Chamfered cube

The chamfered cube, also known as the order-4 truncated rhombic dodecahedron, is one of the near-miss Johnson solids. It has 6 squares and 12 hexagons as faces, 32 order-3 vertices divided into two sets of 8 and 24 each. It can be made equilateral, but the hexagons have two angle sizes.

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

The canonical variant with midradius 1 has two edge lengths: one of length $$sqrt\{2-\sqrt3} ≈ 0.51764$$ and the other of length $$\frac{sqrt\6}{4} ≈ 0.61237$$

It can also be viewed as an order-4-truncated rhombic dodecahedron, or as an octahedrally-symmetric Goldberg polyhedron of index (2,0).

It is the convex core of the uniform quasirhombicuboctahedron and of the compound rhombihexahedron.

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