Dodecahedron atop icosidodecahedron

Dodecahedron atop icosi-dodecahedron, or doaid, is a convex regular-faced polytope segmentochoron (designated as K-4.77 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and an icosidodecahedron as bases, connected by 20 tetrahedra and 12 pentagonal antiprisms.

It is a segment of the hexacosichoron, with the icosidodecahedral base lying on the hexacosichoron's equator.

Vertex coordinates
The vertices of a dodecahedron atop icosidodecahedron segmentochoron of edge length 1 are given by:
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac12,\,0,\,\frac{1+\sqrt5}{4}\right),$$ and all permutations of first three coordinates
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right)$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,0\right)$$ and all permutations of first three coordinates
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,0,\,0\right)$$ and all permutations of first three coordinates