# Projective symplectic group:PSp(4,3)

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## Definition

This group is defined in the following equivalent ways:

1. It is the projective symplectic group of degree four (i.e., arising from $4 \times 4$ matrices) over field:F3. This is denoted $PSp(4,3)$ and the Chevalley notation is $C_2(3)$ -- however, due to collision with another family (see definition (3)) it is denoted $B_2(3)$.
2. It is the projective special unitary group of degree four (i.e., arising from $4 \times 4$ matrices) over field:F2. This is denoted $PSU(4,2)$ and the Chevalley notation is PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE].
3. It is the Chevalley group of type B $B_2(3)$, i.e., the group $\Omega_5(3)$. This is kernel of the spinor norm on the special orthogonal group $SO(5,3)$.

## Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 25920 groups with same order As $PSp(4,q), q = 3: q^4(q^4 - 1)(q^2 - 1)/\operatorname{gcd}(2,q-1) = 3^4(3^4-1)(3^2 - 1)/\operatorname{gcd}(2,3-1) = 25920$

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation for function value
number of conjugacy classes 20 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $PSp(4,q), q = 3$ (odd): $(q^2 + 6q + 13)/2 = (3^2 + (6)(3) + 13)/2 = 20$ (more here)
See element structure of projective symplectic group:PSp(4,3)
number of conjugacy classes of subgroups 116 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups See subgroup structure of projective symplectic group:PSp(4,3)
number of subgroups 45649 groups with same order and number of subgroups | groups with same number of subgroups See subgroup structure of projective symplectic group:PSp(4,3)

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group Yes projective symplectic group is simple

## GAP implementation

Description Functions used
PSp(4,3) PSp
NormalSubgroups(SO(5,3)) SO, NormalSubgroups