Augmented truncated dodecahedron

The augmented truncated dodecahedron, or autid, is one of the 92 Johnson solids (J68). It consists of 5+5+5+5+5 triangles, 5 squares, 1 pentagon, and 1+5+5 decagons. It can be constructed by attaching a pentagonal cupola to one of the decagonal faces of the truncated dodecahedron..

Vertex coordinates
An augmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of: Plus the following additional vertices:
 * $$\left(0,\,±\frac12,\,±\frac{5+3\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(\frac{15+13\sqrt5}{20},\,±\frac12,\,3\frac{5+\sqrt5}{10}\right),$$
 * $$\left(\frac{25+13\sqrt5}{20},\,±\frac{1+\sqrt5}{4},\,\frac{25+\sqrt5}{20}\right),$$
 * $$\left(\frac{10+9\sqrt5}{10},\,0,\,\frac{15+\sqrt5}{20}\right).$$