Largest irreducible representation of a finite non-commutative group Let $G$ be a finite non-commutative group of order $k$. Is there any way to determine a number $m$ such that there will necessarily exist an irreducible representation of $G$ of dimension $d \geq m$? If not, what kind of additional information could give an answer? Thanks!
 A: As the comments indicate, the example of the dihedral groups shows that you cannot expect irreps of dimension greater than $2$ without additional hypotheses, and that bounding the size of the abelianization isn't a particularly helpful hypothesis. 
One thing to say is that if a group $G$ has $c(G)$ conjugacy classes and $a(G) = |G/[G, G]|$, then it must also have $c(G)$ irreps (I am working over $\mathbb{C}$ throughout), $a(G)$ of which are $1$-dimensional. Since the dimensions $d_i$ of these irreps must satisfy $\sum d_i^2 = |G|$, you can conclude by the pigeonhole principle that $G$ must have an irrep of dimension at least
$$\sqrt{ \frac{|G| - a(G)}{c(G) - a(G)} }.$$
As you can see, $c(G)$ has a much bigger influence than $a(G)$ on this bound. Conversely, if the maximum dimension of an irrep is $d$, then 
$$c(G) - a(G) \ge \frac{|G| - a(G)}{d^2}.$$
For example, the symmetric group $S_n$ satisfies $c(S_n) = p(n)$ (the partition number of $n$) and $a(S_n) = 2$ ($n \ge 2$), so we conclude that $S_n$ must have an irrep of dimension at least
$$\sqrt{ \frac{n! - 2}{p(n) - 2} }.$$
As a subexample, when $n = 7$ this bound is 
$$\sqrt{ \frac{7! - 2}{p(7) - 2} } = \sqrt{ \frac{5038}{13} } \approx 19.7$$
so we conclude that $S_7$ has an irrep of dimension at least $20$. In fact it has irreps of dimensions $20, 21, 35$. (Of course, for the symmetric groups we know the dimensions of all of their irreps via the hook length formula, so we can say much more here.)
Another thing to say is that the dimensions $d_i$ of the irreps must also divide the size $|G|$ of $G$. This is a strong restriction if $G$ is, for example, a $p$-group, since it implies that the dimensions must themselves be powers of $p$.
