The 38 faces are:
- Six squares
- Eight regular hexagons
- 24 isosceles triangles
This polyhedron also has an interesting dual, which features 24 kites and twelve rhombi as faces (click on it to see it rotate):
I discovered these while “surfing” the universe of polyhedra using software called Stella 4d. You may try this software as a free trial download at http://www.software3d.com/stella.php.
Update: It turns out that this was previously found by the formidable polyhedra-researching team of Craig S. Kaplan and George W. Hart, as you can see at http://www.cgl.uwaterloo.ca/~csk/papers/kaplan_hart_bridges2001.pdf. Finding polyhedra no one has found before is no easy task, and, often, one thinks it has been done, only to find out later that one has already been beaten to the proverbial punch. When I discover that I’ve merely re-discovered what others have found before me, I always give credit to the previous researchers. That’s part of the prevailing etiquette within the polyhedra-research community, and I think it’s an important principle, on ethical grounds.