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.
View other such prime-parametrized groups

Definition

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

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 $p$ .
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.
It is the $p$ -Sylow subgroup of the symmetric group of order $p^{2}$ .

Particular cases

Value of prime $p$	Value $p^{p+1}$ (order of the wreath product)	Wreath product of groups of order $p$	Symmetric group of order $p^{2}$ , in which it is a $p$ -Sylow subgroup
2	8	dihedral group:D8	symmetric group:S4 (see D8 in S4)
3	81	wreath product of Z3 and Z3	symmetric group:S9
5	15625	wreath product of Z5 and Z5	symmetric group:S25

Arithmetic functions

Function	Value	Explanation for function value
order	$p^{p+1}$	As a wreath product, it has order $p^{p}\cdot p=p^{p+1}$ .
prime-base logarithm of order	$p+1$
exponent	$p^{2}$	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
prime-base logarithm of exponent	2
nilpotency class	$p$	The group is a maximal class group.
derived length	2	The base of the semidirect product is an abelian normal subgroup with abelian quotient group.
Frattini length	2

GAP implementation

Assign to $p$ any numerical value of a prime number. Then, the group can be defined as WreathProduct(CyclicGroup(p),CyclicGroup(p)) using the GAP functions WreathProduct and CyclicGroup.