Prismatorhombated hexacosichoron

The prismatorhombated hexacosichoron, or prix, also commonly called the runcitruncated 120-cell, is a convex uniform polychoron that consists of 1200 triangular prisms, 720 decagonal prisms, 600 cuboctahedra, and 120 truncated dodecahedra. 1 triangular prism, 2 decagonal prisms, 1 cuboctahedron, and 1 truncated dodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hecatonicosachoron.

Vertex coordinates
The vertices of a prismatorhombated hexacosichoron of edge length 1 are given by all permutations of: Plus all even permutations of:
 * $$\left(0,\,±1,\,±\frac{7+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±3\frac{2+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac{9+4\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±3\frac{2+\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(0,\,±\frac12,\,±\frac{17+9\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{13+7\sqrt5}{4},\,±3\frac{2+\sqrt5}{2}\right),$$
 * $$\left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{19+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,±3\frac{1+\sqrt5}{4},\,±\frac{17+7\sqrt5}{4},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±(3+\sqrt5),\,±\frac{11+5\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±\frac{5+3\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{19+7\sqrt5}{4},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{17+9\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{17+7\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±(3+\sqrt5)\right),$$
 * $$\left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{2+\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{17+9\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{9+4\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{7+3\sqrt5}{2},\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±\frac{5+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{19+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±(2+\sqrt5),\,±3\frac{2+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{17+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{7+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±(2+\sqrt5)\right).$$

Semi-uniform variant
The prismatorhombated hexacosichoron has a semi-uniform variant of the form x5y3o3z that maintains its full symmetry. This variant uses 120 truncated dodecahedra of form x5y3o, 600 rhombitetratetrahedra of form y3o3z, 1200 triangular prisms of form x z3o, and 720 dipentagonal prisms of form z x5y as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a5b3o3c), its circumradius is given by $$\sqrt{\frac{14a^2+21b^2+3c^2+33ab+11ac+14bc+(6a^2+9b^2+c^2+15ab+5ac+6bc)\sqrt5}{2}}$$.

Related polychora
The prismatorhombated hexacosichoron is the colonel of a 3-member regiment that also includes the small prismatohexacosihecatonicosihecatonicosachoron and the small rhombiprismic hexacosihecatonicosachoron.

The segmentochoron truncated dodecahedron atop great rhombicosidodecahedron can be obtained as a cap of the prismatorhombated hexacosichoron.