Invariants of finite groups

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 $G$?

• Finite reflection groups are some of the most well behaved objects you can possibly imagine. Have you read Chevalley's paper proving algebraic independence in that case? The proof is completely elementary. I was an undergraduate when I first read it and it was mystifying to me that you could do so much using so little. I don't know the answer to this but I don't expect it to have much to do with reflection groups. – Matt Samuel Dec 2 '15 at 4:24