Let $S_{n}$ denote the permutation group on $n$ letters and $G\subset S_{n}$ a transitive subgroup. The inclusion of $G$ in $S_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a generating set of invariant polynomials and relations among these polynomials we may realize the quotient space $\mathbb{C}^{n}/G$ as an algebraic variety.

I have noticed that for $n=2,3,4$ such quotients are always complete intersections. That is, the difference between the order of a minimal generating set for the invariant polynomials, and the order of a minimal generating set for the relations, is equal to $n=\mathrm{dim}(\mathbb{C}^{n}/G)$.

I would like to know if this fact persists for all $n$. If so, what is the property of the group action which makes the quotient a complete intersection? If not what is a counterexample?