Gyroelongated triangular cupola

The gyroelongated triangular cupola is one of the 92 Johnson solids (J22). It consists of 1+3+3+3+6 triangles, 3 squares, and 1 hexagon. It can be constructed by attaching a hexagonal antiprism to the hexagonal base of the triangular cupola.

If a second cupola is attached to the other hexagonal base of the antiprism, the result is the gyroelongated triangular bicupola.

Vertex coordinates
A gyroelongated triangular cupola of edge length 1 has the following vertices:
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,\frac{2\sqrt6+3\sqrt{\sqrt3-1}}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,\frac{2\sqrt6+3\sqrt{\sqrt3-1}}{6}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,\frac{\sqrt{\sqrt3-1}}{2}\right),$$
 * $$\left(±1,\,0,\,\frac{\sqrt{\sqrt3-1}}{2}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,-\frac{\sqrt{\sqrt3-1}}{2}\right),$$
 * $$\left(0,\,±1,\,-\frac{\sqrt{\sqrt3-1}}{2}\right).$$