# Element structure of wreath product of groups of order p

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: wreath product of groups of order p.
View element structure of group families | View other specific information about wreath product of groups of order p

## Summary

Item Value
number of conjugacy classes in the whole group $p^{p-1} + p^2 - 1$ (equals number of irreducible representations; see number of irreducible representations equals number of conjugacy classes and linear representation theory of wreath product of groups of order p)
order of the group $p^{p+1}$
conjugacy class sizes 1 (occurs $p$ times), $p$ (occurs $p^{p-1} - 1$ times), $p^{p-1}$ (occurs $p^2 - p$ times).
order statistics order 1: 1 conjugacy class of size 1 (total 1 element)
order $p$: $p - 1$ conjugacy classes of size 1, $p^{p-1} - 1$ conjugacy classes of size $p$, $p - 1$ conjugacy classes of size $p^{p-1}$ (total: $2p^p - p^{p-1} - 1$ elements)
order $p^2$: $(p - 1)^2$ conjugacy classes of size $p^{p-1}$ (total $p^{p+1} - 2p^p + p^{p-1}$ elements)