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The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over F 2. Therefore a split extension group of the form 2 12 : (S z (8) : 3) : = G ¯ exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of G ¯ by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of G ¯ is a 43 × 43 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7. [ABSTRACT FROM AUTHOR]

The simple symplectic group Sp(8, 2) has 11 conugacy classes of maximal subgroups. The fourth maximal subgroup of Sp(8, 2) is a group of the form 2 10 : A 8 : = G ¯. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford–Fischer Theory. We determined the inertia factor groups of G ¯ and there are 7 such groups having the forms: H 1 = A 8 , H 2 = 2 3 : G L (3 , 2) , H 3 = 2 4 : (S 3 × S 3) , H 4 = 2 3 : S 4 , H 5 = S 5 , H 6 = (S 3 × S 3) : 2 and H 7 = 2 × S 4. The character table of G ¯ is a 81 × 81 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 16. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X be a conjugacy class of G. The rank of X in G, denoted by rank(G : X), is defined to be the minimal number of elements of X generating G. In this paper we establish some general results on the ranks of certain conjugacy classes of elements for simple alternating group A n . We apply these general results together with the structure constants method to determine the ranks of all the non-trivial classes of A 8 and A 9 . [ABSTRACT FROM AUTHOR]