1. Is there an equivalent of the Casimir operator for an irreducible representation of a finite group?
2. Given an invariant operator of a certain group, can I check if it is invariant under only that group (and it's subgroups)? Alternatively, given a group, is there a way I can construct an operator invariant under only that group and nothing bigger?

Basically I am trying to construct Hamiltonians that are invariant under specific finite groups. This was my original question on Physics stack exchange.