Snub cubic antiprism

The omnisnub cubic antiprism or sniccap, also known as the alternated great rhombicuboctahedral prism, is a convex isogonal polychoron that consists of 2 snub cubes, 6 square antiprisms, 8 triangular antiprisms, 12 rhombic disphenoids and 48 irregular tetrahedra obtained through the process of alternating the great rhombicuboctahedral prism. However, it cannot be made uniform.

Vertex coordinates
The vertices of an omnisnub cubic antiprism, assuming that the snub cubes are regular of edge length 1, centered at the origin, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:


 * (c1, c2, c3, c4)
 * (c2, c1, c3, –c4),

where


 * $$c_1=\sqrt{\frac{1}{12}\left(4-\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{17-3\sqrt{33}}\right)},$$
 * $$c_2=\sqrt{\frac{1}{12}\left(2+\sqrt[3]{17+3\sqrt{33}}+\sqrt[3]{17-3\sqrt{33}}\right)},$$
 * $$c_3=\sqrt{\frac{1}{12}\left(4+\sqrt[3]{199+3\sqrt{33}}+\sqrt[3]{199-3\sqrt{33}}\right)}.$$
 * $$c_4=\sqrt{\frac{1}{12}\left(-2+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}\right)}.$$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:


 * (c1, c2, c3, c4)
 * (c2, c1, c3, –c4),

where


 * $$c_1=\text{root}(9472x^8-5504x^6+1136x^4-98x^2+3, 6) ≈ 0.3135135258027234561506493,$$
 * $$c_2=\text{root}(14208x^8-19840x^6+7432x^4-679x^2+8, 7) ≈ 0.6542869462841313854118897,$$
 * $$c_3=\text{root}(14208x^8-20800x^6+3640x^4-159x^2+2, 8) ≈ 1.1264771628934748152619404,$$
 * $$c_4=\text{root}(18944x^8-7936x^6+1120x^4-60x^2+1, 7) ≈ 0.3894987408692678350179804.$$