I have the following problem:
Let G be the nonabelian group of order 57.
(a.) How many 1-dimensional characters does G have?
(b.) What are the dimensions (aka degrees) of the other irreducible characters of G?
For part (a) I have already found that $G$ has 3 irreducible representations of degree 1 by analyzing the Sylow subgroups of $G$, realizing that the 19-Sylow is normal and the 3-Sylow's are not. Hence $G$ has three 1-dimensional characters.
However for part (b) I am not so sure what to do. I know two things that might be useful:
- The number of irreducible representations of $G$ must equal the number of conjugacy classes of $G$.
- The sum of the squares of the degrees of the irreducible representations of $G$ must equal the order of $G$.
So in particular the sum of the squares of the degrees of the remaining representations must equal $57-3 =54$ and none of those may be degree one. However this alone is not enough to give the answer as there are multiple ways of expressing 54 as a sum of squares, for example $54= 36+9+9 = 25+16+9+4$.
Any help would be appreciated!