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

Abstract

The symplectic group \(Sp_{2n}(2)\) has an affine maximal subgroup of structure \(ASp_n=2^{2n-1}{:}Sp_{2n-2}(2)\) which is a split extension of an elementary abelian 2-group \(N=2^{2n-1}\) by \(G=Sp_{2n-2}(2)\). The vector space \(N=2^{2n-1}\) and its dual \(N^{*}\) are not equivalent as \(2n-1\) dimensional G-modules over GF(2). Therefore, a split extension of the form \(\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)\ncong N{:}Sp_{2n-2}(2)\) exists. In this paper, it will be shown that \(\overline{G}_n\cong \text {Aut}(2^{2n-2}{:}Sp_{2n-2}(2))= \left( 2^{2n-2}{:}Sp_{2n-2}(2)\right) {:} 2\) for \(n\ge 3\). Moreover, the ordinary irreducible characters of \(\overline{G}_n\) are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set Irr\((\overline{G}_5)\) of the group \(\overline{G}_5=2^9{:}Sp_{8}(2)\) which is associated with the affine subgroup \(ASp_5=2^9{:}Sp_{8}(2)\) of \(Sp_{10}(2)\).