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][1] was my original question on Physics stack exchange. 

  [1]: https://physics.stackexchange.com/questions/131436/construction-of-a-spin-chain-hamiltonian-invariant-under-a-finite-subgroup-of-so?noredirect=1#comment268031_131436