Direct product of cyclic group of prime-cube order and cyclic group of prime order

Definition
This group is defined as the defining ingredient::external direct product of the defining ingredient::cyclic group of prime-cube order (denoted $$\mathbb{Z}_{p^3}$$ or $$C_{p^3}$$) and the defining ingredient::cyclic group of prime order (denoted $$\mathbb{Z}_p$$ or $$C_p$$), i.e., it is defined as $$\mathbb{Z}_{p^3} \times \mathbb{Z}_p$$ (or, in alternative notation, $$C_{p^3} \times C_p$$). It corresponds to the partition (see classification of finitely generated abelian groups and abelian group of prime power order):

$$\! 4 = 3 + 1$$

For a given prime $$p$$, it can also be defined as the unique (up to isomorphism) abelian group of order $$p^4$$ and exponent $$p^3$$.