Pentagonal pucofastegium

The pentagonal pucofastegium, or pepuf, also sometimes called the pentagonal magnabicupolic ring, is a CRF segmentochoron (designated K-4.165 on Richard Klitzing's list). It consists of 5 triangular prisms, 5 square pyramids, 2 pentagonal cupolas, and 1 decagonal prism.

The pentagonal pucofastegium occurs as the pentagon-first cap of the small rhombated hecatonicosachoron.

Vertex coordinates
The vertices of a pentagonal pucofastegium with edge length 1 are given by:
 * $$\left(±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,\sqrt{\frac{5-2\sqrt5}{20}}\right),$$
 * $$\left(±\frac{1+\sqrt5}4,\,-\sqrt{\frac{5+\sqrt5}{40}},\,0,\,\sqrt{\frac{5-2\sqrt5}{20}}\right),$$
 * $$\left(0,\,-\sqrt{\frac{5+\sqrt5}{10}},\,0,\,\sqrt{\frac{5-2\sqrt5}{20}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12,\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12,\,0\right).$$

Representations
A pentagonal pucofastegium has the following Coxeter diagrams:


 * ox ox5xx&#x (full symmetry)
 * xxx5oxx&#x (H2 symmetry only, seen with decagon atop pentagonal cupola)