Representation theory has very deep ties with physics, leading to incredibly profound and admittedly cool results such as the classification of particles in the Standard Model via mass and spin by looking at the unitary irreducible representations of the Poincaré group; however, most discussions on the topic are quite complicated as the result is pretty high-level (you can tell by the word salad I had to use to even mention the thing), which I would assume is in part due to the fact that we're dealing with Lie groups (and their Lie algebras) as opposed to finite groups.
Indeed, by comparison, the representation theory of finite groups is a lot easier to present, more self-contained, and is overall something that an undergrad in math should not have too many problems understanding given the right presentation. But I do feel a bit sad when I look at representations of Lie algebras having cool applications like the one mentioned above, whereas the representation theory of finite groups can be used to prove... Burnside's theorem on groups of order $p^aq^b$? That's rather underwhelming.
So this leads me to my question: are there any interesting physical applications of finite group characters/representations to physics? Even including those that are just ‘toy’ versions of equally interesting applications in the field of Lie algebra representations, whenever studying the toy example can lead to actual, significant insight towards the more complicated (and useful) situation—if any such examples exist.
