I'm trying to prove that an 8-dimensional module for S4 over a field of complex numbers has a 2 dimensional submodule, I know that The character table says every irreducible representation of S_4 has dimension 1, 2, or 3.
finite dimensional representation of S_4 is a direct sum of irreducible.so I'm trying to consider all possible ways of writing 8 as a sum of 1, 2, and 3, but I'm stucking, I'll be thankful for help to finish