Proper symmetry group/Finite subgroup/Three classes of semiaxes/2/2/Dihedral/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 ${}(2,2,k)$.

Then ${}G$ is

isomorphic to the dihedral group

${}D_{k}$.

Proof, Write another proof