On two groups of the form 28:A9

This paper is dealing with two split extensions of the form 2 8 : A 9 . We refer to these two groups by G ¯ 1 and G ¯ 2 . For G ¯ 1 , the 8-dimensional GF (2)-module is in fact the deleted permutation module for A 9 . We firstly determine the conjugacy classes of G ¯ 1 and G ¯ 2 using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of G ¯ 1 are A 9 , A 8 , S 7 , ( A 6 × 3 ) : 2 and ( A 5 × A 4 ) : 2 , while the inertia factor groups of G ¯ 2 are A 9 , P S L ( 2 , 8 ) : 3 and 2 3 : G L ( 3 , 2 ) . We then determine the Fischer matrices for these two groups and apply the Clifford–Fischer theory to compute the ordinary character tables of G ¯ 1 and G ¯ 2 . The Fischer matrices of G ¯ 1 and G ¯ 2 are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of G ¯ 1 and G ¯ 2 are 84 × 84 and 40 × 40 complex valued matrices respectively.