Pentagonal-truncated tetrahedral duoprism

The pentagonal-truncated tetrahedral duoprism or petut is a convex uniform duoprism that consists of 5 truncated tetrahedral prisms, 4 pentagonal-hexagonal duoprisms, and 4 triangular-pentagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-pentagonal duoprism, and 2 pentagonal-hexagonal duoprisms.

Vertex coordinates
The vertices of a pentagonal-truncated tetrahedral duoprism of edge length 1 are given by all permutations and even sign changes of the last three coordinates of:
 * $$\left(0,\, \sqrt{\frac{5+\sqrt5}{10}},\,\frac{3\sqrt2}4,\,\frac{\sqrt2}4,\,\frac{\sqrt2}4\right),$$
 * $$\left(±\frac{1+\sqrt5}4,\, \sqrt{\frac{5-\sqrt5}{40}},\,\frac{3\sqrt2}4,\,\frac{\sqrt2}4,\,\frac{\sqrt2}4\right),$$
 * $$\left(±\frac12,\, -\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{3\sqrt2}4,\,\frac{\sqrt2}4,\,\frac{\sqrt2}4\right).$$