Prove that if $H$ is an abelian subgroup of a group $G$ then $\langle H, Z(G)\rangle$ is abelian. Give an explicit example of an abelian subgroup $H$ of a group $G$ such that $\left\langle H, C_{G}(H)\right\rangle$ is not abelian.
The proof of the first line is ready.
However I wonder what is the conterexample that makes $\left\langle H, C_{G}(H)\right\rangle$ non-abelian.