The tetrahemihexahedron is one of the uniform polyhedra. Its faces are four equilateral triangles, as well as three squares. The squares are shown in yellow in the model above, and pass through the center of the polyhedron. It has tetrahedral symmetry, and seven faces in total. I know of no other polyhedra which have seven faces and any form of polyhedral symmetry.

If you stellate this polyhedron once, you get a tetrahedron.

The tetrahemihexahedron only has two stellations, and I really like the second one, shown below.

I made these models using Stella 4d: Polyhedron Navigator, which you can try for free at this website: http://www.software3d.com/Stella.php.