[Math] an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.

Question

I am wondering if my claim is correct, and how to formalize/write it? Thank you.

Give an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.

So I am thinking of the dihedral group of triangle, with $3$ rotations and $2$ reflections. This is not abelian; but if I quotient $3$ rotations, the group has only two elements, which must be abelian.

Answer 1

Perhaps the cheapest sort of example would be the direct product of any non-abelian group with any non-trivial abelian group.