In the book Character Theory Of Finite Groups by I.Martin Issacs as exercise 2.14
Let $G$ be a finite group with commutator subgroup $G'$. Let $H \subset G' \cap Z(G)$ be cyclic of order n and let m be the maximum of the orders of elements of G/H. Assume that n is a prime power and show that $|G| \geq n²m$.
I have tried, following the hints given, to show, taking $\chi$ to be an irreducible character whose kernel intersects trivially with $H$, that $\chi(1) \geq n$. But, thus far as I am concerned, I see no idea of how to show the inequality, not to mention the latter part of the proof, as suggested, to deploy the Problem 2.9(b) which states that $\chi(1) \leq |G:A|$ for any abelian subgroup A of G.
I have asked my teacher, who replied that one is to show that $|G/H| \geq nm$, of which the proof I have no idea either. Of course, if one can demonstrate the existence of one subgroup of $G/H$ order $\geq nm$, then the problem should be resolved, as the teacher suggested. But this still bewilders me at present.