TITLE: Small example of Zappa-Szep product of finite groups
QUESTION [5 upvotes]: Does anyone know a good place to find an example of a group, $G$, of "small" order in particular finite such that there are subgroups $H,K$ of $G$ with $HK=G$ and $G$ a Zappa-Szep product of $H$ and $K$ (in particular neither $H$ nor $K$ is normal?

REPLY [4 votes]: The smallest nontrivial examples have order $16$.  One of them is
$$
\mathbb{Z}_2 \times D_8 \,=\, \langle a, r,s \mid a^2=r^4=s^2=[a,r]=[a,s]=1,sr=r^{-1}s\rangle,
$$
which is the Zappa-Szép product of the subgroups $\{1,a,s,as\}$ and $\{1,ar^2,rs,ar^3s\}$.
Other examples include $S_4$ (as Derek Holt mentions in the comments) and $D_{24}$, which is the Zappa-Szép product of the subgroups $\langle r^4,s\rangle \cong D_6$ and $\langle r^6,rs\rangle \cong V$.