Definition 44.2.10 (Wreath Product).label Let $H$ be a non-trivial group, $\Gamma$ be a group acting on a set $I$, $H^{(I)}= \bigoplus_{i \in I}H$, and $\Gamma$ act on $H^{(I)}$ by left translation ($(\gamma \xi)_{i} = \xi_{\gamma^{-1}i}$), then the generalised wreath product $H \wr_{I} \Gamma$ is the semidirect product $H^{(I)}\rtimes \Gamma$. The wreath product $H \wr \Gamma$ is the special case where $I = \Gamma$, and $\Gamma$ acts by left translation.