What I'm really looking for is a good set of examples for a semi-direct product of two (coprime) cyclic groups, with a non-trivial (not $1$ and not everything) center. What resource would you recommend for that purpose?



List of centers of finite groups
If you would like a complete classification following Jack's strategy, say your group is $G=C\_n\rtimes C\_m$ with $n,m$ coprime, $C\_n=\langle c\rangle$, $C\_m=\langle x\rangle$ and $xcx^{-1}=c^j$. The order of $j$ in $({\mathbb Z}/n{\mathbb Z})^\times$ has to divide $m$, and every such triple $(n,m,j)$ gives a valid semi-direct product $G$. Now



> 
> $G$ has a non-trivial centre if and only if either
> 
> 
> * The order of $j$ is less than $m$, or
> * $j$ is congruent to 1 modulo a prime $p|n$.
> 
> 
> 


In the first case, $C\_m$ does not act faithfully on $C\_n$, and the elements in the kernel commute with both $C\_n$ and $C\_m$, so they are in the centre. In the second case, $C\_p\lt C\_n$ is central. If neither condition holds, then the centre must be contained in $C\_n$ (by faithfullness of the action) and cannot contain any $C\_p$ for $p|n$ (by the failure of the second condition), so it has to be trivial.


