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A group G is said to be (2 , 3) -generated if it can be generated by an involution x and an element y of order three. For G a sporadic simple group, it was proved by the third author Woldar (1989) [26] that G is (2 , 3) -generated if and only if G ∉ { M 11 , M 22 , M 23 , M c L }. In this paper, we investigate all possible (2 , 3) -generations of Fischer's largest sporadic simple group F i 24 ′ under the assumption that the product xy has prime order. [ABSTRACT FROM AUTHOR]

A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer's largest sporadic simple group Fi′24 by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi′24| with 3 < q < r. [ABSTRACT FROM AUTHOR]

A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer's largest sporadic simple group Fi′24 by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi′24| with 3 < q < r. [ABSTRACT FROM AUTHOR]

A group G is said to be (2,3,t)-generated if it can be generated by two elements x and y such that o(x) = 2, o(y) = 3 and o(xy) = t. In the present article, we investigate (2,3,t)-generations for the Held's sporadic simple group He, where t is a divisor of ∣He∣. Most of the computations were carried out with the aid of computer algebra system GAP [23]. [ABSTRACT FROM AUTHOR]

The Fischer group $Fi_{24}={\rm Aut} (Fi_{24}^{\prime})$ is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA. [ABSTRACT FROM AUTHOR]

The Fischer group $Fi_{24}={\rm Aut} (Fi_{24}^{\prime})$ is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA. [ABSTRACT FROM AUTHOR]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji, where i = 1; 2; 3; 4. [ABSTRACT FROM AUTHOR]

The Harada-Norton group HN is a sporadic simple group of order 273030912000000 equal to 214 x 36 x 56 x 7 x 11 x 19. It has 14 conjugacy classes of maximal subgroups and in the present paper we construct the Fischer-Clifford matrices and hence character table for one of these maximal subgroups, that is the non-split extension 26·U4(2), of index 164587500. There is not that many known examples of the application of Fischer-Clifford theory to non-split extensions, and hence this maximal subgroup of Hard-Norton's group serves as a good candidate. Most of the computations were carried out using the computer algebra systems GAℙ [10] and MAGMA [4]. [ABSTRACT FROM AUTHOR]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji, where i = 1; 2; 3; 4. [ABSTRACT FROM AUTHOR]