Point and hyperplane stabilizers are almost conjugate

In a finite general linear group, the permutation representations on nonzero vectors and on nonzero dual vectors have equal characters: an element fixes $q^{\dim \ker (t-I)}-1$ vectors in each action because a matrix and its transpose have the same rank. In characteristic two the stabilizers of a nonzero vector and a hyperplane are therefore almost conjugate subgroups. In dimension three over ${\mathbb{F}}_2$, both have order $24$ and index $7$ in a group of order $168$, but they are not conjugate: a point stabilizer preserves no plane. A triangle cover construction for Sunada surfaces with generator orders $(7,7,7)$ gives quotient surfaces of genus three.