# Maximal unipotent subgroup of symplectic group:Sp(4,p)

## Contents

## Definition

### Definition in terms of wreath product

Let be a prime number greater than . This group is a group of order defined as:

where is the wreath product of groups of order p and is the fourth member of the lower central series of .

### Definition in terms of symplectic group

Let be a prime number greater than . This group is defined as the -Sylow subgroup of the symplectic group of degree four over the prime field . Another way of saying this is that it is a maximal unipotent subgroup of symplectic group of degree four for a field of prime size .

## Particular cases

This group definition works only for :

Value for prime number | Corresponding group |
---|---|

3 | wreath product of Z3 and Z3 |

5 | quotient of wreath product of Z5 and Z5 by fourth member of lower central series |

## GAP implementation

### Group ID

This finite group has order p^4 and has ID 7 among the group of order p^4 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(p^4,7)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(p^4,7);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [p^4,7]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

This group can be described using GAP as:

gap> K := WreathProduct(CyclicGroup(p),CyclicGroup(p)); <group of size 15625 with 2 generators> gap> G := K/CommutatorSubgroup(CommutatorSubgroup(DerivedSubgroup(K),K),K);