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

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

This article describes the element structure of the symplectic group of degree four over a finite field of size $q$ , denoted $S p (4, q)$ .

Summary

Item	Value
order of the group	$q^{4} (q^{4} - 1) (q^{2} - 1)$
number of conjugacy classes	Case $q$ even (e.g., $q = 2, 4, 8, 16, \dots$ ): $q^{2} + 2 q + 3$ Case $q$ odd (e.g., $q = 3, 5, 7, 9, 11, \dots$ ): $q^{2} + 5 q + 10$

Particular cases

$q$ (field size)	$p$ (underlying prime, field characteristic)	Case on $q$	group $S p (4, q)$	order of the group (= $q^{4} (q^{4} - 1) (q^{2} - 1)$ )	conjugacy class sizes (ascending order)	number of conjugacy classes (= $q^{2} + 2 q + 3$ if $q$ even, $q^{2} + 5 q + 10$ if $q$ odd)	element structure page
2	2	even	symmetric group:S6	720	1,15,15,40,40,45,90,90,120,120,144	11	element structure of symmetric group:S6
3	3	odd	symplectic group:Sp(4,3)	51840	1,1,40,40,40,40,90,240,240,360,360,360,360, 480,480, 540,540,540,1440,1440,1440,1440, 2160,2160,2160,2160,2880,2880,2880,2880,4320,5184,5184,6480	34	element structure of symplectic group:Sp(4,3)
4	2	even	projective symplectic group:PSp(4,4)	979200	1,255,255,3264,3264,3264,3264,3825, 5440,5440,30600,30600,39168, 48960,48960,48960,48960,57600,57600,57600,57600, 65280,65280,65280,65280,81600,81600	27	element structure of projective symplectic group:PSp(4,4)