On a group of the form 214:Sp(6,2)

Abstract

The symplectic group Sp(6,2) has a 14-dimensional absolutely irreducible module over F₂: Hence a split extension group of the form G = 2¹⁴:Sp(6,2) does exist. In this paper we first determine the conjugacy classes of G using the coset analysis technique. The structures of inertia factor groups were determined. The inertia factor groups are Sp(6,2); (2¹⁺⁴ x 2²):(S₃ x S₃); S₃ x S₆; PSL(2,8); (((2² x Q₈):3):2):2, S₃ x A₅; and 2 x S₄ x S₃: We then determine the Fischer matrices and apply the Clifford-Fischer theory to compute the ordinary character table of G: The Fischer matrices of G are all integer valued, with size ranging from 4 to 16. The full character table of G is a 186 x 186 complex valued matrix.

Keywords: Group extensions, Clifford theory, inertia groups, Fischer matrices, character table