 Presentation of a semidirect product

Let $G=⟨S\mid R⟩$ and $H=⟨T\mid U⟩$ be group presentations, and let $\varphi :H\to \mathrm{Aut}(G)$ be a group homomorphism. Choose a word $w_{t,s}(S)$ representing $\varphi (t)(s)$ for each pair of generators. Then the semidirect product has presentationThe cross-relations allow every word to be written in factor order $gh$, and the multiplication rule matches the prescribed action. The natural homomorphisms to and from the semidirect product are inverse on the generators. Finite factor presentations yield a finite group presentation.