Invariants of finite groups

Question

Let $G$ be a finite group acting linearly on $\mathbb{C}^n$ and $\mathbb{C}[X]^G$ be the ring of invariant polynomials. If $G$ is a group generated by reflections, this ring is generated by $n$ algebraically independent polynomials. If $G$ is not a finite reflection group, the ring of invariant polnomials is a finite module over a ring generated by $n$ algebraically independent polynomials (called primary invariants). I would like to understand the role of these primary invariants. Do they come from some group generated by reflection $G'$ containing $G$?