I made this using multiple stellations, some of them with tetrahedral symmetry, to modify the cubohemioctahedron, one of the uniform polyhedra. I did this using Stella 4d, which you can try for yourself, free, at http://www.software3d.com/Stella.php.

Also, here’s what the cubohemioctahedron looks like, without modification. It has ten faces: six squares, and four interpenetrating regular hexagons.