Finite symmetry group/One class of semiaxes/Numerical condition/Exercise

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of the group of proper linear isometries in ${}\mathbb {R} ^{3}$. Suppose that there is only one class of semiaxes ${}K$. What numerical relation would hold between ${}{\#\left(G\right)}$, ${}{\#\left(K\right)}$, and ${}{\#\left(G_{H}\right)}$ (${}H\in K$)? Conclude that such a symmetry group does not exist.

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