 Quadratic cone invariant ring

The scalar sign action of a cyclic group of order two on ${\mathbb{C}}^2$ has invariant ring $\mathbb{C}[X^2,XY,Y^2]\cong \mathbb{C}[a,b,c]/(ac-b^2)$. It is not factorial: $a,b,c$ are pairwise nonassociate irreducibles, since every nonconstant invariant has degree at least two, while $ac=b^2$ gives two distinct factorizations.