This is a particular case of rings of invariants of permutation groups. The $n=|G|$ elementary symmetric polynomials $\sigma_k$ form a system of parameters for $R:=\mathbb{C}[V_G]^G$, the ring in question, and $R$ is a free module of rank $(n-1)!$ over the subring generated by the $\sigma_k$.

See Sect. 2.7 of B.Sturmfels' "Algorithms in invariant theory" for details.

This is a particular case of rings of invariants of permutation groups. The $n=|G|$ elementary symmetric polynomials $\sigma_k$ form a system of parameters for $R:=\mathbb{C}[V_G]^G$, the ring in question, and $R$ is a free module of rank $(n-1)!$ over the subring generated by the $\sigma_k$.

See Sect. 2.7 of B.Sturmfels' "Algorithms in invariant theory", ISBN 978-3-211-77416-8, for details.