Proper isometry group/Finite subgroup/Numerical properties/Fact

Let ${\displaystyle {}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}}$ be a finite subgroup of order ${\displaystyle {}n}$ of the group of proper, linear isometries of ${\displaystyle {}\mathbb {R} ^{3}}$. Let ${\displaystyle {}K_{1},\ldots ,K_{m}}$ be the different classes of semiaxes of ${\displaystyle {}G}$, and for every class, let ${\displaystyle {}n_{i}}$, ${\displaystyle {}i=1,\ldots ,m}$, be the order of the stabilizer group ${\displaystyle {}G_{H}}$, ${\displaystyle {}H\in K_{i}}$, which are, due to fact, independent of ${\displaystyle {}H\in K_{i}}$.