Dodecagonal-truncated icosahedral duoprism

The dodecagonal-truncated icosahedral duoprism or twati is a convex uniform duoprism that consists of 12 truncated icosahedral prisms, 20 hexagonal-dodecagonal duoprisms, and 12 pentagonal-dodecagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-dodecagonal duoprism, and 2 hexagonal-dodecagonal duoprisms.

Vertex coordinates
The vertices of a dodecagonal-truncated icosahedral duoprism of edge length 1 are given by all even permutations of the last three coordinates of:
 * $$\left(±\frac{1+\sqrt3}2,\,±\frac{1+\sqrt3}2,\,0,\,±\frac12,\,±3\frac{1+\sqrt5}4\right),$$
 * $$\left(±\frac{1+\sqrt3}2,\,±\frac{1+\sqrt3}2,\,±\frac12,\,±\frac{5+\sqrt5}4,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(±\frac{1+\sqrt3}2,\,±\frac{1+\sqrt3}2,\,±\frac{1+\sqrt5}4,\,±1,\,±\frac{2+\sqrt5}2\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt3}2,\,0,\,±\frac12,\,±3\frac{1+\sqrt5}4\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt3}2,\,±\frac12,\,±\frac{5+\sqrt5}4,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt3}2,\,±\frac{1+\sqrt5}4,\,±1,\,±\frac{2+\sqrt5}2\right),$$
 * $$\left(±\frac{2+\sqrt3}2,\,±\frac12,\,0,\,±\frac12,\,±3\frac{1+\sqrt5}4\right),$$
 * $$\left(±\frac{2+\sqrt3}2,\,±\frac12,\,±\frac12,\,±\frac{5+\sqrt5}4,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(±\frac{2+\sqrt3}2,\,±\frac12,\,±\frac{1+\sqrt5}4,\,±1,\,±\frac{2+\sqrt5}2\right).$$

Representations
A dodecagonal-truncated icosahedral duoprism has the following Coxeter diagrams:
 * x12o o5x3x (full symmetry)
 * x6x o5x3x (dodecagons as dihexagons)