Dodecahedron atop small rhombicosidodecahedron

Dodecahedron atop small rhombicosidodecahedron, or doasrid, is a CRF segmentochoron (designated K-4.152 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and a small rhombicosidodecahedron as bases, connected by 20 tetrahedra, 30 triangular prisms, and 12 pentagonal prisms.

It is also sometimes referred to as a dodecahedral cupola, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

It can be obtained as a dodecahedron-first cap of the small disprismatohexacosihecatonicosachoron.

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