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

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## Definition

### Definition in terms of wreath product

Let $p$ be a prime number greater than $2$. This group $G$ is a group of order $p^4$ defined as: $\! G := K/[[[K,K],K],K]$

where $K$ is the wreath product of groups of order p and $[[[K,K],K],K]$ is the fourth member of the lower central series of $K$.

### Definition in terms of symplectic group

Let $p$ be a prime number greater than $2$. This group is defined as the $p$-Sylow subgroup of the symplectic group of degree four over the prime field $\mathbb{F}_p$. 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 $p$.

## Particular cases

This group definition works only for $p \ge 3$:

Value for prime number $p$ 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 $G$:

gap> G := SmallGroup(p^4,7);

Conversely, to check whether a given group $G$ 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);```