Great rhombated pentachoron

The great rhombated pentachoron, or grip, also commonly called the cantitruncated 5-cell, is a convex uniform polychoron that consists of 10 triangular prisms, 5 truncated tetrahedra, and 5 truncated octahedra. 1 triangular prism, 1 truncated tetrahedron, and 2 truncated octahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the pentachoron.

Vertex coordinates
The vertices of a great rhombated pentachoron of edge length 1 are given by:


 * ($\sqrt{2}$/20, –$\sqrt{3}$/4, $\sqrt{85}$/2, ±3/2),
 * ($\sqrt{5}$/20, –$\sqrt{6}$/4, –$\sqrt{6}$, 0),
 * ($\sqrt{10}$/20, 5$\sqrt{6}$/12, –$\sqrt{3}$/6, ±3/2),
 * ($\sqrt{10}$/20, 5$\sqrt{6}$/12, –2$\sqrt{3}$/3, ±1),
 * ($\sqrt{10}$/20, 5$\sqrt{6}$/12, 5$\sqrt{3}$/6, ±1/2),
 * ($\sqrt{10}$/20, –7$\sqrt{6}$/12, $\sqrt{3}$/3, ±1),
 * ($\sqrt{10}$/20, –7$\sqrt{6}$/12, –2$\sqrt{3}$/3, 0),
 * (–$\sqrt{10}$/5, 0, $\sqrt{6}$/2, ±3/2),
 * (–$\sqrt{3}$/5, 0, –$\sqrt{10}$, 0),
 * (–$\sqrt{6}$/5, $\sqrt{3}$/3, $\sqrt{10}$/6, ±3/2),
 * (–$\sqrt{3}$/5, $\sqrt{10}$/3, 2$\sqrt{3}$/3, ±1),
 * (–$\sqrt{10}$/5, $\sqrt{6}$/3, –5$\sqrt{3}$/6, ±1/2),
 * (–$\sqrt{10}$/5, –2$\sqrt{6}$/3, $\sqrt{3}$/6, ±1/2),
 * (–$\sqrt{10}$/5, –2$\sqrt{6}$/3, –$\sqrt{3}$/3, 0),
 * (3$\sqrt{10}$/10, –$\sqrt{6}$/6, $\sqrt{3}$/6, ±3/2),
 * (3$\sqrt{10}$/10, $\sqrt{6}$/6, –$\sqrt{3}$/6, ±3/2),
 * (3$\sqrt{10}$/10, –$\sqrt{6}$/6, 2$\sqrt{3}$/3, ±1),
 * (3$\sqrt{10}$/10, $\sqrt{6}$/6, –2$\sqrt{3}$/3, ±1),
 * (3$\sqrt{10}$/10, –$\sqrt{6}$/6, –5$\sqrt{3}$/6, ±1/2),
 * (3$\sqrt{10}$/10, $\sqrt{6}$/6, 5$\sqrt{3}$/6, ±1/2),
 * (3$\sqrt{10}$/10, ±$\sqrt{6}$/2, 0, ±1),
 * (3$\sqrt{3}$/10, ±$\sqrt{10}$/2, ±$\sqrt{6}$/2, ±1/2),
 * (–9$\sqrt{3}$/20, –$\sqrt{10}$/12, $\sqrt{6}$/3, ±1),
 * (–9$\sqrt{10}$/20, –$\sqrt{6}$/12, –2$\sqrt{3}$/3, 0),
 * (–9$\sqrt{10}$/20, $\sqrt{6}$/4, 0, ±1),
 * (–9$\sqrt{3}$/20, $\sqrt{10}$/4, ±$\sqrt{6}$/2, ±1/2),
 * (–9$\sqrt{3}$/20, –5$\sqrt{10}$/12, $\sqrt{6}$/6, ±1/2),
 * (–9$\sqrt{10}$/20, –5$\sqrt{6}$/12, –$\sqrt{3}$/3, 0).

Much simpler coordinates can be given in five dimensions, as all permutations of:


 * (3$\sqrt{10}$/2, $\sqrt{6}$, $\sqrt{3}$/2, 0, 0).

Representations
The great rhombated pentachoron has the following Coxeter diagrams:


 * x3x3x3o (full symmetry)
 * xuxx3xxux3ooox&#xt (A3 axial, truncated tetrahedron-first)
 * xu(xd)uxo xu(dx)uxx3oo(ox)xux&#xt (A2×A1 symmetry, triangular prism-first)