Great disprismatohexacosihecatonicosachoron

The great disprismatohexacosihecatonicosachoron, or gidpixhi, also commonly called the omnitruncated 120-cell, is a convex uniform polychoron that consists of 1200 hexagonal prisms, 720 decagonal prisms, 600 truncated octahedra, and 120 great rhombicosidodecahedra. 1 of each type of cell join at each vertex. It is the omnitruncate of the H4 family of uniform polychora, and could also be considered to be the omnitruncated 600-cell. It is therefore the most complex of the non-prismatic convex uniform polychora.

This polychoron can be alternated into a snub hecatonicosachoron, although it cannot be made uniform.

Vertex coordinates
Vertex coordinates for a great disprismatohexacosihecatonicosachoron of edge length 1 are given by all permutations of: plus all even permutations of:
 * $$\left(±\frac12,\,±\frac12,\,±\frac{4+3\sqrt5}{2},\,±\frac{12+5\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{7+4\sqrt5}{2},\,±\frac{11+4\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac{11+6\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac32,\,±\frac{9+4\sqrt5}{2},\,±\frac{9+4\sqrt5}{2}\right),$$
 * $$\left(±1,\,±1,\,±2(2+\sqrt5),\,±(5+2\sqrt5)\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{5+\sqrt5}{2},\,±2(2+\sqrt5),\,±2(2+\sqrt5)\right),$$
 * $$\left(±\frac{4+\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac{7+4\sqrt5}{2},\,±\frac{9+4\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+2\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+4\sqrt5}{2},\,±\frac{7+4\sqrt5}{2}\right),$$
 * $$\left(±(2+\sqrt5),\,±(2+\sqrt5),\,±(3+2\sqrt5),\,±2(2+\sqrt5)\right),$$
 * $$\left(±\frac12,\,±5\frac{3+\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{7+3\sqrt5}{2},\,±\frac{17+7\sqrt5}{4},\,±\frac{17+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±1,\,±\frac{7+5\sqrt5}{4},\,±\frac{23+11\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±3\frac{7+3\sqrt5}{4},\,±(3+2\sqrt5)\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{25+9\sqrt5}{4},\,±\frac{5+3\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{23+9\sqrt5}{4},\,±\frac{11+7\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{11+6\sqrt5}{2},\,±\frac{4+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\, ±\frac{7+\sqrt5}{4},\,±\frac{17+9\sqrt5}{4},\,±2(2+\sqrt5)\right),$$
 * $$\left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{25+9\sqrt5}{4},\,±(3+\sqrt5)\right),$$
 * $$\left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{23+11\sqrt5}{4},\,±\frac{5+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±(1+\sqrt5),\,±\frac{23+9\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±3\frac{3+\sqrt5}{4},\,±\frac{17+9\sqrt5}{4},\,±3\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±(2+\sqrt5),\,±\frac{19+9\sqrt5}{4},\,±\frac{17+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{11+6\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{5+\sqrt5}{4},\,±\frac{19+9\sqrt5}{4},\,±\frac{7+4\sqrt5}{2}\right),$$
 * $$\left(±1,\,±\frac{2+\sqrt5}{2},\,±\frac{25+9\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±1,\,±3\frac{1+\sqrt5}{4},\,±\frac{23+9\sqrt5}{4},\,±3\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{12+5\sqrt5}{2},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{4+\sqrt5}{2},\,±\frac{17+9\sqrt5}{4},\,±\frac{17+7\sqrt5}{4}\right),$$
 * $$\left(±1,\,±\frac{3+2\sqrt5}{2},\,±3\frac{7+3\sqrt5}{4},\,±5\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{13+5\sqrt5}{4},\,±\frac{7+4\sqrt5}{2},\,±3\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±2(2+\sqrt5),\,±\frac{17+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac32,\,±(2+\sqrt5),\,±\frac{23+11\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{11+6\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±(1+\sqrt5),\,±\frac{23+11\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{11+3\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±3\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±(5+2\sqrt5),\,±\frac{17+5\sqrt5}{4}\right),$$
 * $$\left(±\frac32,\,±\frac{2+\sqrt5}{2},\,±\frac{12+5\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),$$
 * $$\left(±\frac32,\,±\frac{3+\sqrt5}{2},\,±\frac{19+9\sqrt5}{4},\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±\frac32,\,±\frac{5+3\sqrt5}{4},\,±3\frac{7+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±(3+\sqrt5),\,±2(2+\sqrt5),\,±3\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{17+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{11+6\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},,\,±\frac{7+\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{23+11\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±(1+\sqrt5),\,±\frac{12+5\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±\frac{5+\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±\frac{17+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+\sqrt5}{4},\,±(2+\sqrt5),\,±\frac{11+4\sqrt5}{2},\,±5\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4},\,±(3+2\sqrt5),\,±\frac{17+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±5\frac{3+\sqrt5}{4},\,±\frac{11+7\sqrt5}{4},\,±2(2+\sqrt5)\right),$$
 * $$\left(±\frac{7+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{12+5\sqrt5}{2},\,±(2+\sqrt5)\right),$$
 * $$\left(±\frac{7+\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±(5+2\sqrt5),\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{7+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{11+4\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±(3+\sqrt5),\,±\frac{7+4\sqrt5}{2},\,±\frac{17+7\sqrt5}{4}\right),$$
 * $$\left(±3\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±5\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{25+9\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{2},\,±\frac{7+5\sqrt5}{4},\,±\frac{11+4\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{4+3\sqrt5}{2},\,±(5+2\sqrt5),\,±\frac{13+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±(3+2\sqrt5),\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{11+7\sqrt5}{4},\,±\frac{7+4\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{4+\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{25+9\sqrt5}{4}\right),$$
 * $$\left(±\frac{5+3\sqrt5}{4},\,±\frac{11+3\sqrt5}{4},\,±\frac{7+4\sqrt5}{2},\,±2(2+\sqrt5)\right),$$
 * $$\left(±\frac{4+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{11+4\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{4+\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{23+9\sqrt5}{4},\,±\frac{11+3\sqrt5}{4}\right),$$
 * $$\left(±\frac{4+\sqrt5}{2},\,±\frac{7+5\sqrt5}{4},\,±3\frac{7+3\sqrt5}{4},\,±(3+\sqrt5)\right),$$
 * $$\left(±(1+\sqrt5),\,±\frac{11+5\sqrt5}{4},\,±\frac{7+4\sqrt5}{2},\,±\frac{15+7\sqrt5}{4}\right),$$
 * $$\left(±(1+\sqrt5),\,±\frac{5+3\sqrt5}{2},\,±2(2+\sqrt5),\,±\frac{7+3\sqrt5}{2}\right),$$
 * $$\left(±\frac{7+3\sqrt5}{4},\,±(3+\sqrt5),\,±\frac{4+3\sqrt5}{2},\,±\frac{19+9\sqrt5}{4}\right),$$
 * $$\left(±\frac{7+3\sqrt5}{4},\,±\frac{5+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{23+9\sqrt5}{4}\right),$$
 * $$\left(±\frac{7+3\sqrt5}{4},\,±3\frac{3+\sqrt5}{4},\,±(3+2\sqrt5),\,±\frac{9+4\sqrt5}{2}\right),$$
 * $$\left(±\frac{5+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±3\frac{7+3\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+2\sqrt5}{2},\,±\frac{4+3\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+2\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±\frac{11+7\sqrt5}{4},\,±(5+2\sqrt5)\right),$$
 * $$\left(±\frac{3+2\sqrt5}{2},\,±\frac{11+3\sqrt5}{4},\,±\frac{5+3\sqrt5}{2},\,±\frac{19+9\sqrt5}{4}\right),$$
 * $$\left(±(2+\sqrt5),\,±\frac{4+3\sqrt5}{2},\,±\frac{17+9\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),$$
 * $$\left(±(2+\sqrt5),\,±\frac{7+5\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{11+7\sqrt5}{4}\right),$$
 * $$\left(±\frac{7+5\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{17+9\sqrt5}{4}\right).$$

Semi-uniform variant
The great disprismatohexacosihecatonicosachoron has a semi-uniform variant of the form a5b3c3d that maintains its full symmetry. This variant uses 120 great rhombicosidodecahedra of form a5b3c, 600 great rhombitetratetrahedra of form b3c3d, 720 dipentagonal prisms of form d a5b, and 1200 ditrigonal prisms of form a c3d as cells, with 4 edge lengths.

With edges of length a, b, c, and d (such that it forms a5b3c3d), its circumradius is given by $$\sqrt{\frac{14a^2+21b^2+10c^2+3d^2+33ab+22ac+11ad+28bc+14bd+10cd+(6a^2+9b^2+4c^2+d^2+15ab+10ac+5ad+12bc+6bd+4cd)\sqrt5}{2}}$$.