Symplectic quotient of the binary subset module
= Symplectic quotient of the binary subset module
{title2=$\mathbf1^\perp/\langle\mathbf1\rangle$}
For even n, the even-weight subspace of $F_2^n$ has radical $\langle\mathbf1\rangle$ 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 $S_6\cong Sp_4(2)$.