= Exterior-square realization of PSL4 over F2
{title2=$PSL_4(2)\cong A_8$}
On the six-dimensional <exterior square> of $\mathbb F_2^4$, the <Pfaffian> $Q=x_{12}x_{34}+x_{13}x_{24}+x_{14}x_{23}$ is a nonsingular plus-type <quadratic form>. The action of $GL_4(2)$ preserves it and is faithful: fixing every decomposable exterior line fixes every two-space and then every one-space. The even-weight binary subset module on eight letters, modulo the all-one vector, gives another six-dimensional plus-type space with $Q=\operatorname{wt}(x)/2\bmod2$. Coordinate permutations embed $S_8$ faithfully into its orthogonal group. That orthogonal group has order $40320$, while $GL_4(2)$ has order $20160$, identifying its index-two image with the <alternating group> $A_8$.
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