Wreath product

Suppose $𝐺$ is a group, $Γ$ is a set, and $𝐻$ is a group acting on $Γ$ . Then the wreath product is the semidirect product group

𝐺 Γ ⋊ 𝐻

where the action of $𝐻$ on $𝐺 Γ$ is induced by that on $Γ$ .