Element structure of general semiaffine group of degree one over a finite field

This article describes the element structure of the general semiaffine group of degree one over a finite field of size $$q$$, where $$q = p^r$$, with $$p$$ the field characteristic and $$r$$ a positive integer. This group is denoted $$\Gamma A(1,q)$$ or $$A\Gamma L(1,q)$$.

Note that if $$r = 1$$, the group is the same as the general affine group of degree one $$GA(1,q)$$.

Number of conjugacy classes formulas
For every fixed value of $$r$$, the number of conjugacy classes simply becomes a polynomial in $$p$$. The values of these polynomials for small $$r$$ are listed below:

Case r = 2
We have $$q = p^2$$. In our discussion, we distinguish between conjugacy classes in the additive group, conjugacy classes in GA(1,q) that are outside the additive group, and conjugacy classes outside GA(1,q). Note that the additive group has size $$q = p^2$$, the group $$GA(1,q)$$ contains it and has size $$q(q - 1) = p^2(p^2 - 1)$$, and the whole group has size $$2q(q - 1) = 2p^2(p^2 - 1)$$.