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.

This compound is sometimes considered to be regular, but it is not flag-transitive, despite the fact it is vertex, edge, and face-transitive.

Vertex coordinates
The vertices of a chiricosahedron of edge length 1 are given by:

Plus all even permutations of:
 * (±$\sqrt{6}$/4, ±$\sqrt{6}$/4, ±$\sqrt{2}$/4)
 * (±($\sqrt{2}$+$\sqrt{2}$)/8, ±$\sqrt{2}$/4, ±($\sqrt{2}$–$\sqrt{10}$)/8)