I'm trying to prove the following statement:
Let's assume $ G $ is a finite group. Let $ Z(G) $ denote its center, $(G : Z(G))$ the index of $ Z(G) $ in $ G $ and $ [G; G] $ the commutator subgroup of $ G $. Prove that if
$$ (G : Z(G)) ^2 < \left| [G; G] \right| $$
then $ [G; G] $ cannot consist of commutators (elements of form $ g^{-1}h^{-1}gh $) only.
I don't really know where to begin with - finding an example of a group which's commutator subgroup isn't just commutators is not obvious and frankly not very helpful either. I would appreciate some help on that