Square pyramidal prism

The square pyramidal prism, or squippyp, is a CRF segmentochoron (designated K-4.12 on Richard Klitzing's list). It consists of 2 square pyramids, 1 cube, and 4 triangular prisms.

As the name suggests, it is a prism based on the square pyramid. As such, it is a segmentochoron between two square pyramids. It can also be viewed as a segmentochoron between a cube and a dyad, or between a triangular prism and a square.

Two square pyramidal prisms can be joined at their cubes to form an octahedral prism. By rotating one of the square pyramidal prisms before joining, one can instead form a dyadic gyrotegmipucofastegium.

Vertex coordinates
Coordinates of the vertices of a square pyramidal prism of edge length 1 centered at the origin are given by:
 * $$\left(±\frac12,\,±\frac12,\,0,\,±\frac12\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt2}{2},\,±\frac12\right).$$

Representations
A square pyramidal prism has the following Coxeter diagrams:


 * xx ox4oo&#x (full symmetry)
 * xx ox ox&#x (A1×A1×A1 symmetry, rectangular pyramidal prism)
 * oxx xxx&#x (A1×A1 axial only, trapezoidal pyramidal prism)
 * oxxo4oooo&#xr (BC2 axial only, square pyramid atop square pyramid)