Finite symmetry group/Cube from numerical condition/Permutation group/Fact

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of the group of proper linear isometries in ${}\mathbb {R} ^{3}$ of type ${}(2,3,4)$.

Then ${}G$ is isomorphic to the

permutation group ${}S_{4}$, and to the

cube group.

Proof, Write another proof