Tetrahemihexahedron

The tetrahemihexahedron, or thah, is a quasiregular polyhedron and one of 10 uniform hemipolyhedra. It consists of 4 equilateral triangles and 3 "hemi" squares passing through the center, with two of each joining at a vertex. It can be derived as a facetorectified tetrahedron. Its triangular faces are parallel to those of a tetrahedron, and its hemi square faces are parallel to those of a cube or hexahedron: hence the name.

It is the only uniform polyhedron that only has A3 symmetry in its highest symmetry form. It's also the only uniform polyhedron, other than prisms whose bases have an odd amount of sides, with an odd amount of faces.

The visible portion of this solid resembles an octahedron with four triangular pyramids carved out. In fact the four triangular faces are a tetrahedral subset of those of an octahedron, while the squares are the 3 equatorial planes of the octahedron.

In vertex figures
The tetrahemihexahedron appears as a vertex figure in one uniform polychoron, that being the tesseractihemioctachoron. It has an edge length of 1.