Gyroelongated square pyramid

The gyroelongated square pyramid is one of the 92 Johnson solids (J10). It consists of 4+4+4 triangles and 1 square. It can be constructed by attaching a square antiprism to the base of the square pyramid.

If a second pyramid is attached to the other base of the square antiprism, the result is the gyroelongated square bipyramid.

Vertex coordinates
A gyroelongated square pyramid of edge length 1 has the following vertices:
 * $$\left(0,\,0,\,\frac{2\sqrt2+\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,-\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,-\frac{\sqrt[4]{8}}{4}\right).$$