Chiricosahedron

The chiricosahedron, ki, or compound of five tetrahedra is a uniform polyhedron compound. It consists of 20 triangles, 3 joining at each vertex. As the name suggests, it is chiral, and has faces in planes parallel to those of the convex icosahedron.

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 tetrahedral pentachoroorthowedge, which is seven-dimensional.

Vertex coordinates
Coordinates for the vertices of a chiricosahedron of edge length 1 are given by:

plus all even permutations of:
 * $$\left(\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2}{4}\right),$$
 * $$\left(\pm\frac{\sqrt2+\sqrt{10}}{8},\,\pm\frac{\sqrt{10}-\sqrt2}{8},\,0\right).$$

Related polyhedra
The icosicosahedron is a compound of the two opposite chiral forms of the chiricosahedron.