Representation theory applied to a group besides the symmetric group A very common application of representation theory is the representation of the symmetric group. There are a lot to be said about the representation of the symmetric group.

I am wondering if the representation of a more exotic finite group $G$ has been studied deeply? 

On the Wikipedia page of the representation of the symmetric group we can see that it has application in symmetric functions problems and in quantum mechanic. If you have an example of such a group $G$ are there applications of the representation of this group $G$? 
 A: If $G$ is a finite group there is a bijection between conjugacy classes and irreducible representations but it's not clear how to do it in practice. The only case is for $S_n$ where everything is really explicit, and indexed by partitions of $n$.
However, it was a great advance to discover that one can realize such a bijection geometrically, by Springer theory. Representations of a general Weyl group can also be constructed using similar geometric methods, but in this case there is no natural bijection anymore (as Stephen mentionned in the comments).
The idea is that there is a geometric action of the Weyl group on the cohomology of the Springer fibers, i.e the fibers of the Springer resolution of the nilpotent cone. To my knowledge, people do not study intensively the representation theory of the Weyl groups today, but this was the starting point of a really fruitful theory which helped to understand a lot of algebras (for example the construction also works for universal enveloping algebras and their affine version) using some geometric constructions. A reference for that is the book by Chriss and Ginzburg, "Complex geometry and representation theory". 
Also as mentioned in the comment by Tobias Kildetoft, variations and deformations of $\Bbb C[W]$ are still studied, for example Hecke algebras, rational Cherednik algebras, ... 
A: The book Group Representations in Probability and Statistics, by Diaconis, is all about applications of group representations.  
