Given a finite group $G$ acting on a vector space $V$ (in characteristic zero), is there a known algorithm to find a $G$-invariant basis for $V$? As a concrete example to work with, one of the systems I'm looking at is the dihedral group $D_6$ acting on $\mathbb{Q}^{10}$.

**EDIT**: All the representations I'm working with are going to be permutation representations.