Square antiwedge

The square antiwedge, or squaw, also sometimes called the square gyrobicupolic ring, is a CRF segmentochoron (designated K-4.64 on Richard Klitzing's list). It consists of 1 square antiprism, 2 square cupolas, and 8 square pyramids.

The square antiwedge can be thought of as a piece of the larger segmentochoron cuboctahedron atop small rhombicuboctahedron, with one base square being a face of the cuboctahedron, and the opposite square cupola being part of the small rhombicuboctahedron.

Vertex coordinates
The vertices of a square antiwedge with edge length 1 are given by:
 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4},\,\sqrt{\frac{4-\sqrt2}{8}}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4},\,\sqrt{\frac{4-\sqrt2}{8}}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4},\,\sqrt{\frac{4-\sqrt2}{8}}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,0,\,0\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,0,\,0\right).$$

Representations
A square antiwedge has the following Coxeter diagrams:


 * os2xo8os&#x (full symmetry)
 * xxo4oxx&#x (BC2 symmetry only, seen with square atop gyro square cupola)