= 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.
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