First a disclaimer: I have not taught any graduate level involving group theory is over twenty years, therefore I have not done any active work in this for that period. That said this may help you.

Any Abelian group is its own center. Thus you are looking for a non-Abelian group of order twelve that has a center of order four.

There has been quite a lot of work done on finite groups. If I were you that is where I would start.

Now there may very well be a completely clear solution that I don't see.