Direct product of cyclic group of prime-square order and elementary abelian group of prime-square order

Definition
Let $$p$$ be a prime number.

This group is defined in the following equivalent ways:


 * 1) $$\mathbb{Z}_{p^2} \times E_{p^2}$$: The defining ingredient::external direct product of the defining ingredient::cyclic group of prime-square order and the defining ingredient::elementary abelian group of prime-square order.
 * 2) $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p \times \mathbb{Z}_p$$: The external direct product of the cyclic group of prime-square order and two copies of the defining ingredient::cyclic group of prime order.

This group is thus the abelian group of prime power order corresponding to the partition (see also structure theorem for finitely generated abelian groups):

$$\! 2 + 1 + 1$$

GAP implementation
The exception is the case $$p = 2$$, in which case the group is $$(p^4,10)$$.