Finite symmetry group/Tetrahedron from numerical condition/Fact

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

Then ${\displaystyle {}G}$ is the tetrahedral group,

and isomorphic to the alternating group ${\displaystyle {}A_{4}}$.