Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the group $SO(3)$ of rotations of $\mathbb{R}^3$. The *octahedral group* $Oct$ is the symmetry group of the (regular) octahedron, i.e., it is the maximal subgroup of $SO(3)$ that maps the octahedron to itself. The *binary octahedral group* $2Oct$ is simply the double cover of the octahedral group in the spin group.

**Question**: what is the quotient $Spin(3)/2Oct$ as a topological space?

Equivalently, what is the topology of $SO(3)/Oct$? (And is there a natural geometry associated with this space?) Note that $Oct$ is not a normal subgroup of $SO(3)$, hence the quotient does not exist in the sense of groups---only in the sense of topological manifolds.

A closely related example is the quotient of $Spin(3)$ by the *icosahedral* group, i.e., the symmetries of the icosahedron in $\mathbb{R}^3$. In that case, the quotient is the PoincarĂ© sphere. I am basically looking for the analogous object in the octahedral case.

Thanks!