Augmented truncated tetrahedron

The augmented truncated tetrahedron, or autut, is one of the 92 Johnson solids (J65). It consists of 1+1+3+3 triangles, 3 squares, and 3 hexagons. It can be constructed by attaching a triangular cupola to one of the hexagonal faces of the truncated tetrahedron..

Vertex coordinates
An augmented truncated tetrahedron of edge length 1 has vertices given by all even sign changes of: Plus all permutations of:
 * $$\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right),$$
 * $$\left(-\frac{11\sqrt2}{12},\,-\frac{5\sqrt2}{12},\,-\frac{5\sqrt2}{12}\right).$$