Square pucofastegium

The square pucofastegium, or squipuf, also sometimes called the square magnabicupolic ring, is a CRF segmentochoron (designated K-4.105 on Richard Klitzing's list). It consists of 4 triangular prisms, 4 square pyramids, 2 square cupolas, and 1 octagonal prism.

The square pucofastegium occurs as the square-first cap of the small rhombated tesseract. In fact diminishing 8 such caps from the small rhombated tesseract results in the octagonal duoprism.

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

Representations
A square pucofastegium has the following Coxeter diagrams:


 * ox ox4xx&#x (full symmetry)
 * xxx4oxx&#x (BC2 symmetry only, seen with octagon atop square cupola)