Give an example of a group $G$ such that $G/Z(G)$ is not abelian.

Question

Give an example of a group $G$ such that $G/Z(G)$ is not abelian.

I am having trouble understanding what the group $G/Z(G)$ looks like. Because of that I am having troubles answering the above question.

Answer 1

Hint $:$ Take $G=S_3.$ What is $Z(S_3)$?

Note that in order to show that $G/Z(G)$ is non-abelian we need only to find two elements $\sigma,\tau \in G$ such that $\sigma \tau {\sigma}^{-1} {\tau}^{-1} \notin Z(G).$

Answer 2

Hint: For any group $G$, we have $$G/Z(G)\cong \operatorname{Inn}(G),$$ where $\operatorname{Inn}(G)$ is the group of inner automorphisms of $G$.