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

This article gives specific information, namely, element structure, about a family of groups, namely: projective symplectic group of degree four.
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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.

Summary

Item	Value
order of the group	Case $q$ even (e.g., $q=2,4,8,16,\dots$): $q^{4}(q^{2}-1)(q^{4}-1)$ Case $q$ odd (e.g., $q=3,5,7,9,11,\dots$): $q^{4}(q^{2}-1)(q^{4}-1)/2$
number of conjugacy classes	Case $q$ a power of 2 (e.g., $q=2,4,8,16,\dots$): $q^{2}+2q+3$ Case $q$ odd (e.g., $q=3,5,7,9,11,\dots$): $(q^{2}+6q+13)/2$

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:

Value of $\operatorname {gcd} (2,q-1)$	Corresponding congruence classes mod 2	Number of conjugacy classes (polynomial of degree 2)	Additional comments
1	0 (e.g., $q=2,4,8,\dots$)	$q^{2}+2q+3$	In this case, the projective symplectic group coincides with the symplectic group, because the symplectic group is centerless
2	1 (e.g., $q=3,5,7,9,\dots$)	$(q^{2}+6q+13)/2$