Great rhombated cubic honeycomb

The great rhombated cubic honeycomb, or grich, also known as the cantitruncated cubic honeycomb, is a convex uniform honeycomb. 1 truncated octahedron, 1 cube, and 2 great rhombicuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the cantitruncation of the cubic honeycomb.

This honeycomb can be alternated into a snub rectified cubic honeycomb, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a cantic snub cubic honeycomb, which is also nonuniform.

Vertex coordinates
The vertices of a great rhombated cubic honeycomb of edge length 1 are given by all permutations of:


 * $$\left(±\frac12+(1+2\sqrt2)i,\,±\frac{1+\sqrt2}{2}+(1+2\sqrt2)j,\,±\frac{1+2\sqrt2}{2}+(1+2\sqrt2)k\right),$$

Where i, j, and k range over the integers.

Representations
A great rhombated cubic honeycomb has the following Coxeter diagrams:


 * x4x3x4o (regular)
 * x3x3x *b4x (S4 symmetry)
 * s4x3x4x (as alternated faceting)