Applications of representation theory of finite group in physics? Well, I have just finished my study on basic representation theory of finite group from a pure math course. After tortured a lot by abstract constructions, I would like to know the real application of this theory, however, it seems to me not many topics in physics are related to representations of finite group. Anyone could provide some examples on the application?
 A: The moonshine phenomenon is a deep subject involving conformal field theory and string theory that is based on the observation of some relationships between the representation theory of finite groups of "big size"(but finite) and modular forms (obtained as partition functions of some CFTs).
References:
Miranda Cheng - Progress on moonshine
TASI lectures on moonshine
Miranda Cheng - Umbral Moonshine and String Theory
A: In the study of equilibrium configurations and vibrational properties of molecules, clusters  and solids, finite groups representations play a key role to extract all consequences of point symmetries of the average atomic positions.
Also electronic states in molecules, clusters and solids provide  a different example of applied representation group theory.
In both cases, the classification of vibrational and electronic states in terms of irreducible representation of the relevant symmetry group allows a complete characterization of such states and it is pivotal for the analysis of experimental data.
