Icositetrachoric antiprism

The icositetrachoric antiprism or icoap, also known as the icositetrachoric alterprism, is a convex scaliform polyteron that consists of 2 icositetrachora, 48 octahedral pyramids, and 192 pentachora. Each vertex joins 1 icositetrachoron, 7 octahedral pyramids, and 18 pentachora. It can be constructed by lacing two icositetrachora in opposite orientations together.

Vertex coordinates
The vertices of an icositetrachoric antiprism of edge length 1 are given by all permutations excluding the last coordinate of:
 * $$\left(0,\,0,\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,\frac{\sqrt{\sqrt2-1}}{2}\right),$$
 * $$\left(0,\,0,\,0,\,±1,\,-\frac{\sqrt{\sqrt2-1}}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12,\,-\frac{\sqrt{\sqrt2-1}}{2}\right).$$