On groups associated with the affine subgroups of Sp2n(2)

The symplectic group S p 2 n (2) has an affine maximal subgroup of structure A S p n = 2 2 n - 1 : S p 2 n - 2 (2) which is a split extension of an elementary abelian 2-group N = 2 2 n - 1 by G = S p 2 n - 2 (2) . The vector space N = 2 2 n - 1 and its dual N ∗ are not equivalent as 2 n - 1 dimensional G-modules over GF(2). Therefore, a split extension of the form G ¯ n = N ∗ : S p 2 n - 2 (2) ≇ N : S p 2 n - 2 (2) exists. In this paper, it will be shown that G ¯ n ≅ Aut (2 2 n - 2 : S p 2 n - 2 (2)) = 2 2 n - 2 : S p 2 n - 2 (2) : 2 for n ≥ 3 . Moreover, the ordinary irreducible characters of G ¯ n are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set Irr (G ¯ 5) of the group G ¯ 5 = 2 9 : S p 8 (2) which is associated with the affine subgroup A S p 5 = 2 9 : S p 8 (2) of S p 10 (2) . [ABSTRACT FROM AUTHOR]