This answer of 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.

$$\begin{array}{c|c|c|c|c} d &n&min&max & G \newline \hline 3 &2&60&168&A_5, \mathrm{PSL}(2,7) \newline \hline 4 &1&60&60&A_5 \newline \hline 5 &4 &60 &25920&A_5, A_6, \mathrm{PSL}(2,11), \mathrm{PSp}(4,3) \newline \hline 6 &4 &168 &25920&\mathrm{PSL}(2,7), A_7, \mathrm{PSU}(3,3), \mathrm{PSp}(4,3) \newline \hline 7 &5 &168 &20160&\mathrm{PSL}(2,7), \mathrm{PSL}(2,8), \mathrm{PSL}(2,13), \mathrm{PSU}(3,3), A_8 \newline \hline 8 &4 &168 &181440& \mathrm{PSL}(2,7), A_6, \mathrm{PSL}(2,8), A_9 \newline \hline 9 &4 &360 &3420& A_6, \mathrm{PSL}(2,8), \mathrm{PSL}(2,17), \mathrm{PSL}(2,19) \newline \hline 10 &5 &360 &25920& A_6, \mathrm{PSL}(2,11), A_7, M_{11}, \mathrm{PSp}(4,3) \newline \hline 11 &4 &660 &95040& \mathrm{PSL}(2,11), \mathrm{PSL}(2,23), M_{11}, M_{12} \newline \hline 12 &4 &660 &62400& \mathrm{PSL}(2,11),\mathrm{PSL}(2,13), \mathrm{PSL}(3,3), \mathrm{PSU}(3,4) \newline \hline 13 &5 &1092 &62400& \mathrm{PSL}(2,13), \mathrm{PSL}(3,3), \mathrm{PSL}(2,25), \mathrm{PSL}(2,27), \mathrm{PSU}(3,4) \newline \hline 14 &6 &1092 &604800& \mathrm{PSL}(2,13), A_7, \mathrm{PSU}(3,3), A_8, \mathrm{Sz}(8), J_2 \end{array} $$