Below I am referring to complex representations.

We know that if $G$ is a finite group with $m=(G:Z(G))$, then every irreducible representation has size at most $\sqrt{m}$. One cannot hope for this to be exact up to $O(1)$ as $|G|\to \inf$ since there are finitely many groups with a given number of conjugacy classes. I am interested in interested in constructions/proof of existence of $G$, where this bound is as close as possible to being tight.

An easy nonsatisfying lower bound is by finding $G$ with a small number of conjugacy classes $C$, since then by averaging there must be a representation with dimension at least $\sqrt{|G|}/C$ (although it's not obvious how to build these?)

I am also very interested in the above when you can make the representation faithful - I thought these are interesting in particular since $G$ that has a large faithful irreducible representation has a small outer automorphism group.

1more comment