 Symplectic quotient of the binary subset module

For even n, the even-weight subspace of $F_2^n$ has radical $⟨1⟩$ for the dot product. Its quotient is a nondegenerate alternating bilinear form space of dimension n−2. The symmetric group acts by coordinate permutations. At n=6 this action is faithful and gives $S_6\cong S{p}_4(2)$.