7 ( 0,0 , 0,0 , 0,1 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 2 E 6720 3072 f 3 = 7 ( 0,0 , 0,0 , 0,0 , 0,1 , 1 ) ( 0,1 , 1,1 , 1,0 , 0,1 , 0 ) 4 A 6720 3072 f 4 = 21 ( 1,0 , 0,0 , 1,1 , 1,0 , 0 ) ( 0,0 , 0,0 , 1,1 , 1,0 , 0 ) 4 B 20160 1024 f 5 = 28 ( 0,0 , 0,0 , 0,0 , 0,0 , 1 ) ( 0,1 , 1,1 , 0,1 , 1,1 , 0 ) 4 C 26880 768  2 B 32 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 2 F 5040 4096 f 2 = 1 ( 0,0 , 0,0 , 0,0 , 0,1 , 1 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 2 G 5040 4096 f 3 = 1 ( 1,1 , 1,1 , 1,1 , 1,1 , 1 ) ( 0,1 , 1,1 , 1,0 , 0,0 , 1 ) 4 D 5040 4096 f 4 = 1 ( 1,1 , 0,0 , 0,0 , 0,1 , 1 ) ( 0,1 , 1,1 , 1,0 , 0,0 , 1 ) 4 E 5040 4096 f 5 = 4 ( 1,1 , 1,0 , 1,0 , 0,1 , 1 ) ( 0,0 , 1,0 , 0,1 , 1,0 , 0 ) 4 F 20160 1024 f 6 = 8 ( 0,1 , 1,1 , 0,1 , 0,1 , 0 ) ( 0,0 , 1,1 , 1,1 , 1,1 , 1 ) 4 G 40320 512 f 7 = 8 ( 1,1 , 1,1 , 1,0 , 1,1 , 1 ) ( 0,0 , 0,0 , 1,1 , 1,1 , 0 ) 4 H 40320 512 f 8 = 8 ( 1,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 1,1 , 0,0 , 0,0 , 1 ) 4 I 40320 512  3 A 8 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 3 A 143360 144 f 2 = 1 ( 1,0 , 0,1 , 0,1 , 1,1 , 1 ) ( 1,0 , 0,1 , 0,1 , 1,1 , 1 ) 6 A 143360 144 f 3 = 3 ( 1,1 , 1,1 , 0,1 , 0,0 , 1 ) ( 1,1 , 1,1 , 0,1 , 0,0 , 1 ) 6 B 430080 48 f 4 = 3 ( 0,0 , 0,0 , 0,0 , 0,1 , 0 ) ( 1,0 , 0,1 , 1,0 , 1,1 , 0 ) 6 C 430080 48  4 A 8 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 4 J 80640 256 f 2 = 1 ( 0,1 , 0,0 , 1,0 , 0,1 , 1 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 4 K 80640 256 f 3 = 2 ( 0,0 , 1,0 , 0,0 , 0,0 , 0 ) ( 0,1 , 1,1 , 1,0 , 0,0 , 1 ) 8 A 161280 128 f 4 = 4 ( 1,0 , 1,0 , 1,0 , 1,1 , 1 ) ( 0,1 , 0,1 , 1,1 , 1,0 , 1 ) 8 B 322560 64  4 B 8 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 4 L 161280 128 f 2 = 1 ( 1,0 , 0,0 , 0,0 , 0,1 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 4 M 161280 128 f 3 = 2 ( 1,0 , 0,0 , 1,1 , 1,1 , 1 ) ( 0,0 , 1,1 , 0,0 , 0,1 , 0 ) 8 C 322560 64 f 4 = 4 ( 1,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 1,1 , 0,1 , 1,0 , 0,1 , 0 ) 8 D 645120 32  4 C 8 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 4 N 161280 128 f 2 = 1 ( 1,0 , 1,1 , 1,0 , 1,1 , 1 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 4 O 161280 128 f 3 = 1 ( 1,0 , 0,0 , 1,1 , 0,1 , 0 ) ( 1,0 , 1,0 , 1,1 , 1,1 , 0 ) 8 E 161280 128 f 4 = 1 ( 0,1 , 1,1 , 1,1 , 1,1 , 0 ) ( 1,0 , 1,0 , 1,1 , 1,1 , 0 ) 8 F 161280 128 f 5 = 2 ( 1,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 1,1 , 0,1 , 1,1 , 0,0 , 1 ) 8 G 322560 64 f 6 = 2 ( 1,0 , 1,0 , 0,1 , 0,0 , 1 ) ( 0,1 , 1,1 , 0,0 , 1,1 , 1 ) 8 H 322560 64  5 A 2 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 5 A 2064384 10 f 2 = 1 ( 1,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 1,1 , 0,0 , 1 ) 10 A 2064384 10  6 A 4 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 6 D 860160 24 f 2 = 1 ( 1,0 , 0,0 , 1,0 , 1,1 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 6 E 860160 24 f 3 = 1 ( 1,0 , 0,1 , 1,0 , 1,1 , 1 ) ( 1,0 , 0,1 , 0,1 , 1,1 , 0 ) 12 A 860160 24 f 4 = 1 ( 1,1 , 1,0 , 1,0 , 1,1 , 1 ) ( 1,0 , 0,1 , 0,1 , 1,1 , 0 ) 12 B 860160 24  7 A 1 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 7 A 1474560 14  7 B 1 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 7 B 1474560 14  8 A 4 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 8 I 645120 32 f 2 = 1 ( 1,0 , 0,1 , 1,1 , 1,1 , 1 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 8 J 645120 32 f 3 = 1 ( 1,1 , 1,0 , 1,0 , 1,1 , 1 ) ( 0,0 , 0,0 , 0,0 , 0,1 , 1 ) 16 A 645120 32 f 4 = 1 ( 1,1 , 1,1 , 1,1 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,1 , 1 ) 16 B 645120 32  14 A 1 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 14 A 1474560 14  14 B 1 f 1 = 1 ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) ( 0,0 , 0,0 , 0,0 , 0,0 , 0 ) 14 B 1474560 14


## 4. The Inertia Groups ofG¯=29:(L3(4):2)

SinceG has four orbits on N, then by Brauer’s Theorem [17]G acts on Irr(N) with the same number of orbits. The lengths of the 4 orbits will be 1,r, s, and t where r + s + t = 511, with corresponding point stabilizers H1, H2, H3, and H4 as subgroups of G such that [G:H1] =1, [G:H2] =r, [G:H3]= s, and [G:H4] = t. We generate G as a permutation group on a set of cardinality 672 within MAGMA. Then the maximal and submaximal subgroups of G are computed. Now, considering the indices of these subgroups in G, the number of the classes of these subgroups, and also the fact that G¯ has 49 conjugacy classes, we deduce that the action of G on Irr(N) has orbits of lengths 1, r=21, s=210, and t=280 with respective point stabilizers H1=L3(4):2, H2=24:S5, H3=24:(2×S3), and H4=32:(Q8:2). Thus we obtain four inertia groups H¯i = 29:Hi, i∈{1,2,3,4}, for 29:(L3(4):2) on Irr(N). Alternatively, we can also determine the inertia factor groups if we let T be the matrix group of dimension 9 over GF(2) formed by the transpose of the generators of G. Then the action of T on the classes of N=29 is the equivalent of G acting on Irr(N). Then with the help of MAGMA or GAP, we can easily verify that the action of T on N has orbits of lengths 1, 21, 210, and 280 with corresponding point stabilizers T, 24:S5, 24:(2×S3), and 32:(Q8:2). The structures of H2 and H4 have been identified by checking the indices of the maximal subgroups of L3(4):2≅L3(4).22 in the ATLAS. The structure of H3 was determined by direct computations in MAGMA. The groups H2, H3, and H4 are constructed from elements within G and the generators are as follows:(i)
H2 =α1,α2, α1∈3A, and α2∈6A where(3)α1=000010100010010100100100111111011100000000110000001000100010010100001100001101010,α2=100011000011110010110010111101000010100001100000001000000001010000001110001101100(ii)
H3 =β1,β2, β1∈4C, and β2∈6A where(4)β1=100000000110101100000111111001111011100111000101000111100101100110101101110110110,β2=100000000100111001100111111010010001100111000101000111001101011110101101110110110(iii)
H4 =γ1,γ2, γ1∈2A, adn γ2∈8A where(5)γ1=000010100010000000100110011100110100100000100000001000000000100000001110001101010,γ2=110100000110001111011111100010000000000001011000011001000110011000011000000010001For the purpose of constructing the character table ofG¯, we use the above generators of the Hi’s to compute their character tables.


## 5. The Fusion ofH2, H3, and H4 into G

We obtain the fusions of the inertia factorsH2, H3, and H4 into G by using direct matrix conjugation in G and their permutation characters in G of degrees 21, 210, and 280, respectively. MAGMA was used for the various computations. The fusion maps of H2, H3, and H4 into G are shown in Tables 3, 4, and 5.Table 3
The fusion ofH2 into L3(4):2.
[ h ] H 2 → [ g ] L 3 ( 4 ) : 2 [ h ] H 2 → [ g ] L 3 ( 4 ) : 2 [ h ] H 2 → [ g ] L 3 ( 4 ) : 2 [ h ] H 2 → [ g ] L 3 ( 4 ) : 2 1 A  1 A 2 C  2 B 4 B  4 B 5 A  5 A 2 A  2 B 3 A  3 A 4 C  4 C 6 A  6 A 2 B  2 A 4 A  4 A 4 D  4 C 8 A  8 ATable 4
The fusion ofH3 into L3(4):2.
[h]H3 → [ g ] L 3 ( 4 ) : 2 [ h ] H 3 → [ g ] L 3 ( 4 ) : 2 [ h ] H 3 → [ g ] L 3 ( 4 ) : 2 [ h ] H 3 → [ g ] L 3 ( 4 ) : 2 1 A  1 A 2 D  2 B 4 A  4 C 4 E  4 B 2 A  2 B 2 E  2 B 4 B  4 C 6 A  6 A 2 B  2 A 2 F  2 A 4 C  4 A 2 C  2 B 3 A  3 A 4 D  4 CTable 5
The fusion ofH4 into L3(4):2.
[h]H4 → [ g ] L 3 ( 4 ) : 2 [h]H4 → [ g ] L 3 ( 4 ) : 2 [h]H4 → [ g ] L 3 ( 4 ) : 2 1 A  1 A 3 A  3 A 6 A  6 A 2 A  2 B 4 A  4 A 8 A  8 A 2 B  2 A 4 B  4 B 8 B  8 A


## 6. The Fischer-Clifford Matrices of29:(L3(4):2)

Having obtained the fusions of the inertia factors intoL3(4):2 and the conjugacy classes of L3(4):2 displayed in the format of Table 2, we can proceed to use the theory and properties discussed in [9] or [14] to help us in the construction of the Fischer-Clifford matrices of 29:(L3(4):2). Note that all the relations hold since 29 is an elementary abelian group.For example, consider the conjugacy class2A of L3(4):2. Then we obtain that M(2A) has the following form with corresponding weights attached to the rows and columns:(6)
M
2
A
=
C
G
¯
2
D
C
G
¯
2
E
C
G
¯
4
A
C
G
¯
4
B
C
G
¯
4
C
C
G
2
A
C
H
2
2
B
C
H
3
2
B
C
H
3
2
F
C
H
4
2
B
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
m
1
m
2
m
3
m
4
m
5
=
21504
3072
3072
1024
768
336
48
48
16
12
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
8
56
56
168
224
.We havea=f=k=p=u=1, b=c=7, d=21, and e=28 by using Theorem 5.2.4 and property (e) of the Fischer-Clifford matrix M(g) (both found in [14]). Thus we obtain the following form:(7)
M
2
A
=
21504
3072
3072
1024
768
336
48
48
16
12
1
7
7
21
28
1
g
h
i
j
1
l
m
n
o
1
q
r
s
t
1
v
w
x
y
8
56
56
168
224
.By the orthogonality relations for columns and rows and remaining properties of the matrixM(g) found in Chapter 5 of [14], we obtain the desired Fischer-Clifford matrix M(2A) of G¯ given below: (8)M2A=111117-1-53-177-1-1-121-391-328-4-4-44.For each class representativeg∈L3(4):2, we construct a Fischer-Clifford matrix M(g). These are listed in Table 6.Table 6
The Fischer-Clifford matrices of29:(L3(4):2).
M ( g ) M ( g ) M(1A)=111121-115-3210502-6280-40-88 M(2A)=111117-1-53-177-1-1-121-391-328-4-4-44  M(2B)=1111111111111-1-114-44-40-2202222200-24-44-402-204444-400088-8-800008-8-880000 M(3A)=11113-3-1133-1-11-11-1  M(4A)=1111111-122-204-400 M(4B)=1111111-122-204-400  M ( 4 C ) = 1 1 1 1 1 1 1 1 - 1 - 1 - 1 1 2 - 2 2 - 2 0 0 1 1 1 1 - 1 - 1 1 1 - 1 - 1 1 - 1 2 - 2 - 2 2 0 0 M ( 5 A ) = 1 1 1 - 1  M ( 6 A ) = 1 1 1 1 1 - 1 - 1 1 1 1 - 1 - 1 1 - 1 1 - 1 M ( 7 A ) = 1  M ( 7 B ) = 1 M(8A)=111111-1-11-1-111-11-1  M ( 14 A ) = 1 M(14B)=(1)


## 7. Character Table of29:(L3(4):2)

Having obtained the Fischer-Clifford matrices, the fusion maps of theHi’s into L3(4):2, and the character tables of the inertia factors Hi, we construct the character table of 29:(L3(4):2) following the methodology discussed in Section 5.2 of [14]. For example, we calculate the partial character table of 29:(L3(4):2) corresponding to the coset of 2A∈L3(4):2. From the Fischer-Clifford matrix M(2A) we obtain that(9)M12A=11111,M22A=7-1-53-1,M32A=77-1-1-121-391-3andM42A=28-4-4-44.LetC1(2A), C2(2A), C3(2A), and C4(2A) be the partial character tables of the inertia factors for the classes which fuse to 2A∈L3(4):2. Then the partial character table of 29:(L3(4):2) on the classes 2D,2E,4A,4B,4C is given by(10)C12AM12A=1-1-66-77-3-3338-80011111=11111-1-1-1-1-1-6-6-6-6-666666-7-7-7-7-777777-3-3-3-3-3-3-3-3-3-3333333333388888-8-8-8-8-80000000000C22AM22A=1-12-2-110-3-33307-1-53-1=7-1-53-1-715-3114-2-106-2-14210-62-715-317-1-53-100000-21315-93-21315-9321-3-159-321-3-159-300000C32AM32A=11-1-11-1-11-20203-1-3-131-3100000-20277-1-1-121-391-3=28480-4-28-4-804-1410-10-2214-10102-2-14-142221414-2-2-2024-12-40-42-18-62642186-2-60-2412400000000000-426-18-2642-6182-6C42AM42A=1-1-11000-2228-4-4-44=28-4-4-44-28444-4-28444-428-4-4-44000000000000000-56888-856-8-8-88Similarly, the partial character table associated with each cosetNg is computed. If necessary, we will restrict some characters of Irr(U6(2):2) to G¯, to ensure that each partial character table corresponding to a coset Ng will give rise to the desired set Irr(G¯).The character table ofG¯ will be partitioned row-wise into 4 blocks Δ1, Δ2, Δ3, and Δ4 where each block corresponds to an inertia group H¯i = 29:Hi. Therefore Irr(29:(L3(4):2))=⋃i=14Δi, where Δ1={χj∣1≤j≤14}, Δ2={χj∣15≤j≤26}, Δ3={χj∣27≤j≤40}, and Δ4={χj∣41≤j≤49}. The character table of 29:(L3(4):2) is shown in Table 7. The consistency and accuracy of the character table of 29:(L3(4):2) have been tested by using the GAP code labelled as Programme E in [18].Table 7
The character table of29:(L3(4):2).
1A    2A    2B  1 A 2 A 2 B 2 C 2 D 2 E 4 A 4 B 4 C 2 F 2 G 4 D  χ 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 χ 3 20 20 20 20 -6 -6 -6 -6 -6 4 4 4 χ 4 20 20 20 20 6 6 6 6 6 4 4 4 χ 5 35 35 35 35 -7 -7 -7 -7 -7 3 3 3 χ 6 35 35 35 35 7 7 7 7 7 3 3 3 χ 7 45 45 45 45 -3 -3 -3 -3 -3 -3 -3 -3 χ 8 45 45 45 45 -3 -3 -3 -3 -3 -3 -3 -3 χ 9 45 45 45 45 3 3 3 3 3 -3 -3 -3 χ 10 45 45 45 45 3 3 3 3 3 -3 -3 -3 χ 11 64 64 64 64 8 8 8 8 8 0 0 0 χ 12 64 64 64 64 -8 -8 -8 -8 -8 0 0 0 χ 13 70 70 70 70 0 0 0 0 0 6 6 6 χ 14 126 126 126 126 0 0 0 0 0 -2 -2 -2  χ 15 21 -11 5 -3 7 -1 -5 3 -1 5 -3 5 χ 16 21 -11 5 -3 -7 1 5 -3 1 5 -3 5 χ 17 84 -44 20 -12 14 -2 -10 6 -2 4 4 4 χ 18 84 -44 20 -12 -14 2 10 -6 2 4 4 4 χ 19 105 -55 25 -15 -7 1 5 -3 1 9 1 9 χ 20 105 -55 25 -15 7 -1 -5 3 -1 9 1 9 χ 21 126 -66 30 -18 0 0 0 0 0 -2 14 -2 χ 22 315 -165 75 -45 -21 3 15 -9 3 11 -13 11 χ 23 315 -165 75 -45 -21 3 15 -9 3 -5 3 -5 χ 24 315 -165 75 -45 21 -3 -15 9 -3 11 -13 11 χ 25 315 -165 75 -45 21 -3 -15 9 -3 -5 3 -5 χ 26 630 -330 150 -90 0 0 0 0 0 -10 6 -10  χ 27 210 50 2 -6 28 4 8 0 -4 18 10 2 χ 28 210 50 2 -6 -28 -4 -8 0 4 18 10 2 χ 29 210 50 2 -6 -14 10 -10 -2 2 2 -6 18 χ 30 210 50 2 -6 14 -10 10 2 -2 2 -6 18 χ 31 420 100 4 -12 -14 -14 2 2 2 20 4 20 χ 32 420 100 4 -12 14 14 -2 -2 -2 20 4 20 χ 33 630 150 6 -18 0 24 -12 -4 0 -10 -2 6 χ 34 630 150 6 -18 -42 -18 -6 2 6 6 14 -10 χ 35 630 150 6 -18 42 18 6 -2 -6 6 14 -10 χ 36 630 150 6 -18 0 -24 12 4 0 -10 -2 6 χ 37 1260 300 12 -36 0 0 0 0 0 -20 -4 12 χ 38 1260 300 12 -36 0 0 0 0 0 12 28 -20 χ 39 1260 300 12 -36 -42 6 -18 -2 6 -4 -20 -4 χ 40 1260 300 12 -36 42 -6 18 2 -6 -4 -20 -4  χ 41 280 -40 -8 8 28 -4 -4 -4 4 8 -8 -8 χ 42 280 -40 -8 8 -28 4 4 4 -4 8 -8 -8 χ 43 280 -40 -8 8 -28 4 4 4 -4 8 -8 -8 χ 44 280 -40 -8 8 28 -4 -4 -4 4 8 -8 -8 χ 45 560 -80 -16 16 0 0 0 0 0 16 -16 -16 χ 46 560 -80 -16 16 0 0 0 0 0 -16 16 16  χ 47 560 -80 -16 16 0 0 0 0 0 -16 16 16 χ 48 2240 -320 -64 64 -56 8 8 8 -8 0 0 0 χ 49 2240 -320 -64 64 56 -8 -8 -8 8 0 0 0  2 B     3 A    4 A  4 E 4 F 4 G 4 H 4 I 3 A 6 A 6 B 6 C 4 J 4 K 8 A 8 B  χ1 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 χ3 4 4 4 4 4 2 2 2 2 0 0 0 0 χ4 4 4 4 4 4 2 2 2 2 0 0 0 0 χ5 3 3 3 3 3 -1 -1 -1 -1 3 3 3 3 χ6 3 3 3 3 3 -1 -1 -1 -1 3 3 3 3 χ7 -3 -3 -3 -3 -3 0 0 0 0 1 1 1 1 χ8 -3 -3 -3 -3 -3 0 0 0 0 1 1 1 1 χ9 -3 -3 -3 -3 -3 0 0 0 0 1 1 1 1 χ10 -3 -3 -3 -3 -3 0 0 0 0 1 1 1 1 χ11 0 0 0 0 0 1 1 1 1 0 0 0 0 χ12 0 0 0 0 0 1 1 1 1 0 0 0 0 χ13 6 6 6 6 6 -2 -2 -2 -2 -2 -2 -2 -2 χ14 -2 -2 -2 -2 -2 0 0 0 0 -2 -2 -2 -2  χ15 -3 1 -3 1 1 3 -3 1 -1 1 1 1 -1 χ16 -3 1 -3 1 1 3 -3 1 -1 1 1 1 -1 χ17 4 4 -4 -4 4 3 -3 1 -1 0 0 0 0 χ18 4 4 -4 -4 4 3 -3 1 -1 0 0 0 0 χ19 1 5 -7 -3 5 -3 3 -1 1 1 1 1 -1 χ20 1 5 -7 -3 5 -3 3 -1 1 1 1 1 -1 χ21 14 6 -2 -10 6 0 0 0 0 -2 -2 -2 2 χ22 -13 -1 -5 7 -1 0 0 0 0 -1 -1 -1 1 χ23 3 -1 3 -1 -1 0 0 0 0 3 3 3 -3 χ24 -13 -1 -5 7 -1 0 0 0 0 -1 -1 -1 1 χ25 3 -1 3 -1 -1 0 0 0 0 3 3 3 -3 χ26 6 -2 6 -2 -2 0 0 0 0 -2 -2 -2 2  χ27 -6 -2 2 -2 -2 3 3 -1 -1 2 2 -2 0 χ28 -6 -2 2 -2 -2 3 3 -1 -1 2 2 -2 0 χ29 10 -2 2 -2 -2 3 3 -1 -1 -2 -2 2 0 χ30 10 -2 2 -2 -2 3 3 -1 -1 -2 -2 2 0 χ31 4 -4 4 -4 -4 -3 -3 1 1 0 0 0 0 χ32 4 -4 4 -4 -4 -3 -3 1 1 0 0 0 0 χ33 14 10 -2 2 -6 0 0 0 0 -2 -2 2 0 χ34 -2 10 -2 2 -6 0 0 0 0 2 2 -2 0 χ35 -2 10 -2 2 -6 0 0 0 0 2 2 -2 0 χ36 14 10 -2 2 -6 0 0 0 0 -2 -2 2 0 χ37 28 -12 -4 4 4 0 0 0 0 4 4 -4 0 χ38 -4 -12 -4 4 4 0 0 0 0 -4 -4 4 0 χ39 -20 4 4 -4 4 0 0 0 0 0 0 0 0 χ40 -20 4 4 -4 4 0 0 0 0 0 0 0 0  χ41 8 0 0 0 0 1 -1 -1 1 4 -4 0 0 χ42 8 0 0 0 0 1 -1 -1 1 4 -4 0 0 χ43 8 0 0 0 0 1 -1 -1 1 4 -4 0 0 χ44 8 0 0 0 0 1 -1 -1 1 4 -4 0 0 χ45 16 0 0 0 0 2 -2 -2 2 -8 8 0 0 χ46 -16 0 0 0 0 2 -2 -2 2 0 0 0 0 χ47 -16 0 0 0 0 2 -2 -2 2 0 0 0 0 χ48 0 0 0 0 0 -1 1 1 -1 0 0 0 0 χ49 0 0 0 0 0 -1 1 1 -1 0 0 0 0  4 B     4 C    5 A  4 L 4 M 8 C 8 D 4 N 4 O 8 E 8 F 8 G 8 H 5 A 10 A  χ 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 χ 3 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 χ 4 0 0 0 0 2 2 2 2 2 2 0 0 χ 5 -1 -1 -1 -1 1 1 1 1 1 1 0 0 χ 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 χ 7 1 1 1 1 1 1 1 1 1 1 0 0 χ 8 1 1 1 1 1 1 1 1 1 1 0 0 χ 9 1 1 1 1 -1 -1 -1 -1 -1 -1 0 0 χ 10 1 1 1 1 -1 -1 -1 -1 -1 -1 0 0 χ 11 0 0 0 0 0 0 0 0 0 0 -1 -1 χ 12 0 0 0 0 0 0 0 0 0 0 -1 -1 χ 13 2 2 2 2 0 0 0 0 0 0 0 0 χ 14 -2 -2 -2 -2 0 0 0 0 0 0 1 1  χ 15 1 1 1 -1 3 -1 1 -3 -1 1 1 -1 χ 16 1 1 1 -1 -3 1 -1 3 1 -1 1 -1 χ 17 0 0 0 0 2 2 -2 -2 -2 2 -1 1 χ 18 0 0 0 0 -2 -2 2 2 2 -2 -1 1 χ 19 1 1 1 -1 1 -3 3 -1 1 -1 0 0 χ 20 1 1 1 -1 -1 3 -3 1 -1 1 0 0 χ 21 -2 -2 -2 2 0 0 0 0 0 0 1 -1 χ 22 -1 -1 -1 1 -1 3 -3 1 -1 1 0 0 χ 23 -1 -1 -1 1 3 -1 1 -3 -1 1 0 0 χ 24 -1 -1 -1 1 1 -3 3 -1 1 -1 0 0 χ 25 -1 -1 -1 1 -3 1 -1 3 1 -1 0 0 χ 26 2 2 2 -2 0 0 0 0 0 0 0 0  χ 27 2 2 -2 0 4 0 -2 2 0 -2 0 0 χ 28 2 2 -2 0 -4 0 2 -2 0 2 0 0 χ 29 -2 -2 2 0 -2 2 0 -4 2 0 0 0 χ 30 -2 -2 2 0 2 -2 0 4 -2 0 0 0 χ 31 0 0 0 0 -2 -2 2 2 -2 2 0 0 χ 32 0 0 0 0 2 2 -2 -2 2 -2 0 0 χ 33 2 2 -2 0 0 -4 -2 2 0 2 0 0 χ 34 -2 -2 2 0 2 -2 -4 0 2 0 0 0 χ 35 -2 -2 2 0 -2 2 4 0 -2 0 0 0 χ 36 2 2 -2 0 0 4 2 -2 0 -2 0 0  χ 37 0 0 0 0 0 0 0 0 0 0 0 0 χ 38 0 0 0 0 0 0 0 0 0 0 0 0 χ 39 0 0 0 0 2 2 2 2 -2 -2 0 0 χ 40 0 0 0 0 -2 -2 -2 -2 2 2 0 0  χ 41 4 -4 0 0 0 0 0 0 0 0 0 0 χ 42 -4 4 0 0 0 0 0 0 0 0 0 0 χ 43 4 -4 0 0 0 0 0 0 0 0 0 0 χ 44 -4 4 0 0 0 0 0 0 0 0 0 0 χ 45 0 0 0 0 0 0 0 0 0 0 0 0 χ 46 0 0 0 0 0 0 0 0 0 0 0 0 χ 47 0 0 0 0 0 0 0 0 0 0 0 0 χ 48 0 0 0 0 0 0 0 0 0 0 0 0 χ 49 0 0 0 0 0 0 0 0 0 0 0 0  6 A  7 A 7 B   8 A  14 A 14 B  6 D 6 E 12 A 12 B 7 A 7 B 8 I 8 J 16 A 16 B 14 A 14 B  χ 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 χ 3 0 0 0 0 -1 -1 0 0 0 0 1 1 χ 4 0 0 0 0 -1 -1 0 0 0 0 -1 -1 χ 5 -1 -1 -1 -1 0 0 1 1 1 1 0 0 χ 6 1 1 1 1 0 0 -1 -1 -1 -1 0 0 χ 7 0 0 0 0 A B -1 -1 -1 -1 - A - B χ 8 0 0 0 0 B A -1 -1 -1 -1 - B - A χ 9 0 0 0 0 A B 1 1 1 1 A B χ 10 0 0 0 0 B A 1 1 1 1 B A χ 11 -1 -1 -1 -1 1 1 0 0 0 0 1 1 χ 12 1 1 1 1 1 1 0 0 0 0 -1 -1 χ 13 0 0 0 0 0 0 0 0 0 0 0 0 χ 14 0 0 0 0 0 0 0 0 0 0 0 0  χ 15 1 -1 -1 1 0 0 1 1 -1 -1 0 0 χ 16 -1 1 1 -1 0 0 -1 -1 1 1 0 0 χ 17 -1 1 1 -1 0 0 0 0 0 0 0 0 χ 18 1 -1 -1 1 0 0 0 0 0 0 0 0 χ 19 -1 1 1 -1 0 0 1 1 -1 -1 0 0 χ 20 1 -1 -1 1 0 0 -1 -1 1 1 0 0 χ 21 0 0 0 0 0 0 0 0 0 0 0 0 χ 22 0 0 0 0 0 0 1 1 -1 -1 0 0 χ 23 0 0 0 0 0 0 -1 -1 1 1 0 0 χ 24 0 0 0 0 0 0 -1 -1 1 1 0 0 χ 25 0 0 0 0 0 0 1 1 -1 -1 0 0 χ 26 0 0 0 0 0 0 0 0 0 0 0 0   χ 27 1 1 -1 -1 0 0 0 0 0 0 0 0 χ 28 -1 -1 1 1 0 0 0 0 0 0 0 0 χ 29 1 1 -1 -1 0 0 0 0 0 0 0 0 χ 30 -1 -1 1 1 0 0 0 0 0 0 0 0 χ 31 1 1 -1 -1 0 0 0 0 0 0 0 0 χ 32 -1 -1 1 1 0 0 0 0 0 0 0 0 χ 33 0 0 0 0 0 0 0 0 0 0 0 0 χ 34 0 0 0 0 0 0 0 0 0 0 0 0 χ 35 0 0 0 0 0 0 0 0 0 0 0 0 χ 36 0 0 0 0 0 0 0 0 0 0 0 0 χ 37 0 0 0 0 0 0 0 0 0 0 0 0 χ 38 0 0 0 0 0 0 0 0 0 0 0 0 χ 39 0 0 0 0 0 0 0 0 0 0 0 0 χ 40 0 0 0 0 0 0 0 0 0 0 0 0  χ 41 1 -1 1 -1 0 0 2 -2 0 0 0 0 χ 42 -1 1 -1 1 0 0 2 -2 0 0 0 0 χ 43 -1 1 -1 1 0 0 -2 2 0 0 0 0 χ 44 1 -1 1 -1 0 0 -2 2 0 0 0 0 χ 45 0 0 0 0 0 0 0 0 0 0 0 0 χ 46 0 0 0 0 0 0 0 0 C - C 0 0 χ 47 0 0 0 0 0 0 0 0 - C C 0 0 χ 48 1 -1 1 -1 0 0 0 0 0 0 0 0 χ 49 -1 1 -1 1 0 0 0 0 0 0 0 0 A = ( - 1 - 7 i ) / 2, B=(-1+7i)/2, and C=22i.The information about the conjugacy classes found in Table2 can be used to compute the power maps for the elements of G¯ and then with the aid of Programme E in [18] we can verify that we obtained the unique p-power maps listed in Table 8 for our Table 7.Table 8
The power maps of the elements of29:(L3(4):2).
[ g ] G [ x ] G ¯ 2 3 5 7 [ g ] G [ x ] G ¯ 2 3 5 7 1 A 1 A     2 A 2 D 1 A 2 A 1 A     2 E 1 A 2 B 1 A     4 A 2 B 2 C 1 A     4 B 2 B 4 C 2 B  2 B 2 F 1 A    3 A 3 A  1 A 2 G 1 A     6 A 3 A 2 C 4 D 2 A     6 B 3 A 2 A 4 E 2 A     6 C 3 A 2 B 4 F 2 B 4 G 2 B 4 H 2 B 4 I 2 B  4 A 4 J 2 F    4 B 4 L 2 F 4 K 2 F     4 M 2 F 8 A 4 E     8 C 4 E 8 B 4 F     8 D 4 I  4 C 4 N 2 F    5 A 5 A   1 A 4 O 2 F     10 A 5 A  2 A 8 E 4 F 8 F 4 F 8 G 4 D 8 H 4 F  6 A 6 D 3 A 2 D   7 A 7 A   1 A 6 E 3 A 2 E 12 A 6 C 4 C 12 B 6 C 4 A  7 B 7 B    1A 8 A 8 I 4 J 8 J 4 J 16 A 8 A 16 B 8 A  14 A 14 A 7 A   2 D 14 B 14 B 7 B   2 D


## 8. The Fusion of29:(L3(4):2) into U6(2):2

SinceG¯ is a maximal subgroup of U6(2):2 of index 891, then the action of U6(2):2 on the cosets of G¯ gives rise to a permutation character χ(U6(2):2∣G¯) of degree 891. We deduce from the character table of U6(2):2 found in GAP that χ(U6(2):2∣G¯)=1a+22a+252a+616a, where 1a, 22a, 252a, and 616a are irreducible characters of U6(2):2 of degrees 1, 22, 252, and 616, respectively.We are able to obtain the partial fusion ofG¯ into U6(2):2, using the information provided by the values of χ(U6(2):2∣G¯) on the classes of G¯ and the power maps of G¯ and U6(2):2. Then, the technique of set intersections for characters (see [9, 14, 19]) is applied to restrict some ordinary irreducible characters of U6(2):2 of small degrees to G¯, to determine fully the fusion of the classes of G¯ into U6(2):2.Letζ be the character afforded by the regular representation of L3(4):2. We obtain that ζ = Σi=114αiΦi, where Φi∈Irr(L3(4):2) and αi=deg(Φi). Then ζ can be regarded as a character of 29:(L3(4):2) which contains 29 in its kernel such that (11)ζx=L34:2ifx∈290otherwise. If ϕ is a character of U6(2):2 than we have that (12)ζ,ϕG¯=129:L34:2ζ1Aϕ1A+21ζ2Aϕ2A+210ζ2Bϕ2B+280ζ2Cϕ2C=129:(L34:2L34:2ϕ1A+21ϕ2A+210ϕ2B+280ϕ2C=1512ϕ1A+21ϕ2A+210ϕ2B+280ϕ2C=ϕ29,129. Here 129 is the identity character of 29 and ϕ29 is the restriction of ϕ to 29. We obtain that (13)ϕ29=a1θ1+a2θ2+a3θ3+a4θ4,where ai∈N∪0 and θi are the sums of the irreducible characters of 29 which are in the same orbit under the action of L3(4):2 on Irr(29), for i∈{1,2,3,4}. Let φj∈Irr(29), where j∈{1,2,3,…,512}. Then we obtain that(14)θ1=φ1,degθ1=1θ2=∑j=222φj,degθ2=21θ3=∑j=23232φj,degθ3=210θ4=∑j=233512φj,degθ4=280.Hence(15)ϕ29=a1φ1+a2∑j=222φj+a3∑j=23232φj+a4∑j=233512φj,and therefore(16)ϕ29,ϕ29=a12+21a22+210a32+280a42=1512ϕ1Aϕ1A+21ϕ2Aϕ2A+210ϕ2Bϕ2B+280ϕ2Cϕ2C,where a1= ζ,ϕ29:(L3(4):2).We apply the above results to some of the irreducible characters ofU6(2):2 of small degrees, which in this case are ϕ1 = 22a, ϕ2 = 22b, ϕ3 = 231a, ϕ4 = 231b, ϕ5 = 440a, and ϕ6 = 440b. Their respective degrees are 22, 22, 231, 231, 440, and 440. For ϕ1 we calculate that (17)ζ,ϕ129:L34:2=151222+21-10+2106+280-2=1.Now a1 + 21a2 + 210a3 + 280a4 = 22, since deg⁡ϕ1 = 22. Since a1 = 1, we must have that a2 = 1 and a3 =a4 = 0. Note that 29:(L3(4):2) does not have irreducible characters of degree 22. We obtain that (ϕ1)29:(L3(4):2 = χ1 + χ15 if the partial fusion of 29:(L3(4):2) into U6(2):2 is taken into consideration. Similarly, for ϕ3 and ϕ5 we calculate that (18)ζ,ϕ329:L34:2=1512231+2139+2107+280-9=0and (19)ζ,ϕ529:L34:2=1512440+21120+21024+2808=20.Since the respective degrees of ϕ3 and ϕ5 are 231 and 440, we have to solve the equations (i) a1 + 21a2 + 210a3 + 280a4 = 231 and (ii) a1 + 21a2 + 210a3 + 280a4 = 440, separately. If we are taking into account the fact that the set Irr(G¯) (see Table 7) does not have any irreducible characters of degrees 231 and 440 and also that ζ,ϕ329:(L3(4):2)=0 and ζ,ϕ529:(L3(4):2)=20, we deduce that the two sets of values {a1=a4=0,a2=a3=1} and {a1=20,a2=10,a3=1,a4=0} are the only possibilities that satisfy equation (i) and (ii), respectively, hence we obtained that (ϕ3)29:(L3(4):2) = χ16 + χ27 and (ϕ5)29:(L3(4):2) = χ4 + χ32. Similar computations were carried out to restrict the characters ϕ2, ϕ4, and ϕ6 to G¯ and we found that (ϕ2)29:(L3(4):2) = χ2+χ16, (ϕ4)29:(L3(4):2) = χ15+χ28, and (ϕ6)29:(L3(4):2)= χ3 + χ31.By making use of the values ofϕ1, ϕ2, ϕ3, ϕ4, ϕ5, and ϕ6 on the classes of U6(2):2 and the values of (ϕ1)29:(L3(4):2), (ϕ2)29:(L3(4):2), (ϕ3)29:(L3(4):2), (ϕ4)29:(L3(4):2), (ϕ5)29:(L3(4):2), and (ϕ6)29:(L3(4):2) on the classes of 29:(L3(4):2) together with the partial fusion, the complete fusion map of 29:(L3(4):2) into U6(2):2 is given in Table 9.Table 9
The fusion of29:(L3(4):2) into U6(2):2.
[ g ] L 3 ( 4 ) : 2 [ x ] 2 9 : ( L 3 ( 4 ) : 2 ) → [ y ] U 6 ( 2 ) : 2 [ g ] L 3 ( 4 ) : 2 [ x ] 2 9 : ( L 3 ( 4 ) : 2 ) → [y]U6(2):2 1 A 1 A 1 A 2 A 2 D 2D 2 A 2 A  2 E 2E 2 B 2 B  4 A 4G 2 C 2 C  4 B 4H 4 C 4 I  2 B 2 F 2 B 3 A 3 A 3C 2 G 2 C  6 A 6G 4 D 4 A  6 B 6E 4 E 4 B  6 C 6F 4 F 4 C 4 G 4 E 4 H 4 F 4 I 4 D  4 A 4 J 4 C 4 B 4 L 4D 4 K 4 F  4 M 4 F 8 A 8 A  8 C 8 A 8 B 8 B  8 D 8 C  4 B 4 N 4 H 5 A 5 A 5A 4 O 4 I  10 A 10 A 8 E 8 F 8 F 8 G 8 G 8 D 8 H 8 E  6 A 6 D 6 K 7 A 7 A 7A 6 E 6 L 12 A 12 K 12 B 12 J  7 B 7 B 7 A 8 A 8 I 8E 8 J 8F 16 A 16 A 16 B 16 B  14 A 14 A 14 A 14 B 14 B 14A



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*Source: 9382525-2019-02-25.xml*