I am working on a project on applications of group theory, starting from point groups and molecular symmetry. Coming up on representations of each point groups, I started studying Representations of Finite Group. Also, I am reading upon Platonic Solids. For a cube, I determined the elements of the Rotational Symmetry group, of order 24. The elements of this group is a permutation of the four main diagonals of the cube. So, is the set of four diagonals representation of the cube? Can we use these four diagonals to find the elements of the total symmetry group of the cube? Applying Symmetry on Platonic solids to study its structure is okay, but what exactly does it tell us? Can it be extended to understand the nature and properties of something else? What is the essence of this study?
I need some good references for symmetric groups of Platonic solids. I want to do some problem solving (self - research) on this related topic. It can be computational, theoretical or simulation. But I am unable to determine the proper link between Platonic Solids and Group Theory( or rather can't find a problem statement).