Pentagonal-heptagonal duoprismatic prism

The pentagonal-heptagonal duoprismatic prism or pehep, also known as the pentagonal-heptagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal-heptagonal duoprisms, 5 square-heptagonal duoprisms, and 7 square-pentagonal duoprisms. Each vertex joins 2 square-pentagonal duoprisms, 2 square-heptagonal duoprisms, and 1 pentagonal-heptagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates
The vertices of a pentagonal-heptagonal duoprismatic prism of edge length 2sin(π/7) are given by: where j = 2, 4, 6.
 * $$\left(0,\,2\sqrt{\frac{5+\sqrt5}{10}}\sin\frac\pi7,\,1,\,0,\,±\sin\frac\pi7\right),$$
 * $$\left(±\frac{(1+\sqrt5)\sin\frac\pi7}2,\,\sqrt{\frac{5-\sqrt5}{10}}\sin\frac\pi7,\,1,\,0,\,±\sin\frac\pi7\right),$$
 * $$\left(±\sin\frac\pi7,\,-\sqrt{\frac{5+2\sqrt5}5}\sin\frac\pi7,\,1,\,0,\,±\sin\frac\pi7\right),$$
 * $$\left(0,\,2\sqrt{\frac{5+\sqrt5}{10}}\sin\frac\pi7,\,\cos\frac{j\pi}7,\,±\sin\frac{j\pi}7,\,±\sin\frac\pi7\right),$$
 * $$\left(±\frac{(1+\sqrt5)\sin\frac\pi7}2,\,\sqrt{\frac{5-\sqrt5}{10}}\sin\frac\pi7,\,\cos\frac{j\pi}7,\,±\sin\frac{j\pi}7,\,±\sin\frac\pi7\right),$$
 * $$\left(±\sin\frac\pi7,\,-\sqrt{\frac{5+2\sqrt5}5}\sin\frac\pi7,\,\cos\frac{j\pi}7,\,±\sin\frac{j\pi}7,\,±\sin\frac\pi7\right),$$

Representations
A pentagonal-heptagonal duoprismatic prism has the following Coxeter diagrams:
 * x x5o x7o (full symmetry)
 * xx5oo xx7oo&#x (pentagonal-heptagonal duoprism atop pentagonal-heptagonal duoprism)