For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation of the compact symplectic group $Sp_n$.

There is a $3$-dimensional real representation naturally associated to such a space. This can be given by the action of the isometry group permuting the holonomy subbundles within their common $Sp_nSp_1$-enlargement, or equivalently by rotating the $2$-sphere of parallel complex structures. This is summarized in the exact sequence
\begin{equation*}
Tri(\mathcal{M},g) \to Isom(\mathcal{M},g) \stackrel{\phi}\to SO(Im \mathbb{H})
\end{equation*}
where the middle entry is the isometry group and on the left are the isometries that preserve all complex structures (*tri-holomorphic*). I would like to know about the representation $\phi$.

A Killing vector field on a compact Ricci-flat space is parallel. Therefore, provided the dimension is larger than one, the isometry group of a locally-irreducible compact Ricci-flat space is discrete. It is furthermore finite by compactness.

The finite subgroups of $SO_3$ are known to be of various forms: cyclic groups, dihedral groups, and the rotation groups of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron.

(Fundamental domains of $C_4$, $D_4$ and the three polyhedral subgroups, images from Wikipedia.)

**Q)** Are examples known of compact hyper-Kähler manifolds that realise all of these types of subgroup through $\phi$?

I would appreciate any related remarks, as I have not been able to deduce much from the examples of compact hyper-Kähler manifolds that I have seen.

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