Icosidodecahedron atop truncated icosahedron

Icosidodecahedron atop truncated icosahedron, or idati, is a CRF segmentochoron (designated K-4.158 on Richard Klitzing's list). As the name suggests, it consists of an icosidodecahedron and a truncated icosahedron as bases, connected by 12 pentagonal prisms and 20 triangular cupolas.

It can be obtained as an icosidodecahedron-first cap of the small rhombated hexacosichoron.

Vertex coordinates
The vertices of an icosidodecahedron atop truncated icosahedron segmentochoron of edge length 1 are given by all permutations of the first three coordinates of: Plus all even permutations of the first three coordinates of:
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12,\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,0\right).$$