TITLE: Dimensions and number of complex irreducible representations for SL3(Z/pZ)
QUESTION [12 upvotes]: This is my first post here :)
I have the following two related questions. While looking in Conway's ATLAS (the 1985 one) for $SL_3(Z/pZ)$, where he does the cases $p=2,3,5,7$, I saw that the dimension of the biggest irreducible representation is ~$p^3$. More precise, as the order of the group is $p^3(p^3-1)(p^2-1)$, it looks like the dimension of the biggest irreducible representation is a polynomial of degree 3 in p, that "obviously" dividers the order. 
For p=2, the biggest one has dimension $8=2^3$
For p=3, the biggest one has dimension $39 = 3(3^2+3+1)$
For p=5, the biggest one has dimension $186=(5+1)(5^2+5+1)$
For p=7, the biggest one has dimension $456=(7+1)(7^2+7+1)$
Is this true in general? Are there infinitely many primes for which the biggest irr rep of $SL_3(Z/pZ)$ has dimension $O(p^3)$ (or even weaker bounds, like $o(p^4)$). 
The second question is about the number of irreducible representations. While looking in the ATLAS, looks like there are quite few of them, relative to the dimension of the group. Is there a result that computes the growth rate of the number of conjugacy classes/number of irr rep, as $p\to\infty$?
I am interested in a result that somehow combines both. Something in the spirit "small", but "few" representations.

REPLY [6 votes]: The question of the maximal dimension of irreducible representations of simple finite groups (including those of Lie type) is considered in "The largest irreducible representations of simple groups" by Larsen, Malle and Tiep (arXiv:1010.5628); for finite groups of Lie type (not necessarily simple) over large enough field, it is a theorem of Seitz ("Cross-characteristic embeddings of finite groups of Lie type", Proc. LMS 1990, 166-200, section 2) that the maximal dimension is at most $|G:T|_{p'}$, where $T$ is a maximail torus of smallest possible order (i.e., maximally split).  In particular, if there is a split maximal torus, this shows that the principal series representations are indeed those with maximal dimension.