Great hexacronic icositetrahedron

The great hexacronic icositetrahedron is a uniform dual polyhedron. It consists of 24 kites.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the short edges of the kites will measure $$2\frac{\sqrt{2\left(26-17\sqrt2\right)}}{7} ≈ 0.56545$$, and the long edges will be $$2\sqrt{2-\sqrt2} ≈ 1.53073$$. The kite faces will have length $$2\frac{\sqrt{31+8\sqrt2}}{7} ≈ 1.85854$$, and width $$2\left(\sqrt2-1\right) ≈ 0.82843$$. The kites have two interior angles of $$\arccos\left(\frac14-\frac{\sqrt2}{2}\right) ≈ 117.20057^\circ$$, one of $$\arccos\left(-\frac14+\frac{\sqrt2}{8}\right) ≈ 94.19914^\circ$$, and one of $$\arccos\left(\frac12+\frac{\sqrt2}{4}\right) ≈ 31.39971^\circ$$.

Vertex coordinates
A great hexacronic icositetrahedron with dual edge length 1 has vertex coordinates given by all permutations of:
 * $$\left(±\left(2-\sqrt2\right),\,0,\,0\right),$$
 * $$\left(±\sqrt2,\,0,\,0\right),$$
 * $$\left(±\frac{4-\sqrt2}{7},\,±\frac{4-\sqrt2}{7},\,±\frac{4-\sqrt2}{7}\right).$$