Element structure of projective semilinear group of degree two over a finite field

This article discusses the element structure of the projective semilinear group of degree two over a finite field.

We denote the field size by $$q$$ and the field characteristic by $$p$$. We define $$r = \log_p q$$, so the field $$\mathbb{F}_q$$ is a Galois extension of $$\mathbb{F}_p$$ of degree $$r$$ with cyclic Galois group.

Number of conjugacy classes
For $$p \ne 2$$, the number of conjugacy classes can be expressed as a polynomial of degree $$r$$ in $$p$$, where the polynomial depends only on $$r$$ and not on $$p$$. Below are the first few cases: