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

The symplectic group \(Sp_{2n}(2)\) $S{p}_{2n}(2)$ has an affine maximal subgroup of structure \(ASp_n=2^{2n-1}{:}Sp_{2n-2}(2)\) $AS{p}_n=2^{2n-1}:S{p}_{2n-2}(2)$ which is a split extension of an elementary abelian 2-group \(N=2^{2n-1}\) $N=2^{2n-1}$ by \(G=Sp_{2n-2}(2)\) $G=S{p}_{2n-2}(2)$ . The vector space \(N=2^{2n-1}\) $N=2^{2n-1}$ and its dual \(N^{*}\) $N^{\ast }$ are not equivalent as \(2n-1\) $2n-1$ dimensional G-modules over GF(2). Therefore, a split extension of the form \(\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)\ncong N{:}Sp_{2n-2}(2)\) ${\overline{G}}_n=N^{\ast }:S{p}_{2n-2}(2)\ncong N:S{p}_{2n-2}(2)$ exists. In this paper, it will be shown that \(\overline{G}_n\cong \text {Aut}(2^{2n-2}{:}Sp_{2n-2}(2))= \left( 2^{2n-2}{:}Sp_{2n-2}(2)\right) {:} 2\) ${\overline{G}}_n\cong \text{Aut}(2^{2n-2}:S{p}_{2n-2}(2))=\left(2^{2n-2},:,S,p_{2n-2},(2)\right):2$ for \(n\ge 3\) $n\ge 3$ . Moreover, the ordinary irreducible characters of \(\overline{G}_n\) ${\overline{G}}_n$ are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set Irr \((\overline{G}_5)\) $({\overline{G}}_5)$ of the group \(\overline{G}_5=2^9{:}Sp_{8}(2)\) ${\overline{G}}_5=2^9:S{p}_8(2)$ which is associated with the affine subgroup \(ASp_5=2^9{:}Sp_{8}(2)\) $AS{p}_5=2^9:S{p}_8(2)$ of \(Sp_{10}(2)\) $S{p}_{10}(2)$ .