John Conway has given an explanation how a regular Hecatohedron (polyhedron with 100 faces) looks like: http://www.ics.uci.edu/~eppstein/junkyard/hecatohedron.html
Here's a hecatohedron with full tetrahedral symmetry: Form the "16-reticulated cube", by dividing each face of a cube into 18 smaller "square" faces in the obvious way, giving a 96-hedron. Then tetrahedrally truncate this. I suppose I'd call it the "semi-trivalently-truncated 16-reticulated cube" !
Unfortunatly I was not able to find a way how implement this into Mathematica. Can somebody give me some more hints or explanations? Thanks!