Hypertruncated cube

The hypertruncated cube is a semi-uniform polyhedron. It consists of 8 triangles and 6 tetrapods and has 1 triangle and 2 tetragons at a vertex. It is very similar to the antitruncated cube, the only difference being that the triangles connect to the larger sides of tetragons in the hypertruncated cube, and the smaller sides in the antitruncated cube. The hypertruncated cube is both the hypertruncation of the cube and a faceting of the truncated cube.

Vertex coordinates
The hypertruncated cube shares its vertices with the truncated cube, being all permutations of:
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$

Variations
A teepee of semi-uniform polyhedra can be formed by adjusting the size of the tetrapods.