ON A GROUP OF THE FORM 3 7 :Sp(6, 2) Communicated by Mohammad Reza Darafsheh

Transcription

1 International Journal of Group Theory ISSN (print): , ISSN (on-line): Vol 5 No (016), pp c 016 University of Isfahan wwwtheoryofgroupsir wwwuiacir ON A GROUP OF THE FORM 3 7 :Sp(6, ) AYOUB B M BASHEER AND JAMSHID MOORI Communicated by Mohammad Reza Darafsheh Abstract The purpose of this paper is the determination of the inertia factors, the computations of the Fischer matrices and the ordinary character table of the split extension G = 3 7 :Sp(6, ) by means of Clifford-Fischer Theory We firstly determine the conjugacy classes of G using the coset analysis method The determination of the inertia factor groups of this extension involved looking at some maximal subgroups of the maximal subgroups of Sp(6, ) The Fischer matrices of G are all listed in this paper and their sizes range between and 10 The character table of G, which is a C-valued matrix, is available in the PhD thesis of the first author, which could be accessed online 1 Introduction Let G = Sp(6, ) be the symplectic group of order By the electronic Atlas [16], the group G has a 7 dimensional (absolutely) irreducible module over GF (3) = F 3 = {0, 1, ξ}, where ξ is a primitive element of the field F 3 Consequently a split extension of the form 3 7 :Sp(6, ) does exist Using the two 7 7 matrices over F 3 that generate Sp(6, ), supplied by the electronic Atlas, we were able to construct G and then G inside GAP [10] In fact we constructed the group G in GAP, in terms of 8 8 matrices over F 3 The following two elements g 1 and g are 8 dimensional matrices over F 3 that generate G MSC(010): Primary: 0C15; Secondary: 0C40 Keywords: Group extensions, symplectic group, character table, inertia groups, Fischer matrices Received: 5 September 014, Accepted: 1 February 015 Corresponding author 41

2 4 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 1 1 ξ 1 1 ξ ξ 0 0 ξ ξ 0 ξ 1 ξ ξ ξ 1 ξ ξ g 1 = ξ 1 ξ ξ 0 ξ ξ ξ ξ 0 0 ξ with o(g 1 ) = 45, o(g ) = 45 and o(g 1 g ) = 6, g = ξ ξ ξ ξ 0 ξ ξ 0 ξ 0 0 ξ 0 0 ξ ξ ξ 0 ξ 0 0 ξ 1 0 ξ 0 ξ ξ 0 1 ξ ξ 1 ξ ξ ξ ξ 1, Corollary 11 G SL(8, 3) with index Proof This is readily verified since det(g 1 ) = det(g ) = 1 and consequently det(g) = 1 for all g G Using GAP, one can easily check all the normal subgroups of G In fact the only proper normal subgroup of G is a group of order 187 and thus must be isomorphic to the elementary abelian group N = 3 7 The following elements n 1, n,, n 7 are 8 8 matrices over F 3 that are generators of N n 1 =, n =, n 3 =, ξ 1 1 ξ ξ 1 ξ ξ ξ 0 ξ n 4 =, n 5 =, n 6 =, n 7 = ξ ξ 0 ξ ξ 1 ξ 1 0 ξ ξ ξ ξ ξ 1 ξ 0 1 In terms of 8 8 matrices over F 3, the group Sp(6, ) is generated by the following two elements g 1 and g : 0 ξ g 1 =, g =, 0 0 ξ ξ ξ ξ ξ 0 ξ ξ ξ ξ ξ ξ 0 0 ξ ξ 0 0 ξ 1 ξ with o(g 1 ) =, o(g ) = 7 and o(g 1 g ) = 9 Note that the group Sp(6, ) = g 1, g together with the mentioned generators of N, gives the split extension G = 3 7 :Sp(6, ) For the notation used in this paper and the description of Clifford-Fischer theory technique, we follow [1, ]

3 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 43 Conjugacy Classes of G = 3 7 :Sp(6, ) In this section we calculate the conjugacy classes of G using the coset analysis technique (see [1] or [1, 13] for more details) as we are interested to organize the classes of G corresponding to the classes of Sp(6, ) We have used GAP to build a small subroutine to find the values of k i s, which we list in Table 1 We supplied the values of χ(sp(6, ) 3 7 ) on each of the 30 conjugacy classes of Sp(6, ) In fact the subroutine we have used to find the values of k i s can be developed further to find the values of f ij s for each coset corresponding to [g i ] G A complete set of the f ij s and representatives for the conjugacy classes of G are given in Table 1 To each class of G, we have attached some weight m ij, which will be used later in computing the Fischer matrices of the extension These weights are computed through the formula (1) m ij = [N G (Ng i ) : C G (g ij )] = N C G(g i ) C G (g ij ) Example 1 Consider the identity coset N = 3 7 as this coset is so important Recall that N is abelian and thus each orbit of the action of N on itself consists of singleton Therefore k 1 = N = 178 Since we can present G and N in GAP in terms of 8 8 matrices over F 3, it is easy for G to act on N In fact this action yielded six orbits of lengths 1, 56, 16, 576, 67 and 756 with representatives g 11, g 1, g 13, g 14, g 15 and g 16 defined as follows: g 11 =, g 1 =, g 13 =, ξ ξ ξ g 14 =, g 15 =, g 16 = ξ ξ ξ ξ ξ ξ 0 0 ξ 1 Thus the identity coset affords six conjugacy classes in G These classes have sizes equal to the orbits lengths of G on N with respective representatives g 11, g 1, g 13, g 14, g 15 and g 16 Also the values of the f 1j s are same as lengths of the corresponding conjugacy classes for all 1 j 6 Clearly g 11 = 1 G and thus o(g 11 ) = 1 Since g 1 = 1 Sp(6,) and o(g 1 ) = 1, it follows by applications of Proposition 33 of [1] that o(g 1 ) = o(g 13 ) = o(g 14 ) = o(g 15 ) = o(g 16 ) = 3 The orders of the preceding elements can also be seen directly since N is an elementary abelian 3 group Similar arguments can be applied to all the other cosets Ng i, i 30 We list the conjugacy classes of G in Table 1

4 44 Int J Group Theory, 5 no (016) A B M Basheer and J Moori Table 1: The conjugacy classes of G = 3 7 :Sp(6, ) [g i ] G k i f ij m ij [g ij ] G o(g ij ) [g ij ] G C G (g ij ) f 11 = 1 m 11 = 1 g f 1 = 56 m 1 = 56 g g 1 = 1A k 1 = 187 f 13 = 16 m 13 = 16 g f 14 = 576 m 14 = 576 g f 15 = 67 m 15 = 67 g f 16 = 756 m 16 = 756 g g = A k = 3 f 1 = 1 m 1 = 79 g f = m = 1458 g f 31 = 1 m 31 = 81 g g 3 = B k 3 = 7 f 3 = 6 m 3 = 486 g f 33 = 8 m 33 = 648 g f 34 = 1 m 34 = 97 g f 41 = 1 m 41 = 9 g f 4 = m 4 = 18 g f 43 = 8 m 43 = 7 g f 44 = 16 m 44 = 144 g g 4 = C k 4 = 43 f 45 = 16 m 45 = 144 g f 46 = 4 m 46 = 16 g f 47 = 3 m 47 = 88 g f 48 = 3 m 48 = 88 g f 49 = 48 m 49 = 43 g f 4,10 = 64 m 4,10 = 576 g 4, f 51 = 1 m 51 = 81 g g 5 = D k 5 = 7 f 5 = 6 m 5 = 486 g f 53 = 8 m 53 = 648 g f 54 = 1 m 54 = 97 g f 61 = 1 m 61 = 9 g f 6 = 0 m 6 = 180 g f 63 = 30 m 63 = 70 g g 6 = 3A k 6 = 43 f 64 = 30 m 64 = 70 g f 65 = 1 m 65 = 108 g f 66 = 30 m 66 = 70 g f 67 = 10 m 67 = 1080 g f 71 = 1 m 71 = 81 g g 7 = 3B k 7 = 7 f 7 = 8 m 7 = 648 g f 73 = 9 m 73 = 79 g f 74 = 9 m 74 = 79 g f 81 = 1 m 81 = 81 g f 8 = m 8 = 16 g f 83 = m 83 = 16 g g 8 = 3C k 8 = 7 f 84 = 4 m 84 = 34 g f 85 = 6 m 85 = 486 g f 86 = 1 m 86 = 97 g f 91 = 1 m 91 = 81 g g 9 = 4A k 9 = 7 f 9 = 6 m 9 = 486 g f 93 = 8 m 93 = 648 g f 94 = 1 m 94 = 97 g f 10,1 = 1 m 10,1 = 81 g 10, g 10 = 4B k 10 = 7 f 10, = 6 m 10, = 486 g 10, continued on next page

5 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 45 Table 1 (continued from previous page) [g i ] G k i f ij m ij [g ij ] G o(g ij ) [g ij ] G C G (g ij ) f 10,3 = 8 m 10,3 = 648 g 10, f 10,4 = 1 m 10,4 = 97 g 10, g 11 = 4C k 11 = 3 f 11,1 = 1 m 11,1 = 79 g 11, f 11, = m 11, = 1458 g 11, g 1 = 4D k 1 = 3 f 1,1 = 1 m 1,1 = 79 g 1, f 1, = m 1, = 1458 g 1, f 13,1 = 1 m 13,1 = 81 g 13, f 13, = m 13, = 16 g 13, f 13,3 = m 13,3 = 16 g 13, g 13 = 4E k 13 = 7 f 13,4 = m 13,4 = 16 g 13, f 13,5 = 4 m 13,5 = 34 g 13, f 13,6 = 4 m 13,6 = 34 g 13, f 13,7 = 4 m 13,7 = 34 g 13, f 13,8 = 8 m 13,8 = 648 g 13, f 14,1 = 1 m 14,1 = 81 g 14, f 14, = 3 m 14, = 43 g 14, f 14,3 = 3 m 14,3 = 43 g 14, g 14 = 5A k 14 = 7 f 14,4 = m 14,4 = 16 g 14, f 14,5 = 6 m 14,5 = 486 g 14, f 14,6 = 6 m 14,6 = 486 g 14, f 14,7 = 6 m 14,7 = 486 g 14, g 15 = 6A k 15 = 3 f 15,1 = 1 m 15,1 = 79 g 15, f 15, = m 15, = 1458 g 15, f 16,1 = 1 m 16,1 = 81 g 16, f 16, = 4 m 16, = 34 g 16, g 16 = 6B k 16 = 7 f 16,3 = 4 m 16,3 = 34 g 16, f 16,4 = 6 m 16,4 = 486 g 16, f 16,5 = 1 m 16,5 = 97 g 16, f 17,1 = 1 m 17,1 = 79 g 17, g 17 = 6C k 17 = 3 f 17, = 1 m 17, = 79 g 17, f 17,3 = 1 m 17,3 = 79 g 17, f 18,1 = 1 m 18,1 = 81 g 18, f 18, = m 18, = 16 g 18, f 18,3 = m 18,3 = 16 g 18, g 18 = 6D k 18 = 7 f 18,4 = 4 m 18,4 = 34 g 18, f 18,5 = m 18,5 = 16 g 18, f 18,6 = 4 m 18,6 = 34 g 18, f 18,7 = 4 m 18,7 = 34 g 18, f 18,8 = 8 m 18,8 = 648 g 18, g 19 = 6E k 19 = 3 f 19,1 = 1 m 19,1 = 79 g 19, f 19, = m 19, = 1458 g 19, g 0 = 6F k 0 = 3 f 0,1 = 1 m 0,1 = 79 g 0, f 0, = m 0, = 1458 g 0, g 1 = 6G k 1 = 3 f 1,1 = 1 m 1,1 = 79 g 1, f 1, = m 1, = 1458 g 1, f,1 = 1 m,1 = 79 g, g = 7A k = 3 f, = 1 m, = 79 g, f,3 = 1 m,3 = 79 g, continued on next page

6 46 Int J Group Theory, 5 no (016) A B M Basheer and J Moori Table 1 (continued from previous page) [g i ] G k i f ij m ij [g ij ] G o(g ij ) [g ij ] G C G (g ij ) g 3 = 8A k 3 = 3 f 3,1 = 1 m 3,1 = 79 g 3, f 3, = m 3, = 1458 g 3, f 4,1 = 1 m 4,1 = 79 g 4, g 4 = 8B k 4 = 3 f 4, = 1 m 4, = 79 g 4, f 4,3 = 1 m 4,3 = 79 g 4, f 5,1 = 1 m 5,1 = 79 g 5, g 5 = 9A k 5 = 3 f 5, = 1 m 5, = 79 g 5, f 5,3 = 1 m 5,3 = 79 g 5, f 6,1 = 1 m 6,1 = 79 g 6, g 6 = 10A k 6 = 3 f 6, = 1 m 6, = 79 g 6, f 6,3 = 1 m 6,3 = 79 g 6, g 7 = 1A k 7 = 3 f 7,1 = 1 m 7,1 = 79 g 7, f 7, = m 7, = 1458 g 7, g 8 = 1B k 8 = 3 f 8,1 = 1 m 8,1 = 79 g 8, f 8, = m 8, = 1458 g 8, f 9,1 = 1 m 9,1 = 79 g 9, g 9 = 1C k 9 = 3 f 9, = 1 m 9, = 79 g 5, f 9,3 = 1 m 9,3 = 79 g 9, f 30,1 = 1 m 30,1 = 79 g 30, g 30 = 15A k 30 = 3 f 30, = 1 m 30, = 79 g 30, f 30,3 = 1 m 30,3 = 79 g 30, Inertia Factor Groups of G = 3 7 :Sp(6, ) In this section, through some computations, we determine the inertia factor groups of 3 7 :Sp(6, ) This determination is achieved by investigating the number of irreducible characters and fusions of conjugacy classes of some of the maximal subgroups of the maximal subgroups of Sp(6, ) (sometimes we may go further and look at some of the maximal subgroups of the maximal subgroups of the maximal subgroups of Sp(6, )) We have seen in Section that the action of G = 3 7 :Sp(6, ) (or just G = Sp(6, )) on 3 7 produces six orbits of lengths 1, 56, 16, 576, 67 and 756 By a theorem of Brauer (for example see Theorem 515 of [14]) the number of orbits of G (or just G) on Irr(3 7 ) will also be 6 Since N = 3 7 is a vector space, the action of G on Irr(3 7 ) can be viewed as the action of G on N, where N is the dual space of N In fact we have found that the orbit lengths of G on Irr(3 7 ) are 1, 56, 16, 576, 67 and 756 Let H 1, H,, H 6 be the respective inertia factor groups of the representatives of characters from the previous orbits We notice that these inertia factors have indices 1, 56, 16, 576, 67 and 756 respectively in Sp(6, ) Clearly H 1 = Sp(6, ) By looking at the ATLAS [5], the group Sp(6, ) has 8 conjugacy classes of maximal subgroups Let M[1], M[],, M[8] be representatives of these classes That is M[1] = U 4 ():, M[] = S 8, M[3] = 5 :S 6, M[4] = U 3 (3):, M[5] = 6 :L 3 (),

7 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 47 M[6] = ( 1+4 ):(S 3 S 3 ), M[7] = S 3 S 6 and M[8] = P SL(, 8):3 In Table we give some information about the maximal subgroups of M[1], M[],, M[8] Now by considering the indices of the maximal subgroups of Sp(6, ), we get the following possibilities for H 1, H,, H 6 : H 1 = Sp(6, ), H M[1] = U 4 (): with index, H 3 M[3] = 5 :S 6 with index, H 4 M[] = S 8 with index 16, H 5 M[1] = U 4 (): with index 4 or a subgroup of M[7] = S 3 S 6 with index, H 6 M[1] with index 7, or a subgroup of M[] of index 1, or a subgroup of M[3] with index 1 Recall from Table 1 that the total number of conjugacy classes of G is 118 We deduce that the total contribution of irreducible characters from the six inertia factor groups must also be 118 That is (31) Irr(H 1 ) + Irr(H ) + Irr(H 3 ) + Irr(H 4 ) + Irr(H 5 ) + Irr(H 6 ) = First, Second, Third and Fourth Inertia Factor Groups We recall that in Section 1, we represented the group G = Sp(6, ) in terms of 8 dimensional matrices For the sake of convenience in computations with GAP, we use 6 dimensional representations over F of Sp(6, ) We have used the following 6 dimensional matrices α 1 and α over F, that generate Sp(6, ) (see [16]) to represent Sp(6, ) in GAP and then locate the maximal subgroups and the other required subgroups α 1 = , α = We have mentioned that the first inertia factor group H 1 is Sp(6, ), which has 30 irreducible characters Since H has an index in U 4 ():, it is readily verified that H = M[11] = U 4 (), which has 0 irreducible characters generated by σ 1 and σ, where As a 6 dimensional subgroup of Sp(6, ) over F, the group H is σ 1 = , σ = The character table of H = U 4 () is available in the ATLAS and also appears as Table 111 of [1] The third inertia factor group H 3 has an index in 5 :S 6 In Table we list some information on the maximal subgroups of the maximal subgroups of Sp(6, )

8 48 Int J Group Theory, 5 no (016) A B M Basheer and J Moori Table Some information on the maximal subgroups of the maximal subgroups of Sp(6, ) Maximal Subgroups M[ij] M[ij] [M[i] : M[ij]] Irr(M[ij]) of Sp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y checking the indices of the maximal subgroups of M[3] = 5 :S 6 (supplied in Table ) we can see that H 3 is in the conjugacy class of maximal subgroups of M[3] containing M[31] = 5 :A 6 Thus we can take H 3 = M[31] Note that M[31] = 5 :A 6 = ξ 1, ξ, where

9 Int J Group Theory, 5 no (016) A B M Basheer and J Moori ξ 1 = , ξ = The character table of H 3 = 5 :A 6 appears as Table 11 of [1] or can easily be obtained using GAP Next we determine the fourth inertia factor group H 4, which sits inside M[] = S 8 Note that [S 8 : H 4 ] = 16 and thus (using Table ) H 4 is either a subgroup of M[1] = A 8 with index 8, or it is a subgroup of M[] = S 7 with index In Table 3, we give some information about the maximal subgroups of M[1] and M[] Table 3 Some information on the maximal subgroups of M[1] and M[] Maximal Subgroups M[ijk] M[ijk] [M[ij] : M[ijk]] Irr(M[ijk]) of M[1] & M[] M[11] = A M[1] = 3 :P SL(3, ) M[1] = A 8 M[13] = 3 :P SL(3, ) M[14] = S M[15] = ((A 4 A 4 ):): M[16] = GL(, 4): M[1] = A M[] = S M[] = S 7 M[3] = S M[4] = S 4 S M[5] = (7:3): From Table 3, we can see that a subgroup of M[1] = A 8 of index 8 must be isomorphic to A 7, while a subgroup of M[] = S 7 of index must also be isomorphic to A 7 Note that M[11] = A 7 = β 1, β and M[1] = A 7 = γ 1, γ, where β 1 = , β = , γ 1 = , γ = Thus for the construction of the character table of G, it will not make difference to which A 7 we choose Hence we may take H 4 to be M[11] = A 7 and note that Irr(H 4 ) = Irr(A 7 ) = 9 The character table of A 7 appears as Table 113 of [1] 3 Fifth and Sixth Inertia Factor Groups From the last subsection we have seen that Irr(H 1 ) = 30, Irr(H ) = 0, Irr(H 3 ) = 3 and Irr(H 4 ) = 9 Substituting these into Equation (31), we get that Irr(H 5 ) + Irr(H 6 ) = 36 Recall that H 5 is either an index 4 subgroup of M[1] = U 4 (): or it is an index subgroup of M[7] = S 3 S 6 If H 5 U 4 (): with index 4, then the only possibility (see Table ) is that

10 50 Int J Group Theory, 5 no (016) A B M Basheer and J Moori H 5 U 4 () with [U 4 () : H 5 ] = 1 However by looking at the ATLAS we can see that the group U 4 () does not contain a subgroup of index 1 This leaves us with the other possibility, that is H 5 S 3 S 6 and [S 3 S 6 : H 5 ] = From Table, we can see that there are three classes of nonconjugate maximal subgroups of S 3 S 6, such that a subgroup of each class has an index Therefore H 5 is either M[71] = A 6 S 3, M[7] = (3 A 6 ): or M[73] = 3 S 6 Hence by the last column of Table, it follows that Irr(H 5 ) {1, 18, 33} We take this point into our considerations and we look at the group H 6 The index of the sixth inertia factor group H 6 in Sp(6, ) is 756 This forces H 6 to be either a subgroup of U 4 (): with index 7, or a subgroup of S 8 of index 1, or a subgroup of 5 :S 6 with index 1 However the second possibility (H 6 S 8 ) is not feasible since S 8 does not contain a subgroup of index that is a divisor of 7 (see Table ) If H 6 is a subgroup of U 4 (): of index 7, then it must be in the conjugacy class of maximal subgroups of U 4 (): containing M[1] = ( 4 :A 5 ): The group M[1] = ( 4 :A 5 ): is generated by π 1 and π, where π 1 = , π = and Irr(H 6 ) = 18 On the other hand if H 6 5 :S 6 such that [ 5 :S 6 : H 6 ] = 1, then three possibilities arise (see Table ): H 6 M[31] = 5 :A 6 with index 6, H 6 M[3] = ( 5 :A 5 ): with index or H 6 M[33] = (( 4 :A 5 ):) with index In Table 4, we provide some information on the maximal subgroups of M[31], M[3] and M[33] Table 4 Some information on the maximal subgroups of M[31], M[3] and M[33] Maximal Subgroups M[ijk] M[ijk] [M[ij] : M[ijk]] Irr(M[ijk]) of M[31], M[3] & M[33] M[311] = 5 :A M[31] = ( 4 :A 5 ) M[31] = 5 :A 6 M[313] = ((A 4 A 4):4) M[314] = ( (((( D 8):):3):)): M[315] = ((( ( 4 :)):):3): M[316] = A M[317] = A M[31] = 5 :A M[3] = ((( ( 4 :)):):3): M[3] = ( 5 :A 5 ): M[33] = (( 4 :5):4) M[34] = ((( 4 :3):):) M[35] = S M[36] = S M[331] = ( 4 :A 5): M[33] = ( 4 :A 5) M[33] = (( 4 :A 5):) M[333] = ( 4 :A 5): M[334] = ((((( D 8 ):):3):):) M[335] = (( 4 :5):4) M[336] = S 4 D M[337] = S

11 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 51 From Table 4, we can see that there are 6 possibilities for H 6 This together with the option H 6 = M[1] = ( 4 :A 5 ): gives that H 6 {M[1], M[311], M[31], M[31], M[331], M[33], M[333]} and we notice that Irr(H 6 ) {18, 16, 4, 16, 18, 4, 18} respectively We recall that Irr(H 5 ) {1, 18, 33} Therefore possible pairs representing (H 5, H 6 ) are (H 5, H 6 ) {(M[71], M[1]), (M[71], M[311]), (M[71], M[31]), (M[71], M[31]), (M[71], M[331]), (M[71], M[33]), (M[71], M[333]), (M[7], M[1]), (M[7], M[311]), (M[7], M[31]), (M[7], (M[31]), (M[7], M[331]), (M[7], M[33]), (M[7], M[333]), (M[73], M[1]), (M[73], (M[311]), (M[73], M[31]), (M[73], M[31]), (M[71], M[331]), (M[71], M[33]), (M[73], M[333])} and it follows respectively that ( Irr(H 5 ), Irr(H 6 ) ) {(1, 18), (1, 16), (1, 4), (1, 16), (1, 18), (1, 4), (1, 18), (18, 18), (18, 16), (18, 4), (18, 16), (18, 18), (18, 4), (18, 18), (33, 18), (33, 16), (33, 4), (33, 16), (33, 18), (33, 4), (33, 18)} Since Irr(H 5 ) + Irr(H 6 ) = 36, we deduce that ( Irr(H 5 ), Irr(H 6 ) ) {(18, 18), (18, 18), (18, 18)}, that is (H 5, H 6 ) {(M[7], M[1]), (M[7], M[331]), (M[7], M[333])} Hence H 5 = M[7] = (3 A 6 ): (in all the cases) and H 6 = M[1] = ( 4 :A 5 ):, M[331] = ( 4 :A 5 ): or M[333] = ( 4 :A 5 ): Note that the group M[7] = (3 A 6 ): is generated by µ 1 and µ, where µ 1 = , µ = The full character table of H 5 appears as Table 114 of [1] or can easily be obtained using GAP Next we determine the group H 6 The groups M[1], M[331] and M[333] are generated as follows: M[1] = θ 1, θ, M[331] = ϵ 1, ϵ and M[333] = δ 1, δ, where θ 1 = , θ = ,

12 5 Int J Group Theory, 5 no (016) A B M Basheer and J Moori ϵ 1 = , ϵ = , δ 1 = , δ = By looking at the fusion of the conjugacy classes of M[1] = θ 1, θ, we see that the unique conjugacy class of involutions with size 0, fuses into the class g = A of Sp(6, ) From Table we see that c(g ) = and hence from the properties of the Fischer matrices (Proposition 36 of t 6 []), we have c(g ik ) = c(g i ) and for i = we get c(g k ) = c(g ) = From Table 611 of k=1 [1], we can see that the classes A of H 1 = Sp(6, ) and a of H 3 are both fusing into the class A of Sp(6, ) Therefore if M[1] is the sixth inertia factor group, then we get a contradiction (the Fischer matrix F corresponds to g = A will be of size 3 contradicting Proposition 36(i) of []) By similar arguments we can show that the group M[333] can not be H 6 Hence we deduce that M[331] = ( 4 :A 5 ):, with the generators ϵ 1 and ϵ, is the sixth inertia factor group H 6 The fusion of classes of H 6 = ( 4 :A 5 ): into classes of Sp(6, ) can be viewed in Table 611 of [1] The full character table of H 6 appears as Table 115 of [1] This completes our determination of the inertia factor groups of G = 3 7 :Sp(6, ) Note 31 If θ i is an orbit representative of the action of G on Irr(N), then H i can be obtained by GAP as G θi, the set stabilizer of θ i in G 33 Fusions of the Inertia Factor Groups into Sp(6, ) In this section we determine the fusions of classes of the inertia factor groups H, H 3, H 4, H 5 and H 6 into classes of Sp(6, ) We have used the permutation characters of Sp(6, ) on the inertia factor groups and the centralizer sizes to determine these fusions We have found the following proposition is very helpful in calculating the permutation characters χ(sp(6, ) H i ), i 6 Proposition 3 Let K 1 K K 3 and let ψ be a class function on K 1 Then (ψ K K 1 ) K 3 K = ψ K 3 K 1 More generally if K 1 K K n is a nested sequence of subgroups of K n and ψ is a class function on K 1, then (ψ K K 1 ) K 3 K K n K n 1 = ψ K n K 1 k=1 Proof See Proposition 356 of [1] The decompositions of the permutation characters χ(sp(6, ) M[i]), 1 i 8 are all given in the ATLAS Also it is not difficult to calculate χ(m[i] M[ij]), 1 i 8, j is the number of conjugacy classes of maximal subgroups of M[i] Where it is needed, it is also not difficult to calculate the permutation character χ(m[ij] M[ijk]), 1 i 8, j is the number of conjugacy classes of maximal subgroups of M[i], k is the number of conjugacy classes of maximal subgroups of M[ij] Thus with

13 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 53 the applications of Proposition 3, the values of χ(sp(6, ) H i ), i 6 are easy to calculate In fact we have listed these values at the bottom of Tables 66, 67, 68, 69 and 610 of [1] respectively Now with the aid of the permutation characters, centralizer sizes and matrix conjugation in Sp(6, ), we are able to determine all the fusions of classes of the inertia factor groups into classes of Sp(6, ) We list these fusions in Table 5 Table 5 The fusions of conjugacy classes of the inertia factor groups into classes of Sp(6, ) Inertia Factor Class of Class of Class of Class of Groups H, H 3, H 4, H 5 & H 6 H i Sp(6, ) H i Sp(6, ) 1a = g 11 1A 6a = g 16,1 6B a = g 41 C 6b = g 16, 6B b = g 31 B 6c = g 17,1 6C 3a = g 61 3A 6d = g 17, 6C H = M[11] = U 4 () 3b = g 71 3B 6e = g 0,1 6F 3c = g 7 3B 6f = g 18,1 6D 3d = g 81 3C 9a = g 5,1 9A 4a = g 13,1 4E 9b = g 5, 9A 4b = g 91 4A 1a = g 9,1 1C 5a = g 14,1 5A 1b = g 9, 1C 1a = g 131 1A 4e = g 10,31 4B a = g 31 A 5a = g 14,31 5A b = g 331 B 5b = g 14,3 5A c = g 431 C 6a = g 16,31 6B d = g 43 C 6b = g 18,31 6D H 3 = M[31] = 5 :A 6 e = g 531 D 6c = g 15,31 6A 3a = g 631 3A 6d = g 19,31 6E 3b = g 831 3C 8a = g 4,31 8B 4a = g 1,31 4D 8b = g 4,3 8B 4b = g 931 4A 10a = g 6,31 10A 4c = g 13,31 4E 10b = g 6,3 10A 4d = g 13,3 4E 1a = g 141 1A 5a = g 14,41 5A a = g 441 C 6a = g 18,41 6D H 4 = M[11] = A 7 3a = g 841 3C 7a = g,41 7A 3b = g 641 3A 7b = g,4 7A 4a = g 13,41 4E 1a = g 151 1A 4a = g 10,51 4B a = g 451 C 4b = g 13,51 4E b = g 45 C 5a = g 14,51 5A c = g 551 D 6a = g 18,51 6D H 5 = M[7] = (3 A 6): 3a = g 651 3A 6b = g 18,5 6D 3b = g 65 3A 6c = g 1,51 6G 3c = g 851 3C 1a = g 7,51 1A 3d = g 751 3B 15a = g 30,51 15A 3e = g 85 3C 15b = g 30,5 15A 1a = g 161 1A 4c = g 10,61 4B a = g 461 C 4d = g 11,61 4C b = g 361 B 4e = g 13,6 4E c = g 46 C 5a = g 14,61 5A H 6 = M[331] = ( 4 :A 5): d = g 561 D 6a = g 16,61 6B e = g 463 C 6b = g 18,61 6D 3a = g 661 3A 6c = g 18,6 6D 4a = g 961 4A 8a = g 3,61 8A 4b = g 13,61 4E 1a = g 8,61 1B

14 54 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 4 Character Tables of the Inertia Factor Groups We recall that knowledge of the appropriate character tables of inertia factor groups is pivotal in calculating the full character table of any group extension Since in our extension G = 3 7 :Sp(6, ), the normal subgroup 3 7 is abelian and the extension splits, it follows by Mackey s Theorem (see Theorem 514 of Basheer [1] for example), that every character θ k of 3 7 is extendible to a character of its inertia group H k Thus all the factor sets α k are trivial and all the character tables of the inertia factor groups that we will use to construct the character table of G, are the ordinary ones The character table of H 1 = Sp(6, ) is available in the ATLAS We have used GAP to construct the character tables of H = U 4 (), H 3 = 5 :A 6, H 4 = A 7, H 5 = (3 A 6 ): and H 6 = ( 4 :A 5 ): since we know from Section 3 that H = σ 1, σ, H 3 = ξ 1, ξ, H 4 = β 1, β, H 5 = µ 1, µ and H 6 = ϵ 1, ϵ Also the character tables of H, H 3, H 4, H 5 and H 6 are given as Tables 111, 11, 113, 114 and 115 of [1] respectively 5 Fischer Matrices of G = 3 7 :Sp(6, ) The theory of Clifford-Fischer matrices, which is based on Clifford Theory (see [6]), was developed by B Fischer [7, 8, 9] For the general definition of Fischer matrices we refer to [1, ] We recall that we label the top and bottom of the columns of the Fischer matrix F i, corresponding to g i, by the sizes of the centralizers of g ij, 1 j c(g i ) in G and m ij respectively In Table 1 we supplied C G (g ij ) and m ij, 1 i 30, 1 j c(g i ) Also having obtained the fusions of the inertia factor groups H, H 3, H 4, H 5 and H 6 into Sp(6, ), we are able to label the rows of the Fischer matrices as described in [1] We have used the properties of Fischer matrices, given in Proposition 36 of [] to calculate some of the entries of the Fischer matrices and also to build an algebraic system of equations For example since the extension is split, then every coset Ng i (or just Ng i ) is a split coset (see [15]) and it results that a (k,m), for all i {1,,, 30} With the help of the symbolic mathematical i1 = C Sp(6,)(g i ) C Hk (g ikm ) package Maxima [11], we were able to solve these systems of equations and hence we have computed all the Fischer matrices of G, which we list below F 1 g 1 g 11 g 1 g 13 g 14 g 15 g 16 o(g 1j ) C G (g 1j ) (k, m) C Hk (g 1km ) (1, 1) (, 1) (3, ) (4, 1) (5, 1) (6, 1) m 1j

15 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 55 F 3 F g g 1 g o(g j ) 6 C G (g j ) (k, m) C Hk (g km ) (1, 1) (3, 1) m j g 3 g 31 g 3 g 33 g 34 o(g 3j ) C G (g j ) (k, m) C Hk (g km ) (1, 1) (, 1) (3, 1) (6, 1) m j F 4 g 4 g 4,1 g 4, g 4,3 g 4,4 g 4,5 g 4,6 g 4,7 g 4,8 g 4,9 g 4,10 o(g 4j ) C G (g 4j ) (k, m) C Hk (g 4km ) (1, 1) (, 1) (3, 1) (3, ) (4, 1) (5, 1) (5, ) (6, 1) (6, ) (6, 3) m 4j F 5 g 5 g 51 g 5 g 53 g 54 o(g 5j) C G (g 5j) (k, m) C Hk (g 5km ) (1, 1) (3, 1) (5, 1) (6, 1) m 5j F 6 g 6 g 61 g 6 g 63 g 64 g 65 g 66 g 67 o(g 6j ) C G (g 6j ) (k, m) C Hk (g 6km ) (1, 1) (, 1) (3, 1) (4, 1) (5, 1) (5, ) (6, 1) m 6j F 7 g 7 g 71 g 7 g 73 g 74 o(g 7j ) C G (g 7j ) (k, m) C Hk (g 7km ) (1, 1) (, 1) (, ) (5, 1) m 7j

16 56 Int J Group Theory, 5 no (016) A B M Basheer and J Moori F 8 g 8 g 81 g 8 g 83 g 84 g 85 g 86 o(g 8j ) C G (g 8j ) (k, m) C Hk (g 8km ) (1, 1) (, 1) (3, 1) (4, 1) (5, 1) (5, ) m 8j F 9 g 9 g 91 g 9 g 93 g 94 o(g 9j) C G (g 9j) (k, m) C Hk (g 9km ) (1, 1) (, 1) (3, 1) (6, 1) m 9j F 10 g 10 g 10,1 g 10, g 10,3 g 10,4 o(g 10j) C G (g 10j) (k, m) C Hk (g 10km ) (1, 1) (3, 1) (5, 1) (6, 1) m 10j F 11 g 11 g 11,1 g 11, o(g 11j ) 4 1 C G (g 11,j ) (k, m) C Hk (g 11km ) (1, 1) (6, 1) 96 1 m 11j F 1 g 1 g 1,1 g 1, o(g 1j ) 4 1 C G (g 1,j ) (k, m) C Hk (g 1km ) (1, 1) (3, 1) 64 1 m 1j F 13 g 13 g 13,1 g 13, g 13,3 g 13,4 g 13,5 g 13,6 g 13,7 g 13,8 o(g 13j) C G (g 13j) (k, m) C Hk (g 13km ) (1, 1) (, 1) (3, 1) (3, ) (4, 1) (5, 1) (6, 1) (6, ) m 13j F 14 g 14 g 14,1 g 14, g 14,3 g 14,4 g 14,5 g 14,6 g 14,7 o(g 14j ) C G (g 14j ) (k, m) C Hk (g 14km ) (1, 1) (, 1) (3, 1) (3, ) (4, 1) (5, 1) (6, 1) m 14j

17 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 57 F 16 F 15 g 15 g 15,1 g 15, o(g 15j ) 6 6 C G (g 15,j ) (k, m) C Hk (g 15km ) (1, 1) (3, 1) 7 1 m 15j g 16 g 16,1 g 16, g 16,3 g 16,4 g 16,5 o(g 16j ) C G (g 16j ) (k, m) C Hk (g 16km ) (1, 1) (, 1) (, ) (3, 1) (6, 1) m 16j F 17 g 17 g 17,1 g 17, g 17,3 o(g 17j) C G (g 17j) (k, m) C Hk (g 17km ) (1, 1) (, 1) (, ) m 17j F 18 g 18 g 18,1 g 18, g 18,3 g 18,4 g 18,5 g 18,6 g 18,7 g 18,8 o(g 18j ) C G (g 18j ) (k, m) C Hk (g 18km ) (1, 1) (, 1) (3, 1) (4, 1) (5, 1) (5, ) (6, 1) (6, ) m 18j F 19 g 19 g 19,1 g 19, o(g 19j ) 6 6 C G (g 19,j ) (k, m) C Hk (g 19km ) (1, 1) (3, 1) 18 1 m 19j F 0 g 0 g 0,1 g 0, o(g 0j ) 6 18 C G (g 0,j ) (k, m) C Hk (g 0km ) (1, 1) (, 1) 18 1 m 0j F 1 g 1 g 1,1 g 1, o(g 1j) 6 18 C G (g 1,j) (k, m) C Hk (g 1km ) (1, 1) (5, 1) 6 1 m 1j F g g,1 g, g,3 o(g j) C G (g j) (k, m) C Hk (g km ) (1, 1) (4, 1) (4, ) m j

18 58 Int J Group Theory, 5 no (016) A B M Basheer and J Moori F 3 g 3 g 3,1 g 3, o(g 3j ) 8 4 C G (g 3,j ) 48 4 (k, m) C Hk (g 31km ) (1, 1) (6, 1) 8 1 m 3j F 4 g 4 g 4,1 g 4, g 4,3 o(g 4j ) C G (g 4j ) (k, m) C Hk (g 4km ) (1, 1) (3, 1) (3, ) m 4j F 5 g 5 g 5,1 g 5, g 5,3 o(g 5j ) C G (g 5j ) (k, m) C Hk (g 5km ) (1, 1) (, 1) (, ) m 5j F 6 g 6 g 6,1 g 6, g 6,3 o(g 6j ) C G (g 6j ) (k, m) C Hk (g 6km ) (1, 1) (3, 1) (3, ) m 6j F 7 g 7 g 7,1 g 7, o(g 7j ) 1 36 C G (g 7,j ) 7 36 (k, m) C Hk (g 7km ) (1, 1) (5, 1) 1 1 m 7j F 9 g 9 g 9,1 g 9, g 9,3 o(g 9j) C G (g 9j) (k, m) C Hk (g 9km ) (1, 1) (, 1) (, ) m 9j F 8 g 8 g 8,1 g 8, o(g 8j ) 1 1 C G (g 8,j ) 7 36 (k, m) C Hk (g 8km ) (1, 1) (6, 1) 1 1 m 8j F 30 g 30 g 30,1 g 30, g 30,3 o(g 30j) C G (g 30j) (k, m) C Hk (g 30km ) (1, 1) (5, 1) (5, ) m 30j Character Table of G Now we have the conjugacy classes of G = 3 7 :Sp(6, ) (Table 1), the character tables of all the inertia factors (Tables 111, 11, 113, 114 and 115 of [1]), the fusions of classes of the inertia factors into classes of Sp(6, ) (Table 5), the Fischer matrices of G (see Section 5) By Section 3 of [], it follows that the full character table of G can be constructed easily The character table of G is a C valued matrix The full character table of G is available in the PhD thesis [1] of the first author, which could be accessed online This character table is not yet incorporated into the GAP library but our aim is to do so

19 Int J Group Theory, 5 no (016) A B M Basheer and J Moori 59 Acknowledgments The first author would like to thank his supervisor (second author) for his advice and support The financial support from the National Research Foundation (NRF) of South Africa and the North-West University are also acknowledged References [1] A B M Basheer, Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups, PhD Thesis, University of KwaZulu-Natal, Pietermaitzburg, 01 [] A B M Basheer and J Moori, Fischer matrices of Dempwolff group 5 GL(5, ), Int J Group Theory, 1 no 4 (01) [3] A B M Basheer and J Moori, On the non-split extension group 6 Sp(6, ), Bull Iranian Math Soc, 39 (013) [4] A B M Basheer and J Moori, A survey on Clifford-Fischer Theory, London Mathematical Society Lecture Note Series, Groups St Andrews 013, Cambridge University Press, 4 (015), [5] J H Conway, R T Curtis, S P Norton, R A Parker and R A Wilson, Atlas of Finite Groups, Clarendon Press, Oxford University Press, Eynsham, 1985 [6] A H Clifford, Representations induced in an invariant subgroup, Ann of Math (), 38 (1937) [7] B Fischer, Clifford matrizen, manuscript, 198 [8] B Fischer, Unpublished manuscript, 1985 [9] B Fischer, Clifford matrices, Representation theory of finite groups and finite-dimensional Lie algebras (eds G O Michler and C M Ringel; Birkhäuser, Basel, (1991), 1 16 [10] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4410; [11] Maxima, A Computer Algebra System Version 5181; [1] J Moori, On the Groups G + and G of the form 10 :M and 10 :M, PhD Thesis, University of Birmingham, 1975 [13] J Moori, On certain groups associated with the smallest Fischer group, J London Math Soc, (1981) [14] Z E Mpono, Fischer Clifford Theory and Character Tables of Group Extensions, PhD Thesis, University of Natal, Pietermaritzburg, 1998 [15] U Schiffer, Cliffordmatrizen, Diplomarbeit, Lehrstul D Fur Matematik, RWTH, Aachen, 1995 [16] R A Wilson et al, Atlas of finite group representations, Ayoub Basheer Mohammed Basheer School of Mathematical Sciences, North-West University (Mafikeng), P Bag X046, Mmabatho 735, South Africa ayoubbasheer@gmailcom Jamshid Moori School of Mathematical Sciences, North-West University (Mafikeng), P Bag X046, Mmabatho 735, South Africa jamshidmoori@nwuacza