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

International Journal of Group Theory ISSN (print): 51-7650, ISSN (on-line): 51-7669 Vol 5 No (016), pp 41-59 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 118 118 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 145150 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

4 Int J Group Theory, 5 no (016) 41-59 A B M Basheer and J Moori 1 1 ξ 1 1 ξ ξ 0 0 ξ 0 1 1 1 ξ 0 ξ 1 ξ ξ ξ 1 ξ 0 1 1 0 0 ξ 0 1 0 g 1 = 1 0 0 ξ 1 ξ 0 0 0 0 ξ 0 ξ 0 1 0 0 1 1 1 ξ ξ ξ 0 0 ξ 1 0 0 0 0 1 with o(g 1 ) = 45, o(g ) = 45 and o(g 1 g ) = 6, g = ξ 1 1 1 ξ 1 0 0 ξ 0 1 1 0 0 ξ 0 ξ 0 0 1 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 1 0 0 ξ 1 ξ ξ ξ 0 1 0 1 0 0 0 0 0 ξ 1, Corollary 11 G SL(8, 3) with index 30 954 487 010 019 919 360 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 =, 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 ξ 1 1 ξ 0 1 0 ξ 1 ξ 1 0 1 1 0 ξ ξ 0 ξ 1 0 1 n 4 =, n 5 =, n 6 =, 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 n 7 = ξ 0 0 0 0 0 0 1 ξ 0 ξ 0 0 1 ξ 1 ξ 1 0 ξ 1 1 1 1 0 0 0 0 0 0 1 0 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 0 0 0 0 0 0 g 1 =, g =, 0 0 ξ ξ ξ 0 0 0 0 0 0 0 0 0 1 0 0 0 ξ ξ 0 ξ 0 0 0 1 0 ξ ξ ξ 0 0 0 0 ξ ξ 0 0 ξ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ξ 0 0 ξ 1 ξ 1 0 1 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, ]

Int J Group Theory, 5 no (016) 41-59 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 =, 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 1 0 ξ ξ ξ 1 0 1 g 14 =, g 15 =, g 16 = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 ξ 1 0 0 1 1 ξ 1 0 1 ξ 1 1 1 0 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

44 Int J Group Theory, 5 no (016) 41-59 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 11 1 1 317447440 f 1 = 56 m 1 = 56 g 1 3 56 56687040 g 1 = 1A k 1 = 187 f 13 = 16 m 13 = 16 g 13 3 16 519440 f 14 = 576 m 14 = 576 g 14 3 576 551140 f 15 = 67 m 15 = 67 g 15 3 67 47390 f 16 = 756 m 16 = 756 g 16 3 756 4199040 g = A k = 3 f 1 = 1 m 1 = 79 g 1 4597 6910 f = m = 1458 g 6 91854 34560 f 31 = 1 m 31 = 81 g 31 5515 14416 g 3 = B k 3 = 7 f 3 = 6 m 3 = 486 g 3 6 153090 0736 f 33 = 8 m 33 = 648 g 33 6 0410 1555 f 34 = 1 m 34 = 97 g 34 6 306180 10368 f 41 = 1 m 41 = 9 g 41 8505 37348 f 4 = m 4 = 18 g 4 6 17010 18664 f 43 = 8 m 43 = 7 g 43 6 68040 46656 f 44 = 16 m 44 = 144 g 44 6 136080 338 g 4 = C k 4 = 43 f 45 = 16 m 45 = 144 g 45 6 136080 338 f 46 = 4 m 46 = 16 g 46 6 0410 1555 f 47 = 3 m 47 = 88 g 47 6 7160 11664 f 48 = 3 m 48 = 88 g 48 6 7160 11664 f 49 = 48 m 49 = 43 g 49 6 40840 7776 f 4,10 = 64 m 4,10 = 576 g 4,10 6 54430 583 f 51 = 1 m 51 = 81 g 51 306180 10368 g 5 = D k 5 = 7 f 5 = 6 m 5 = 486 g 5 6 1837080 178 f 53 = 8 m 53 = 648 g 53 6 449440 196 f 54 = 1 m 54 = 97 g 54 6 3674160 864 f 61 = 1 m 61 = 9 g 61 3 6048 54880 f 6 = 0 m 6 = 180 g 6 3 10960 644 f 63 = 30 m 63 = 70 g 63 3 181440 17496 g 6 = 3A k 6 = 43 f 64 = 30 m 64 = 70 g 64 3 181440 17496 f 65 = 1 m 65 = 108 g 65 9 7576 43740 f 66 = 30 m 66 = 70 g 66 9 181440 17496 f 67 = 10 m 67 = 1080 g 67 9 75760 4374 f 71 = 1 m 71 = 81 g 71 3 181440 17496 g 7 = 3B k 7 = 7 f 7 = 8 m 7 = 648 g 7 3 145150 187 f 73 = 9 m 73 = 79 g 73 9 163960 1944 f 74 = 9 m 74 = 79 g 74 9 163960 1944 f 81 = 1 m 81 = 81 g 81 3 1088640 916 f 8 = m 8 = 16 g 8 3 17780 1458 f 83 = m 83 = 16 g 83 9 17780 1458 g 8 = 3C k 8 = 7 f 84 = 4 m 84 = 34 g 84 9 4354560 79 f 85 = 6 m 85 = 486 g 85 9 6531840 486 f 86 = 1 m 86 = 97 g 86 9 13063680 43 f 91 = 1 m 91 = 81 g 91 4 306180 10368 g 9 = 4A k 9 = 7 f 9 = 6 m 9 = 486 g 9 1 1837080 178 f 93 = 8 m 93 = 648 g 93 1 449440 196 f 94 = 1 m 94 = 97 g 94 1 3674160 864 f 10,1 = 1 m 10,1 = 81 g 10,1 4 61360 5184 g 10 = 4B k 10 = 7 f 10, = 6 m 10, = 486 g 10, 1 3674160 864 continued on next page

Int J Group Theory, 5 no (016) 41-59 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 10,3 1 4898880 648 f 10,4 = 1 m 10,4 = 97 g 10,4 1 73480 43 g 11 = 4C k 11 = 3 f 11,1 = 1 m 11,1 = 79 g 11,1 4 551140 576 f 11, = m 11, = 1458 g 11, 1 110480 88 g 1 = 4D k 1 = 3 f 1,1 = 1 m 1,1 = 79 g 1,1 4 866860 384 f 1, = m 1, = 1458 g 1, 1 1653370 19 f 13,1 = 1 m 13,1 = 81 g 13,1 4 3674160 864 f 13, = m 13, = 16 g 13, 1 734830 43 f 13,3 = m 13,3 = 16 g 13,3 1 734830 43 g 13 = 4E k 13 = 7 f 13,4 = m 13,4 = 16 g 13,4 1 734830 43 f 13,5 = 4 m 13,5 = 34 g 13,5 1 14696640 16 f 13,6 = 4 m 13,6 = 34 g 13,6 1 14696640 16 f 13,7 = 4 m 13,7 = 34 g 13,7 1 14696640 16 f 13,8 = 8 m 13,8 = 648 g 13,8 1 939380 108 f 14,1 = 1 m 14,1 = 81 g 14,1 5 3919104 810 f 14, = 3 m 14, = 43 g 14, 15 1175731 70 f 14,3 = 3 m 14,3 = 43 g 14,3 15 1175731 70 g 14 = 5A k 14 = 7 f 14,4 = m 14,4 = 16 g 14,4 15 783808 405 f 14,5 = 6 m 14,5 = 486 g 14,5 15 351464 135 f 14,6 = 6 m 14,6 = 486 g 14,6 15 351464 135 f 14,7 = 6 m 14,7 = 486 g 14,7 15 351464 135 g 15 = 6A k 15 = 3 f 15,1 = 1 m 15,1 = 79 g 15,1 6 734830 43 f 15, = m 15, = 1458 g 15, 6 14696640 16 f 16,1 = 1 m 16,1 = 81 g 16,1 6 816480 3888 f 16, = 4 m 16, = 34 g 16, 6 36590 97 g 16 = 6B k 16 = 7 f 16,3 = 4 m 16,3 = 34 g 16,3 6 36590 97 f 16,4 = 6 m 16,4 = 486 g 16,4 6 4898880 648 f 16,5 = 1 m 16,5 = 97 g 16,5 6 9797760 34 f 17,1 = 1 m 17,1 = 79 g 17,1 6 14696640 16 g 17 = 6C k 17 = 3 f 17, = 1 m 17, = 79 g 17, 18 14696640 16 f 17,3 = 1 m 17,3 = 79 g 17,3 18 14696640 16 f 18,1 = 1 m 18,1 = 81 g 18,1 6 449440 196 f 18, = m 18, = 16 g 18, 6 4898880 648 f 18,3 = m 18,3 = 16 g 18,3 6 4898880 648 g 18 = 6D k 18 = 7 f 18,4 = 4 m 18,4 = 34 g 18,4 6 9797760 34 f 18,5 = m 18,5 = 16 g 18,5 18 4898880 648 f 18,6 = 4 m 18,6 = 34 g 18,6 18 9797760 34 f 18,7 = 4 m 18,7 = 34 g 18,7 18 9797760 34 f 18,8 = 8 m 18,8 = 648 g 18,8 18 1959550 16 g 19 = 6E k 19 = 3 f 19,1 = 1 m 19,1 = 79 g 19,1 6 39191040 108 f 19, = m 19, = 1458 g 19, 6 7838080 54 g 0 = 6F k 0 = 3 f 0,1 = 1 m 0,1 = 79 g 0,1 6 39191040 108 f 0, = m 0, = 1458 g 0, 18 7838080 54 g 1 = 6G k 1 = 3 f 1,1 = 1 m 1,1 = 79 g 1,1 6 88179840 36 f 1, = m 1, = 1458 g 1, 18 176359680 18 f,1 = 1 m,1 = 79 g,1 7 151165440 1 g = 7A k = 3 f, = 1 m, = 79 g, 1 151165440 1 f,3 = 1 m,3 = 79 g,3 1 151165440 1 continued on next page

46 Int J Group Theory, 5 no (016) 41-59 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 3,1 8 66134880 48 f 3, = m 3, = 1458 g 3, 4 1369760 4 f 4,1 = 1 m 4,1 = 79 g 4,1 8 66134880 48 g 4 = 8B k 4 = 3 f 4, = 1 m 4, = 79 g 4, 4 66134880 48 f 4,3 = 1 m 4,3 = 79 g 4,3 4 66134880 48 f 5,1 = 1 m 5,1 = 79 g 5,1 9 11757310 7 g 5 = 9A k 5 = 3 f 5, = 1 m 5, = 79 g 5, 9 11757310 7 f 5,3 = 1 m 5,3 = 79 g 5,3 9 11757310 7 f 6,1 = 1 m 6,1 = 79 g 6,1 10 105815808 30 g 6 = 10A k 6 = 3 f 6, = 1 m 6, = 79 g 6, 30 105815808 30 f 6,3 = 1 m 6,3 = 79 g 6,3 30 105815808 30 g 7 = 1A k 7 = 3 f 7,1 = 1 m 7,1 = 79 g 7,1 1 4408990 7 f 7, = m 7, = 1458 g 7, 36 88179840 36 g 8 = 1B k 8 = 3 f 8,1 = 1 m 8,1 = 79 g 8,1 1 4408990 7 f 8, = m 8, = 1458 g 8, 1 88179840 36 f 9,1 = 1 m 9,1 = 79 g 9,1 1 88179840 36 g 9 = 1C k 9 = 3 f 9, = 1 m 9, = 79 g 5, 36 88179840 36 f 9,3 = 1 m 9,3 = 79 g 9,3 36 88179840 36 f 30,1 = 1 m 30,1 = 79 g 30,1 15 7054387 45 g 30 = 15A k 30 = 3 f 30, = 1 m 30, = 79 g 30, 45 7054387 45 f 30,3 = 1 m 30,3 = 79 g 30,3 45 7054387 45 3 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 (),

Int J Group Theory, 5 no (016) 41-59 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 ) = 118 31 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 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 α 1 = 1 1 1 0 0 0 1 1 0 1 0 0, α = 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 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 0 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 σ 1 = 0 1 1 0 1 0 1 1 0 1 0 1, σ = 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 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, )

48 Int J Group Theory, 5 no (016) 41-59 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(6, ) M[11] = U 4 () 590 0 M[1] = ( 4 :A 5): 190 7 18 M[1] = U 4(): M[13] = S 6 1440 36 M[14] = ((((3 (3 :)):):3):): 196 40 M[15] = ((3 1+ :Q 8 ):3): 196 40 18 M[16] = ((((( D 8 ):):3):3):): 115 45 5 M[1] = A 8 0160 14 M[] = S 7 5040 8 15 M[3] = S 6 1440 8 M[] = S 8 M[4] = (S 4 S 4 ): 115 35 0 M[5] = S 5 S 3 70 56 1 M[6] = (((( D 8):):3):): 384 105 0 M[7] = P SL(3, ): 336 10 9 M[31] = 5 :A 6 1150 3 M[3] = ( 5 :A 5): 3840 6 3 M[33] = (( 4 :A 5):) 3840 6 36 M[3] = 5 :S 6 M[34] = ((S 4 S 4 ):) 304 10 40 M[35] = ( (((( D 8 ):):3):)): 1536 15 53 M[36] = (((( ( 4 :)):):3):): 1536 15 40 M[37] = S 6 1440 16 M[38] = S 6 1440 16 M[41] = P SU(3, 3) 6048 14 M[4] = (3 1+ :8): 43 8 14 M[4] = U 3(3): M[43] = P SL(3, ): 336 36 9 M[44] = ((4 :3):): 19 63 14 M[45] = (SL(, 3):4): 19 63 17 M[51] = ((( (( D 8):)):3):): 1536 7 40 M[5] = (((( ( 4 :)):):3):): 1536 7 40 M[5] = 6 :L 3 () M[53] = 3 :P SL(3, ) 1344 8 11 M[34] = 3 P SL(3, ) 1344 8 11 M[55] = ( 6 :7):3 1344 8 16 M[61] = (( ((( D 8 ):):3)):3): 304 36 M[6] = (( ((( D 8 ):):3)):3): 304 33 M[63] = (( ((( D 8 ):):3)):3): 304 39 M[6] = ( 1+4 ):(S 3 S 3) M[64] = ( (((( D 8):):3):)): 1536 3 40 M[65] = ((( (( D 8):)):3):): 1536 3 53 M[66] = ((((( D 8):):3):3):): 115 4 5 M[67] = S 4 S 3 88 16 30 M[71] = A 6 S 3 160 1 M[7] = (3 A 6 ): 160 18 M[73] = 3 S 6 160 33 M[7] = S 3 S 6 M[74] = S 6 1440 3 M[75] = S 3 S 5 70 6 1 M[76] = S 5 S 3 70 6 1 M[77] = ((S 3 S 3):) S 3 43 10 7 M[78] = S 4 S 3 88 15 30 M[79] = S 3 S 4 88 15 30 M[81] = P SL(, 8) 504 3 9 M[8] = ( 3 :7):3 168 9 8 M[8] = P SL(, 8):3 M[83] = (9:3): 54 8 10 M[84] = (7:3): 4 36 7 By 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

Int J Group Theory, 5 no (016) 41-59 A B M Basheer and J Moori 49 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 ξ 1 = 1 0 0 1 0 1, ξ = 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 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 7 50 8 9 M[1] = 3 :P SL(3, ) 1344 15 11 M[1] = A 8 M[13] = 3 :P SL(3, ) 1344 15 11 M[14] = S 6 70 8 11 M[15] = ((A 4 A 4 ):): 576 35 16 M[16] = GL(, 4): 360 56 1 M[1] = A 7 50 9 M[] = S 6 70 7 11 M[] = S 7 M[3] = S 5 40 1 14 M[4] = S 4 S 3 144 35 15 M[5] = (7:3): 4 10 7 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 1 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 0 1 β 1 = 0 1 1 1 0 1 1 0 0 1 0 1, β = 0 0 0 0 1 1 1 0 0 0 0 1, 1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 γ 1 = 1 1 0 1 0 1 1 1 1 0 0 0, γ = 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 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

50 Int J Group Theory, 5 no (016) 41-59 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 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 π 1 = 1 0 0 0 1 0, π = 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 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 5 190 6 16 M[31] = ( 4 :A 5 ) 190 6 4 M[31] = 5 :A 6 M[313] = ((A 4 A 4):4) 115 10 6 M[314] = ( (((( D 8):):3):)): 768 15 31 M[315] = ((( ( 4 :)):):3): 768 15 3 M[316] = A 6 70 16 14 M[317] = A 6 70 16 14 M[31] = 5 :A 5 190 16 M[3] = ((( ( 4 :)):):3): 768 5 3 M[3] = ( 5 :A 5 ): M[33] = (( 4 :5):4) 640 6 M[34] = ((( 4 :3):):) 384 10 8 M[35] = S 5 40 16 14 M[36] = S 5 40 16 14 M[331] = ( 4 :A 5): 190 18 M[33] = ( 4 :A 5) 190 4 M[33] = (( 4 :A 5):) M[333] = ( 4 :A 5): 190 18 M[334] = ((((( D 8 ):):3):):) 768 5 40 M[335] = (( 4 :5):4) 640 6 M[336] = S 4 D 8 384 10 50 M[337] = S 5 40 16 14

Int J Group Theory, 5 no (016) 41-59 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 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 1 1 µ 1 = 0 1 0 1 0 1 1 1 0 0 1 1, µ = 1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 1 1 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 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 θ 1 = 1 0 0 0 1 1, θ = 1 0 0 1 1 1, 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0

5 Int J Group Theory, 5 no (016) 41-59 A B M Basheer and J Moori 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 ϵ 1 = 1 0 0 1 1 1, ϵ = 1 0 0 0 1 1, 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 1 1 1 0 1 δ 1 = 1 1 1 0 0 0 0 0 0 0 0 1, δ = 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 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

Int J Group Theory, 5 no (016) 41-59 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

54 Int J Group Theory, 5 no (016) 41-59 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 ) 1 3 3 3 3 3 C G (g 1j ) 317447440 56687040 519440 551140 47390 4199040 (k, m) C Hk (g 1km ) (1, 1) 145150 1 1 1 1 1 1 (, 1) 590 56 5 0 7 (3, ) 1150 16 45 7 0 9 0 (4, 1) 50 576 7 0 7 18 0 (5, 1) 160 67 4 48 1 3 4 (6, 1) 190 756 7 0 0 7 7 m 1j 1 56 16 576 67 756

Int J Group Theory, 5 no (016) 41-59 A B M Basheer and J Moori 55 F 3 F g g 1 g o(g j ) 6 C G (g j ) 6910 34560 (k, m) C Hk (g km ) (1, 1) 3040 1 1 (3, 1) 1150 1 m j 79 1458 g 3 g 31 g 3 g 33 g 34 o(g 3j ) 6 6 6 C G (g j ) 14416 0736 1555 10368 (k, m) C Hk (g km ) (1, 1) 4608 1 1 1 1 (, 1) 576 8 4 1 (3, 1) 768 6 3 3 0 (6, 1) 384 1 0 3 3 m j 81 486 648 97 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 ) 6 6 6 6 6 6 6 6 6 C G (g 4j ) 37348 18664 46656 338 338 1555 11664 11664 7776 583 (k, m) C Hk (g 4km ) (1, 1) 1536 1 1 1 1 1 1 1 1 1 1 (, 1) 96 16 8 10 8 5 4 4 1 (3, 1) 768 1 1 1 1 1 (3, ) 64 4 4 6 6 6 3 6 3 3 3 (4, 1) 4 64 3 8 8 4 8 4 10 4 5 (5, 1) 48 3 3 4 4 4 4 4 5 4 5 (5, ) 48 3 16 16 8 8 1 4 4 (6, 1) 19 8 8 5 4 5 4 1 1 (6, ) 3 48 4 1 15 1 3 15 6 3 6 (6, 3) 96 16 16 17 19 17 5 19 7 5 7 m 4j 9 18 7 144 144 16 88 88 43 576 F 5 g 5 g 51 g 5 g 53 g 54 o(g 5j) 6 6 6 C G (g 5j) 10368 178 196 864 (k, m) C Hk (g 5km ) (1, 1) 384 1 1 1 1 (3, 1) 64 6 3 3 0 (5, 1) 48 8 4 1 (6, 1) 3 1 0 3 3 m 5j 81 486 648 97 F 6 g 6 g 61 g 6 g 63 g 64 g 65 g 66 g 67 o(g 6j ) 3 3 3 3 9 9 9 C G (g 6j ) 54880 644 17496 17496 43740 17496 4374 (k, m) C Hk (g 6km ) (1, 1) 160 1 1 1 1 1 1 1 (, 1) 108 0 7 10 8 1 (3, 1) 7 30 3 6 3 15 6 3 (4, 1) 36 60 6 1 6 15 6 3 (5, 1) 1080 1 1 1 (5, ) 54 40 14 4 4 10 8 1 (6, 1) 4 90 9 9 18 0 0 0 m 6j 9 180 70 70 108 70 1080 F 7 g 7 g 71 g 7 g 73 g 74 o(g 7j ) 3 3 9 9 C G (g 7j ) 17496 187 1944 1944 (k, m) C Hk (g 7km ) (1, 1) 648 1 1 1 1 (, 1) 648 1 1 1 + 3 1 3 (, ) 648 1 1 1 3 1 + 3 (5, 1) 7 4 3 0 0 m 7j 81 648 79 79

56 Int J Group Theory, 5 no (016) 41-59 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 ) 3 3 9 9 9 9 C G (g 8j ) 916 1458 1458 79 486 43 (k, m) C Hk (g 8km ) (1, 1) 108 1 1 1 1 1 1 (, 1) 54 1 1 1 (3, 1) 18 6 3 6 3 0 0 (4, 1) 9 1 6 6 3 0 0 (5, 1) 7 4 4 1 (5, ) 54 1 1 m 8j 81 16 16 34 486 97 F 9 g 9 g 91 g 9 g 93 g 94 o(g 9j) 4 1 1 1 C G (g 9j) 10368 178 196 864 (k, m) C Hk (g 9km ) (1, 1) 384 1 1 1 1 (, 1) 48 8 4 1 (3, 1) 64 6 3 3 0 (6, 1) 3 1 0 3 3 m 9j 81 486 648 97 F 10 g 10 g 10,1 g 10, g 10,3 g 10,4 o(g 10j) 4 1 1 1 C G (g 10j) 5184 864 648 43 (k, m) C Hk (g 10km ) (1, 1) 19 1 1 1 1 (3, 1) 16 1 0 3 3 (5, 1) 4 8 4 1 (6, 1) 3 6 3 3 0 m 10j 81 486 648 97 F 11 g 11 g 11,1 g 11, o(g 11j ) 4 1 C G (g 11,j ) 576 88 (k, m) C Hk (g 11km ) (1, 1) 19 1 1 (6, 1) 96 1 m 11j 79 1458 F 1 g 1 g 1,1 g 1, o(g 1j ) 4 1 C G (g 1,j ) 384 19 (k, m) C Hk (g 1km ) (1, 1) 18 1 1 (3, 1) 64 1 m 1j 79 1458 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) 4 1 1 1 1 1 1 1 C G (g 13j) 864 43 43 43 16 16 16 108 (k, m) C Hk (g 13km ) (1, 1) 3 1 1 1 1 1 1 1 1 (, 1) 8 4 4 1 1 (3, 1) 16 1 1 1 1 (3, ) 16 1 1 1 1 (4, 1) 4 8 4 4 4 1 (5, 1) 8 4 4 1 1 (6, 1) 16 1 1 1 1 (6, ) 8 4 4 1 1 m 13j 81 16 16 16 34 34 34 648 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 ) 5 15 15 15 15 15 15 C G (g 14j ) 810 70 70 405 135 135 135 (k, m) C Hk (g 14km ) (1, 1) 30 1 1 1 1 1 1 1 (, 1) 5 6 0 0 3 0 3 3 (3, 1) 10 3 3 3 3 3 + 3 3 3 0 0 0 (3, ) 10 3 3 + 3 3 3 3 3 3 0 0 0 (4, 1) 5 6 0 0 3 3 3 0 (5, 1) 15 1 1 1 (6, 1) 5 6 0 0 3 3 0 3 m 14j 81 43 43 16 486 486 486

Int J Group Theory, 5 no (016) 41-59 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 ) 43 16 (k, m) C Hk (g 15km ) (1, 1) 144 1 1 (3, 1) 7 1 m 15j 79 1458 g 16 g 16,1 g 16, g 16,3 g 16,4 g 16,5 o(g 16j ) 6 6 6 6 6 C G (g 16j ) 3888 97 97 648 34 (k, m) C Hk (g 16km ) (1, 1) 144 1 1 1 1 1 (, 1) 36 4 1 3 3 1 + 3 3 1 (, ) 36 4 1 + 3 3 1 3 3 1 (3, 1) 4 6 3 3 3 0 (6, 1) 1 1 3 3 0 3 m 16j 81 34 34 486 97 F 17 g 17 g 17,1 g 17, g 17,3 o(g 17j) 6 18 18 C G (g 17j) 16 16 16 (k, m) C Hk (g 17km ) (1, 1) 7 1 1 1 (, 1) 7 1 1 + 3 1 3 (, ) 7 1 1 3 1 + 3 m 17j 79 79 79 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 ) 6 6 6 6 18 18 18 18 C G (g 18j ) 196 648 648 34 648 34 34 16 (k, m) C Hk (g 18km ) (1, 1) 48 1 1 1 1 1 1 1 1 (, 1) 1 4 1 4 1 (3, 1) 4 1 1 1 1 (4, 1) 1 4 4 1 1 (5, 1) 4 1 1 1 1 (5, ) 6 8 4 4 4 1 (6, 1) 4 1 1 1 1 (6, ) 1 4 4 1 1 m 18j 81 16 16 34 16 34 34 648 F 19 g 19 g 19,1 g 19, o(g 19j ) 6 6 C G (g 19,j ) 108 54 (k, m) C Hk (g 19km ) (1, 1) 36 1 1 (3, 1) 18 1 m 19j 79 1458 F 0 g 0 g 0,1 g 0, o(g 0j ) 6 18 C G (g 0,j ) 108 54 (k, m) C Hk (g 0km ) (1, 1) 36 1 1 (, 1) 18 1 m 0j 79 1458 F 1 g 1 g 1,1 g 1, o(g 1j) 6 18 C G (g 1,j) 36 18 (k, m) C Hk (g 1km ) (1, 1) 1 1 1 (5, 1) 6 1 m 1j 79 1458 F g g,1 g, g,3 o(g j) 7 1 1 C G (g j) 1 1 1 (k, m) C Hk (g km ) (1, 1) 7 1 1 1 (4, 1) 7 1 1 + 3 1 3 (4, ) 7 1 1 3 1 + 3 m j 79 79 79

58 Int J Group Theory, 5 no (016) 41-59 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) 16 1 1 (6, 1) 8 1 m 3j 79 1458 F 4 g 4 g 4,1 g 4, g 4,3 o(g 4j ) 8 4 4 C G (g 4j ) 48 48 48 (k, m) C Hk (g 4km ) (1, 1) 16 1 1 1 (3, 1) 16 1 1 + 3 1 3 (3, ) 16 1 1 3 1 + 3 m 4j 79 79 79 F 5 g 5 g 5,1 g 5, g 5,3 o(g 5j ) 9 9 9 C G (g 5j ) 7 7 7 (k, m) C Hk (g 5km ) (1, 1) 9 1 1 1 (, 1) 9 1 1 + 3 1 3 (, ) 9 1 1 3 1 + 3 m 5j 79 79 79 F 6 g 6 g 6,1 g 6, g 6,3 o(g 6j ) 10 30 30 C G (g 6j ) 30 30 30 (k, m) C Hk (g 6km ) (1, 1) 10 1 1 1 (3, 1) 10 1 1 + 3 1 3 (3, ) 10 1 1 3 1 + 3 m 6j 79 79 79 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) 4 1 1 (5, 1) 1 1 m 7j 79 1458 F 9 g 9 g 9,1 g 9, g 9,3 o(g 9j) 1 36 36 C G (g 9j) 36 36 36 (k, m) C Hk (g 9km ) (1, 1) 1 1 1 1 (, 1) 1 1 1 + 3 1 3 (, ) 1 1 1 3 1 + 3 m 9j 79 79 79 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) 4 1 1 (6, 1) 1 1 m 8j 79 1458 F 30 g 30 g 30,1 g 30, g 30,3 o(g 30j) 15 45 45 C G (g 30j) 45 45 45 (k, m) C Hk (g 30km ) (1, 1) 15 1 1 1 (5, 1) 15 1 1 + 3 1 3 (5, ) 15 1 1 3 1 + 3 m 30j 79 79 79 6 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 118 118 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

Int J Group Theory, 5 no (016) 41-59 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) 43 63 [3] A B M Basheer and J Moori, On the non-split extension group 6 Sp(6, ), Bull Iranian Math Soc, 39 (013) 1189 11 [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), 160 17 [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) 533 550 [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; 007 http://wwwgap-systemorg [11] Maxima, A Computer Algebra System Version 5181; 009 http://maximasourceforgenet [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) 61 67 [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, http://brauermathsqmulacuk/atlas/v3/ Ayoub Basheer Mohammed Basheer School of Mathematical Sciences, North-West University (Mafikeng), P Bag X046, Mmabatho 735, South Africa Email: ayoubbasheer@gmailcom Jamshid Moori School of Mathematical Sciences, North-West University (Mafikeng), P Bag X046, Mmabatho 735, South Africa Email: jamshidmoori@nwuacza