Octahedron atop triangular cupola

Octahedron atop triangular cupola is a CRF segmentochoron (designated K-4.30 on Richard Klitzing's list). As the name suggests, it consists of an octahedron and a triangular cupola as bases, connected by 4 further, 1 further triangular cupola, and 6 square pyramids. It also has a higher symmetry orientation with 3 layers of vertices: a triangle (joining two octahedra) on one side, a hexagon (joining two triangular cupolas) on the other side, and 6 vertices in between in the shape of a non-uniform triangular prism.

An octahedron atop triangular cupola segmentochoron can be cut into two triangular antiwedges. Three octahedron atop triangular cupola segmentochora can be joined around a shared hexagonal face to form the regular icositetrachoron.

Vertex coordinates
The vertices of an octahedron atop triangular cupola segmentochoron of edge length 1 are given by:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,\frac{\sqrt2}{2}\right)$$ and all permutations of first three coordinates,
 * $$±\left(\frac{\sqrt2}{2},\,-\frac{\sqrt2}{2},\,0,\,0\right)$$ and all permutations of first three coordinates,
 * $$\left(\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0,\,0\right)$$ and all permutations of first three coordinates.