On the character table of non-split extension 26.S 8

Problem Statement & Objective : Character tables of maximal subgroups of finite simple groups provide considerable amount of information about the groups. In the present article, our objective is to compute the character table of one maximal subgroup of the orthogonal group P S O 8 + ( 3 ) . Approach : The projective special orthogonal group P S O 8 + ( 3 ) ≅ O 8 + ( 3 ) .2 1 is obtained from the special orthogonal group S O 8 + ( 3 ) on factoring by the group of scalar matrices it contains. The group O 8 + ( 3 ) .2 1 has a maximal subgroup of the form 2 6 . S 8 with index 3838185. The group Q ≅ 2 6 · S 8 is a non-split group extension of an elementary abelian 2-group of order 64 by the symmetric group S 8 . We apply the Fischer-Clifford theory to compute the irreducible characters of the extension 2 6 · S 8 . Results and Conclusion : We produce 64 conjugacy classes of elements as well as 64 irreducible character of the non-split group extension 2 6 · S 8 corresponding to the three inertia factors H 1 = S 8 , H 2 = S 6 × 2 and H 3 = ( S 4 × S 4 ):2.