On groups associated with the affine subgroups of $$Sp_{2n}(2)$$

Abstract

AbstractThe symplectic group $$Sp_{2n}(2)$$ S p 2 n ( 2 ) has an affine maximal subgroup of structure $$ASp_n=2^{2n-1}{:}Sp_{2n-2}(2)$$ 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 $$N=2^{2n-1}$$ N = 2 2 n - 1 by $$G=Sp_{2n-2}(2)$$ G = S p 2 n - 2 ( 2 ) . The vector space $$N=2^{2n-1}$$ N = 2 2 n - 1 and its dual $$N^{*}$$ N ∗ are not equivalent as $$2n-1$$ 2 n - 1 dimensional G-modules over GF(2). Therefore, a split extension of the form $$\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)\ncong N{:}Sp_{2n-2}(2)$$ 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 $$\overline{G}_n\cong \text {Aut}(2^{2n-2}{:}Sp_{2n-2}(2))= \left( 2^{2n-2}{:}Sp_{2n-2}(2)\right) {:} 2$$ 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 $$n\ge 3$$ n ≥ 3 . Moreover, the ordinary irreducible characters of $$\overline{G}_n$$ 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$$(\overline{G}_5)$$ ( G ¯ 5 ) of the group $$\overline{G}_5=2^9{:}Sp_{8}(2)$$ G ¯ 5 = 2 9 : S p 8 ( 2 ) which is associated with the affine subgroup $$ASp_5=2^9{:}Sp_{8}(2)$$ A S p 5 = 2 9 : S p 8 ( 2 ) of $$Sp_{10}(2)$$ S p 10 ( 2 ) .