Multiplicity of eigenvalues of adjacency operator and dimension of represenations

Question

Let $G$ be a finite group, $S\subset G$ a symmetric subset of generators of $G$ (symmetric means $S=S^{-1}$), and $Cay(G,S)$ the corresponding Cayley graph. I'm trying to show that if the minimal dimension of non-trivial $G$-represenations is $m$, then every non-trivial eigenvalue of the adjancesy operator on the graph has multiplicity of $m$ at least. I tried to take an eigenvalue with multiplicity of $m$ and use the corresponding eigenvectors to construct a presentation of dimension $m$, but it didn't work (probably because it is a stronger claim than the question itself). Any other ideas will be welcomed.