Octahedral space

A space obtained from an octahedron by identifying its opposite triangular faces, positioned at an angle of $\pi/3$ to each other. An octahedral space is a three-dimensional manifold and is the orbit space of the action of a binary octahedron group on a three-dimensional sphere. It can be identified with a cube space obtained in an analogous way. The one-dimensional Betti group of an octahedral space is a group of order three.