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.



How to construct groups and large dimension representations? How about faithful ones?
In the infinite family $G\_p=(\mathbb{Z}/p\mathbb{Z})\rtimes (\mathbb{Z}/p\mathbb{Z})^{\times}$, as $p$ runs over prime numbers, the bound is tight up to $O(1)$, and the representation in question is faithful. Indeed, the centre is trivial, and $\#G\_p=p^2-p$, so $p-1\leq \sqrt{m}\leq p$. By Clifford theory, the induction of any faithful character from $\mathbb{Z}/p\mathbb{Z}$ to $G\_p$ is a faithful irreducible representation of $G\_p$ of dimension $p-1$, which is less than $1$ away from $\sqrt{m}$.


