linear representation of the mapping class group ${\operatorname{Mod}}(S_G)$. This representation takes its values in the centralizer of $G$ in the symplectic group ${\operatorname{Sp}}(H^1(S,{{\mathbb Q}}))$ and respects the isotypical decomposition of $H^1(S,{{\mathbb Q}})$ into irreducible ${{\mathbb Q}}G$-modules. Then the above minimality property implies that when $S_G$ has genus $\ge 3$, a ${{\mathbb Q}}G$-isotypical summand of $H^1(S,{{\mathbb Q}})$ is essentially also isotypical for the representation of ${\operatorname{Mod}}(S)^G$ (to be more precise, isotypical for the *connected monodromy group*, that is, the identity component of the Zariski closure of the image of this representation). In particular, the subspace of $H^1(S,{{\mathbb Q}})$ spanned by the finite ${\operatorname{Mod}}(S)^G$-orbits is a sum of ${{\mathbb Q}}G$-isotypical summands. This subspace comes with a natural polarizable Hodge structure which is independent of any complex structure on $S$, but let us add Putman and Wieland conjecture that this subspace is trivial.

Our main applications are through the use of Theorem \[topology1\]. For example, Corollary \[cor:primmon\] states that in case a ${{\mathbb Q}}G$-isotypical summand is not fixed by all the $G$-invariant multi-Dehn twists, and the genus of the quotient curve is at least $3$, then the connected monodromy group acts irreducibly on that summand. Theorem \[topology1\] gives rise to a number of 