Chamfered cube

The chamfered cube is a modification of the cube that can have one edge length but has irregular faces. It has 6 squares and 12 hexagons as faces, and 8 order-3 vertices that can be thought of as coming from the cube as well as 24 new order-3 vertices.

The hexagonal faces have angles of $$\arccos\left(-\frac13\right) ≈ 109.47122^\circ$$ on a pair of opposite vertices, and angles of $$\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439^\circ$$ on the four remaining vertices.

It can be modified such that it has a single inradius, or such that it has a single midradius or "edge radius." The latter version is called the "canonical" version.

It can also be viewed as an order-4-truncated rhombic dodecahedron, or as an octahedrally-symmetric Goldberg polyhedron.

Vertex coordinates
The vertices of a chamfered cube can be given by all changes of sign of: plus all permutations and changes of sign of:
 * $$\left(\frac12+\frac{\sqrt3}{3},\,\frac12+\frac{\sqrt3}{3},\,\frac12+\frac{\sqrt3}{3}\right),$$
 * $$\left(\frac12,\,\frac12,\,\frac12+\frac{2\sqrt3}{3}\right).$$