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Every element of $G{L}_2({\mathbb{F}}_q)$ is conjugate to exactly one representative of one of these types, with the indicated parameter identifications:

The projective line ${\mathbb{P}}^1({\mathbb{F}}_q)$ has $q+1$ points. A line is fixed exactly when it is an eigenline over ${\mathbb{F}}_q$, so the permutation character $\pi$ takes respectively the valuesThe action is two-transitive. Part (a) therefore giveswhere the nontrivial constituent $\mathrm{St}$ is irreducible of degree $q$, the Steinberg representation of GL2 over a finite field. Its values on the four types are