Element structure of projective symplectic group of degree four over a finite field

This article describes the element structure of the projective symplectic group of degree four over a finite field of size $$q$$, denoted $$PSp(4,q)$$. This group is a Chevalley group of type C, and with the Chevalley notation, it is denoted $$C_2(q)$$. Note that the parameter as a Chevalley group is half the order of the matrices.

Number of conjugacy classes
The general theory tells us that number of conjugacy classes in projective symplectic group of fixed degree over a finite field is PORC function of field size. For $$PSp(2m,q)$$, the degree of the polynomial is $$m$$ and the polynomial depends on the value $$\operatorname{gcd}(\operatorname{lcm}(2,m),q-1)$$. In this case, $$m = 2$$, so the polynomials have degree two and they depend on $$\operatorname{gcd}(2,q-1)$$. This is exactly what happens: