= Presentation of a semidirect product
Let $G=\langle S\mid R\rangle$ and $H=\langle T\mid U\rangle$ be <group presentations>, and let $\phi:H\to\operatorname{Aut}(G)$ be a <group homomorphism>. Choose a word $w_{t,s}(S)$ representing $\phi(t)(s)$ for each pair of generators. Then the <semidirect product> has presentation
$$
G\rtimes_\phi H=\langle S\sqcup T\mid R,U,\ tst^{-1}=w_{t,s}\ (t\in T,s\in S)\rangle.
$$
The 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>.
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