Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size

Statement
Suppose $$n$$ is a natural number. Then, there exists a PORC function $$f$$ of degree $$n - 1$$ such that, for any prime power $$q$$, the  number of conjugacy classes in the  projective general linear group $$PGL(n,q)$$ (i.e., the projective general linear group of degree $$n$$ over the finite field of size $$q$$) is $$f(q)$$.

A PORC function is a polynomial on residue classes -- it looks like different polynomial functions on different congruence classes modulo a particular number. In this case, we only need to consider congruence classes modulo $$n$$ to define the PORC function. In fact, for a field size of $$q$$, the polynomial depends only on the value $$\operatorname{gcd}(n,q - 1)$$.

General observations
Below, we list some general observations about the PORC function in $$q$$ giving number of conjugacy classes in $$PGL(n,q)$$.

Related facts

 * Number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size
 * Number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size
 * Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size