Symplectic group of degree four

Definition

Let $K$ be a field. The group $Sp(4,K)$ is defined as the symplectic group of degree four over $K$. Explicitly, it is the group given as:

$\{A\in GL(4,K)\mid A{\begin{pmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\\\end{pmatrix}}A^{T}={\begin{pmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\\\end{pmatrix}}\}$

We could give an alternative definition, which yields a different but conjugate subgroup of $GL(4,K)$ (and hence the same group up to isomorphism):

$\{A\in GL(4,K)\mid A{\begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{pmatrix}}A^{T}={\begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{pmatrix}}\}$

For $q$ a prime power, we denote by $Sp(4,q)$ the group $Sp(4,\mathbb {F} _{q})$ where $\mathbb {F} _{q}$ is the (unique up to isomorphism) field of size $q$.

Arithmetic functions

We list here the arithmetic functions for $Sp(4,q)$ for a prime power $q$.

Function	Value	Similar groups	Explanation
order	$q^{4}(q^{4}-1)(q^{2}-1)$

Particular cases

$q$ (field size)	$p$ (underlying prime, field characteristic)	exponent on $p$ giving $q$	$Sp(4,q)$	order (= $q^{4}(q^{4}-1)(q^{2}-1)$)
2	2	1	symmetric group:S6	720
3	3	1	symplectic group:Sp(4,3)	51840
4	2	2	symplectic group:Sp(4,4)	979200