There is a lot of information about classical/linear invariants of finite groups.
But does it lead to general invariants of group (for example, when we consider some action of our group on finite set)? And, if the answer is yes - how can these invariants can be derived (like invariants of action Rubic's rotations on the Rubic's cube configurations, or parity invariants in 15 puzzle)?
In particular, I need general invariants of the product of two symmetric groups.