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

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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