Prismatorhombated hecatonicosachoron

The prismatorhombated hecatonicosachoron, or prahi, also commonly called the runcitruncated 600-cell, is a convex uniform polychoron that consists of 720 pentagonal prisms, 1200 hexagonal prisms, 600 truncated tetrahedra, and 120 small rhombicosidodecahedra. 1 pentagonal prism, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 small rhombicosidodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hexacosichoron.

Vertex coordinates
The vertices of a prismatorhombated hecatonicosachoron of edge length 1 are given by all permutations of: Plus all even permutations of:
 * $$\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{7+4\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac{8+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{7+\sqrt5}{4},\,±\frac{11+5\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{7+5\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac12,\,±\frac{9+5\sqrt5}{4},\,±5\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{7+4\sqrt5}{2},\,±\frac{5+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±\frac{7+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±(3+\sqrt5)\right),$$
 * $$\left(0,\,±\frac{5+3\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{2},\,±\frac{7+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{11+5\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{4+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±3\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±3\frac{3+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac12,\,±(2+\sqrt5),\,±\frac{11+5\sqrt5}{4},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{7+4\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{13+5\sqrt5}{4},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{3+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{1+\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±5\frac{3+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4},\,±\frac{7+5\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{13+5\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±5\frac{3+\sqrt5}{4},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{2},\,±\frac{4+\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{7+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{7+5\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{7+5\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{5+3\sqrt5}{2}\right).$$

Semi-uniform variant
The prismatorhombated hecatonicosachoron has a semi-uniform variant of the form x5o3y3z that maintains its full symmetry. This variant uses 120 small rhombicosidodecahedra of form x5o3y, 600 truncated tetrahedra of form z3y3o, 720 pentagonal prisms of form z x5o, and 1200 ditrigonal prisms of form x y3z as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a5o3b3c), its circumradius is given by $$\sqrt{\frac{14a^2+10b^2+3c^2+22ab+11ac+10bc+(6a^2+4b^2+c^2+10ab+5ac+4bc)\sqrt5}{2} }$$.

Related polychora
The prismatorhombated hecatonicosachoron is the colonel of a 3-member regiment that also includes the small prismatohexacosidishecatonicosachoron and the small rhombiprismic dishecatonicosachoron.