Computing resolutions of quotient singularities

Let $Gsubseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $Xrightarrow mathbb{C}^n/G$, which is based just on the geometry of the singularity $mathbb{C}^n/G$, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions $Xrightarrow mathbb{C}^3/G$ for all $Gsubseteq GL(3)$ with the aforementioned property and of order $|G|leq 12$. We also provide examples in dimension 4.