# Projective symplectic group:PSp(6,2)

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## Definition

This group is defined in the following equivalent ways:

1. It is the projective symplectic group of degree six (i.e., arising from $6 \times 6$ matrices) over field:F2. This is denoted $PSp(6,2)$ and the Chevalley notation is $C_3(2)$; however, it can also be described as $B_3(2)$ (see below).
2. It is the symplectic group of degree six (i.e., arising from $6 \times 6$ matrices) over field:F2. This is denoted $Sp(6,2)$.
3. It is the Chevalley group of type B $B_3(2)$ over field:F2.

### Equivalence of definitions

The equivalence of (1) and (2) follows from isomorphism between symplectic and projective symplectic group in characteristic two.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 1451520 groups with same order As $Sp(2m,q), m = 3, q = 2$: $q^{m^2} \prod_{i=1}^m (q^{2i} -1)$ becomes $2^{3^2}(2^2 - 1)(2^4 - 1)(2^6 - 1) = (512)(3)(15)(63) = 1451520$

## GAP implementation

Description Functions used
PSp(6,2) PSp
Sp(6,2) Sp