Construct groups that satisfy given conditions

Question

I want to construct groups which satisfy the following conditions.

1. It should be generated by an abelian normal subgroup $M$ and a subgroup $G$.
2. $G$ should be either an Abelian group or Symmetric group.

I have found one set of examples, namely dihedral groups. Since we know that Dihedral group is a semidirect product of $C_{2}$ and $C_{n}$. It satisfies the above conditions.

I want to construct other examples.

Note: I am dealing only with finite groups.

Answer 1

Given groups $A$ and $G$, you can always form a semidirect product which will be different from a direct product if you have a nontrivial action of $G$ on $A$. Thus if $A=Z_n$ and $G=Z_n^{\ast}$ (the group of automorphisms of $A$) then you can always form a nontrivial semidirect product. This gives you a new example already for $n=5$.