Exterior-square realization of PSL4 over F2 ($PS{L}_4(2)\cong A_8$)

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 $G{L}_4(2)$ 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 $Q=\mathrm{wt}(x)/2\mathrm{mod}2$. Coordinate permutations embed $S_8$ faithfully into its orthogonal group. That orthogonal group has order $40320$, while $G{L}_4(2)$ has order $20160$, identifying its index-two image with the alternating group $A_8$.