Irreducible groups with submultiplicative spectrum

Let be a finite group of matrices. The spectrum σ is submultiplicative on if σ(ST) σ(S)σ(T)={λμ:λ σ(S),μ σ(T)}for every S, . We construct examples of irreducible groups of matrices with submultiplicative spectrum over vector spaces of all possible finite dimensions. We show that in the case of even dimensions the divisibility by 8 is required for irreducibility of the group.