Icosagons are polygons with twenty sides, and do not appear in any well-known polyhedra. The first of these three regular-icosagon-based polyhedra has 122 faces.
The second of these polyhedra, each of which bears an overall resemblance to a dodecahedron, has 132 faces.
Finally, the third of these polyhedra has a total of 152 faces.
I used Stella 4d to make each of these virtual polyhedron models — and you may try this program for free at http://www.software3d.com/Stella.php.