Let $X$ be a compact Riemann surface and $G = Aut(X)$ be its group of automorphisms (biholomorphisms between $X$ and $X$). It is known that $G$ acts on the space $Harm(X)$ of all harmonic forms and also on the space $Omega(X)$ of all holomorphic forms.

We know that $Harm(X)$ is a direct sum of $Omega(X)$ and its conjugate.

Now, if we know the representation of $G$ on $Harm(X)$ (by this I mean we have a matrix for each element of $G$), how can we find matrices for the representation of $G$ on $Omega(X)\ ?$

ADDED: I am assuming we have no information about $Omega(X)$ or $Harm(X)$. We just know the genus of $X$, $G$ (with a multiplication table) and the representation of $G$ on $Harm(X)$ (the matrices associated to each element of the group).