Petrial cube

The Petrial cube is a regular skew polyhedron. It consists of 4 skew hexagons, and it is the Petrie dual of the cube. It has a Euler characteristic of 0.

Vertex coordinates
The coordinates for the vertices of a Petrial cube with unit side length are the same of that as a cube, being located at $$\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12\right).$$

Related polyhedra
The rectification of the Petrial cube is the octahemioctahedron, which is uniform.

The Petrial cube is abstractly equivalent to the blended tetrahedron.