There is a 6 dimensional multiplet belonging to an irreducible representation of a unitary group of rank less than 3. How does one check if the states $|i\rangle$ belong to spin 5/2 representation of $SU(2)$ or they belong to 6 dimensional representation of $SU(3)$?
For a given choice of basis states of the Hilbert space, one can, in principle, obtain a representation of any group, the dimension of the representation being the same as that of the Hilbert space. Just by knowing the basis states, I think it may not be possible to identify which group has been represented. Is my conclusion correct, or there is some point I am missing?