The maximal finite subgroups of \(3\times 3\) integral matrices (Q2719950)

Two finite subgroups \(G\) and \(G'\) of the general linear group \(\text{GL}_3(\mathbb{Z})\) over the integers \(\mathbb{Z}\) are said to be arithmetically equivalent (resp., geometrically equivalent) if there exists \(T\in\text{GL}_3(\mathbb{Z})\) (resp., \(T\in\text{GL}_3(\mathbb{Q})\)) such that \(T^{-1}GT=G'\). The authors investigate the maximal finite subgroups of \(\text{GL}_3(\mathbb{Z})\), prove that there are four arithmetically equivalent classes and two geometrically equivalent classes, and give a representative for each class.