Wreath product of groups of order p

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
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Definition

Let $p$ be a prime number. The wreath product of groups of order $p$ is any of the following equivalent things:

1. It is the wreath product of the cyclic group of order $p$ with the cyclic group of order $p$, where the latter is given the regular action on a set of size $p$.
2. It is the semidirect product of the elementary Abelian group of order $p^{p}$ and a cyclic group of order $p$ acting on it by cyclic permutation of coordinates.
3. It is the $p$-Sylow subgroup of the symmetric group of order $p^{2}$.