Prismatorhombated hecatonicosachoric prism

The prismatorhombated hecatonicosachoric prism or prahip is a prismatic uniform polyteron that consists of 2 prismatorhombated hecatonicosachora, 120 small rhombicosidodecahedral prisms, 600 truncated tetrahedral prisms, 1200 square-hexagonal duoprisms, and 720 square-pentagonal duoprisms. 1 prismatorhombated hecatonicosachoron, 1 small rhombicosidodecahedral prism, 1 truncated tetrahedral prism, 1 square-pentagonal duoprism, and 2 square-hexagonal duoprisms join at each vertex. As the name suggests, it is a prism based on the prismatorhombated hecatonicosachoron, which also makes it a convex segmentoteron.

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