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If a group $K$ acts on a finite set $X$, it acts on the direct product of groups $H^X$ by permuting its factors. The resulting semidirect product is the permutation wreath product. For $K=S_m$ acting on $m$ two-point blocks and $H=C_2$, it describes independent within-block swaps combined with a permutation of blocks, and has order $2^{m}m!$. The action on $X$ is part of the definition; it need not be the regular action used in a regular wreath product.

The Poincare group is the semidirect product of spacetime translations by the Lorentz group. It is the isometry group of Minkowski spacetime.

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.

If $K$ is a finitely generated group, then the semidirect product $K⋊H$ is a residually finite group exactly when both $K$ and $H$ are residually finite groups. To separate an element whose $H$ coordinate is nontrivial, use a finite quotient of $H$. For $1\ne k\in K$, choose a finite-index normal subgroup $U$ omitting $k$, and intersect all subgroups of index at most $[K:U]$ to obtain a characteristic finite-index subgroup $C\le U$. The induced map into $(K/C)⋊\mathrm{im}(H\to \mathrm{Aut}(K/C))$ has finite target and separates $k$. More generally, finite generation can be replaced by a separating family of finite-index normal subgroups invariant under the given $H$ action.

If a group homomorphism section $i:G/N\to G$ satisfies $\pi i=\mathrm{id}$, then each $g\in G$ has a unique form $i(q)n$ with $q\in G/N$ and $n\in N$: take $q=\pi (g)$ and $n=i(q{)}^{-1}g$. Thus $(q,n)\mapsto i(q)n$ is a bijection. It is a homomorphism for the ordinary direct product of groups exactly when the section image centralizes $N$. In general it describes a semidirect product with a nontrivial conjugation action.

For $S\subseteq {\mathbb{N}}_{>0}$, form the Right-angled Artin group on generators $b_i$ indexed by $i\in \mathbb{Z}$, imposing $[b_i,b_j]=1$ precisely when $|i-j|\in S$. Translation $b_i\mapsto b_{i+1}$ is an automorphism. Its semidirect product with an infinite cyclic group generated by $t$ is countable and generated by $b_0,t$. If $n\notin S$, killing all $b_i$ except $b_0,b_n$ retracts the graph group onto the rank-two free group, so their commutator is nonidentity. This proves the displayed equivalence. For non-recursively-enumerable $S$, enumerable identities of finitely generated subgroups rules out an embedding of this two-generated group in any finitely presented group.

A group extension is split when its quotient homomorphism admits a group-homomorphic section. Equivalently, the middle group is a semidirect product of the kernel by the quotient.

The Möbius transformations generated by $D_{\lambda }(z)=\lambda z$ with $\lambda \ne 0$ and $I(z)=1/z$ have exactly the two displayed forms. The relations $I^2=1$ and $I{D}_{\lambda }=D_{{\lambda }^{-1}}I$ reduce every word to $D_{\lambda }{I}^{ϵ}$, $ϵ\in \{0,1\}$. This is a semidirect product of ${\mathbb{C}}^{\times }$ with a cyclic group of order two acting by multiplicative inversion. It preserves the unordered pair $\{0,\infty \}$, and only its scaling elements fix zero.

A space group for which an origin can be chosen so that representatives of all point-group operations have zero translation part. It is a semidirect product of its lattice translations and point group. Presence of centring translations in a conventional cell does not alone make a group nonsymmorphic.

This Lie group acts on the plane by $v\mapsto x+\rho v$ with $\rho >0$. Its multiplication is $(\rho ,x)({\rho }^{\prime },x^{\prime })=(\rho {\rho }^{\prime },x+\rho x^{\prime })$, and its inverse is $({\rho }^{-1},-{\rho }^{-1}x)$. It is a semidirect product in which positive dilation acts on the translation subgroup. A dilation generator $D$ and translation generators $T_1,T_2$ have Lie brackets $[D,T_a]=T_a$ and $[T_1,T_2]=0$.

Let $p$ be prime, $a\in {\mathbb{F}}_p^{\times }$, $x,y\in \mathbb{Z}/(p-1)\mathbb{Z}$ and $u,v\in {\mathbb{F}}_p$. Fermat's little theorem makes the displayed multiplication well defined. Its identity is $(0,0)$ and inverse is $(-x,-a^{-x}u)$; associativity follows by expanding both bracketings. Projection onto the first coordinate is a group homomorphism with normal cyclic kernel of order $p$. Under the change $U=a^{-x}u$, multiplication becomes $(x,U)(y,V)=(x+y,U+a^{-x}V)$, the semidirect product of the additive prime field by the cyclic group of order $p-1$. The group is abelian precisely when $a=1$; $a$ need not be a generator of the multiplicative field group.