Small icosicosahedron

The small icosicosahedron, se, or compound of five octahedra is a uniform polyhedron compound. It consists of 40 triangles which form 20 coplanar pairs, combining into golden hexagrams. 4 triangles join at each vertex.

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.

It can be derived as a rectified chiricosahedron. It is also related to the icosicosahedron. If each stella octangula in the icosicosahedron is replaced with the intersection of the two tetrahedra (an octahedron), the result is a small icosicosahedron.

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

Vertex coordinates
The vertices of a small icosicosahedron of edge length 1 are given by all permutations of: Plus all even permutations of:
 * $$\left(\pm\frac{\sqrt2}{2},\,0,\,0\right),$$
 * $$\left(\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2+\sqrt{10}}{8},\,\pm\frac{\sqrt{10}-\sqrt2}{8}\right).$$