examples of semidirect products of groups

Suppose $H=\mathbb{Z}/n\mathbb{Z}$ and let $r$ be a generator for $H$. Let . Define $\theta :Q\to \mathrm{Aut}(H)$ by $\theta (s)(r)=r^{-1}$. Let $G=H{⋊}_{\theta }Q$. Then in $G$,

$$srs=sr{s}^{-1}=\theta (s)(r)=r^{-1}$$

by the canonical equivalence of inner and outer semidirect products. So $G$ has $2n$ elements, two generators $r,s$ satisfying

$$r^n=s^2=1$$

$$srs=r^{-1}$$

and thus $G={𝒟}_{2n}$, the $n^{\mathrm{th}}$ dihedral group.

If instead $H=\mathbb{Z}$, the result is the infinite dihedral group.

As another example, if $G$ is a group, then the holomorph of $G$ is $G⋊\mathrm{Aut}(G)$ under the identity map from $\mathrm{Aut}(G)$ to itself.