Proper symmetry group/Finite subgroup/Two classes of semiaxes/Cyclic/Fact

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of order ${}n$ of the group of proper linear isometries of ${}\mathbb {R} ^{3}$, with two different classes of semiaxes for ${}G$.

Then ${}G$ is the

cyclic group

of rotations around one fixed rotation axis for multiples of the angle ${}2\pi /n$.

Proof, Write another proof