The only simple groups having an irreducible character of degree 3 are $A\_5$ and $PSL\_2(7)$. What is it know for degrees 4 or 5? That is, do we know what simple groups posses an irreducible charcter of degree 4? of degree 5?



What simple groups have irreducible characters of degree 4 or 5?
I think the best modern source for this information is the paper:


G. Hiss and G. Malle.
Low-dimensional representations of quasi-simple groups.
LMS J. Comput. Math. 4 (2001), 22-63.
Corrigenda: LMS J. Comput. Math. 5 (2002), 95-126.


This lists all irreducible representations (in characteristic 0 and coprime to the defining characteristic of the group, if any) of all finite quasisimple groups (perfect groups with $G/Z(G)$ simple) up to dimension 250.


