The wreath product 
of two groups, relative to an action of 
on a set 
, is the semidirect product
![G^X×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]H,](/images/equations/WreathProduct/NumberedEquation1.svg) |
where 
acts on the direct product 
by permuting its coordinates. If 
with the regular action, this is the regular wreath product.
If 
is a permutation
group on a finite set 
, it is the corresponding permutational wreath product.
Wreath products describe automorphism groups of rooted trees and appear in the decomposition of finite automata and permutation groups.
