An example that $G/N$ cyclic does not imply $G$ cyclic.

Question

Let $G$ a group and $N$ a normal subgroup of $G$. A known fact is that if $G$ is cyclic, then the quotient group $G/N$ is cyclic too. Other fact is that if $G/Z(G)$ is cyclic, where $Z(G)$ denoted the center of $G$, then $Z(G) = G$. What would be an example of that $G/N$ is cyclic, but $G$ is not cyclic?

Answer 1

Let $G=S_3$ and $N=A_3$.

Then $G$ is not cyclic (not even Abelian), but $G/N$ has order $2$ and is cyclic.