Element structure of general linear group of degree three over a finite field

This article discusses the element structure of the general linear group of degree three over a finite field. The group is $$GL(3,q)$$ where $$q$$ is the order (size) of the field. We denote by $$p$$ the prime number that is the characteristic of the field.

Conjugacy class structure
There is a total of $$(q^3 - 1)(q^3 - q)(q^3 - q^2) = q^3(q - 1)^3(q + 1)(q^2 + q + 1)$$ elements, and a total of $$q^3 - q = q(q - 1)(q + 1)$$ conjugacy classes.