Character Theory of Finite Groups- Applications

Hi everyone.

For my algebra class I have to give a 50 minute talk on character theory of finite groups. The first 30 minutes I will build up the theory and give some examples. I will also state some of the big group theory results that have proofs using character theory. The last 20 minutes the professor would like me to talk about applications of character theory outside of group theory. Preferably even outside of pure algebra.

At first I began looking at harmonic analysis. However, that draws more on representation theory. I have looked at a few books (Isaacs' Character Theory is my main reference) and everything is group theory.

So I'm turning to the forum for ideas.