[This answer](https://mathoverflow.net/a/243908/34538) of [Geoff Robinson](https://mathoverflow.net/users/14450/geoff-robinson) shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $3$ if and only if it is isomorphic to $A\_5$ or $\mathrm{PSL}(2,7)$.


*Question:* Is there such a classification for every dimension $d$?  

[Or at least, for every *small* dimension, let's say less than $15$?]


*Weaker question*: Are there finitely many finite simple groups with an irrep of dimension $d$?


*If yes*: What are the best known lower/upper bounds for the order of a finite simple group with an irrep of dimension $d$? 


The following table provides the number $n$ of finite simple groups $G$ of order less than $10^6$ with an irrep of a given dimension $d < 15$, together with the minimun and maximum orders. 


$$\begin{array}{c|c|c|c|c} 
d &n&min&max & G \newline \hline
3 &2&60&168&A\_5, \mathrm{PSL}(2,7) \newline \hline
4 &1&60&60&A\_5 \newline \hline
5 &4 &60 &25920&A\_5, A\_6, \mathrm{PSL}(2,11), \mathrm{PSp}(4,3) \newline \hline
6 &4 &168 &25920&\mathrm{PSL}(2,7), A\_7, \mathrm{PSU}(3,3), \mathrm{PSp}(4,3) \newline \hline
7 &5 &168 &20160&\mathrm{PSL}(2,7), \mathrm{PSL}(2,8), \mathrm{PSL}(2,13), \mathrm{PSU}(3,3), A\_8 \newline \hline
8 &4 &168 &181440& \mathrm{PSL}(2,7), A\_6, \mathrm{PSL}(2,8), A\_9 \newline \hline
9 &4 &360 &3420& A\_6, \mathrm{PSL}(2,8), \mathrm{PSL}(2,17), \mathrm{PSL}(2,19) \newline \hline
10 &5 &360 &25920& A\_6, \mathrm{PSL}(2,11), A\_7, M\_{11}, \mathrm{PSp}(4,3) \newline \hline
11 &4 &660 &95040& \mathrm{PSL}(2,11), \mathrm{PSL}(2,23), M\_{11}, M\_{12} \newline \hline
12 &4 &660 &62400& \mathrm{PSL}(2,11),\mathrm{PSL}(2,13), \mathrm{PSL}(3,3), \mathrm{PSU}(3,4) \newline \hline
13 &5 &1092 &62400& \mathrm{PSL}(2,13), \mathrm{PSL}(3,3), \mathrm{PSL}(2,25), \mathrm{PSL}(2,27), \mathrm{PSU}(3,4) \newline \hline
14 &6 &1092 &604800& \mathrm{PSL}(2,13), A\_7, \mathrm{PSU}(3,3), A\_8, \mathrm{Sz}(8), J\_2 
\end{array} $$ 



On the finite simple groups with an irreducible complex representation of a given dimension
I will write this an an answer, though the answer to the basic question is provided by one of the oldest results in finite group theory.


There is a theorem of C. Jordan, proved in the 19th century, that there is a function $f : \mathbb{N} \to \mathbb{N}$ such that whenever $n \in \mathbb{N}$ and $G$ is a finite subgroup of ${\rm GL}(n,\mathbb{C}),$ then there is an Abelian subgroup $A \lhd G$ with $[G:A] \leq f(n)$.


Hence, in particular, for any integer $d >1$, there are only finitely many finite simple groups $G$ with a complex irreducible representation of degree $d$.


Explicit bounds for Jordan's theorem were given by many people over the years, such as Frobenius and Blichfeldt.


The bounds given prior to the classification of finite simple groups (CFSG) were far from optimal, but, using CFSG, B. Weisfeiler outlined a proof of a close to optimal bound in sufficiently large $n$.


Unfortunately, Weisfeiler disappeared before his work was published in complete form. 


Recently, M.J. Collins published a complete proof with the optimal bound $f(n) = (n+1)!$ 
for sufficiently large $n$ ( I think $n > 72$ will do, if my memory is correct). I think Collins also gives the maximal value of $f(n)$ (with the optimal choice of function $f$) for smaller values of $n$, but this is not in general an answer to the first question.


I am not sure of the largest value of $n$ such that all finite irreducible simple subgroups of ${\rm GL}(n,\mathbb{C})$ are known. I believe that people like G.Malle and G.Hiss have done this up to $n =19$ - ( Later edit- see Derek Holt's answer, which indicates a much higher bound due to those authors) ( the cases $n \leq 11$ were understood before CFSG by the work of many authors).


