One can calculate the center of a semidirect product $U\ltimes V$ very explicit. Let $\phi: V\to Aut(U)$ the corresponding homomorphism. Then the following holds: $$(u,v)\in Z(U\ltimes V) \iff v\in Z(V), u\in C_U(V), \kappa_u=\phi(v^{-1})$$ where $\kappa_u$ is the conjugation with $u$. In particular $\phi(v)\in Inn(U)$. If $U$ is abelian, this gives $Z(U\ltimes V)=Z(V) \cap \ker(\phi)$. If $V$ is abelian too, then $Z(U\ltimes V)=\ker(\phi)$. Now all examples you're looking for can be easily written down.
One can calculate the center of a semidirect product $U\ltimes V$ very explicit. Let $\phi: V\to Aut(U)$ the corresponding homomorphism. Then the following holds: $$(u,v)\in Z(U\ltimes V) \iff v\in Z(V), u\in C_U(V), \kappa_u=\phi(v^{-1})$$ where $\kappa_u$ is the conjugation with $u$. In particular $\phi(v)\in Inn(U)$. If $U$ is abelian, this gives $Z(U\ltimes V)=Z(V) \cap \ker(\phi)$ and hence $Z(U\ltimes V)=Z(V)$ if $V$ is abelian too. ker(\phi)$. Now all examples you're looking for can be easily written down. 1 One can calculate the center of a semidirect product$U\ltimes V$very explicit. Let$\phi: V\to Aut(U)$the corresponding homomorphism. Then the following holds: $$(u,v)\in Z(U\ltimes V) \iff v\in Z(V), u\in C_U(V), \kappa_u=\phi(v^{-1})$$ where$\kappa_u$is the conjugation with$u$. In particular$\phi(v)\in Inn(U)$. If$U$is abelian, this gives$Z(U\ltimes V)=Z(V) \cap \ker(\phi)$and hence$Z(U\ltimes V)=Z(V)$if$V\$ is abelian too. Now all examples you're looking for can be easily written down.