Rhombihedron

The rhombihedron, rhom, or compound of five cubes is a uniform polyhedron compound. It consists of 30 squares. The vertices coincide in pairs, leading to 20 vertices where 6 squares join.

It has the same edges as the small ditrigonary icosidodecahedron.

This compound is sometimes considered to be regular, but it is not flag-transitive, despite the fact it is vertex, edge, and face-transitive. It is however regular if you consider conjugacies along with its other symmetries.

Its quotient prismatic equivalent is the cubic pentachoroorthowedge, which is seven-dimensional.

Vertex coordinates
The vertices of a rhombihedron of edge length 1 are given by: along with all even permutations of:
 * $$\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12\right),$$
 * $$\left(0,\,\pm\frac{\sqrt5-1}{4},\,\pm\frac{1+\sqrt5}{4}\right).$$