Finite symmetry group/Tetrahedron from numerical condition/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 ${}(2,3,3)$.

Then ${}G$ is the tetrahedral group,

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

Proof, Write another proof