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

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 defining ingredient::wreath product of groups of order p and $$[K,K],K],K]$ is the fourth member of the [[defining ingredient::lower central series of $$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 $$p \ge 3$$:

Other descriptions
This group can be described using GAP as:

gap> K := WreathProduct(CyclicGroup(p),CyclicGroup(p));  gap> G := K/CommutatorSubgroup(CommutatorSubgroup(DerivedSubgroup(K),K),K);