Holey monster

The holey monster is a quasi-convex Stewart toroid with 830 triangles, 66 squares, 18 pentagons, and 20 hexagons as faces. It can be made by excavating the following from a great rhombicosidodecahedron: 30 of the polyhedron, 12 pentagonal cupolae, 12 pentagonal antiprisms, and a central small rhombicosidodecahedron, then augmenting the inside of the hollowed-out small rhombicosidodecahedron with 6 copies of the  polyhedron that are all connected to a central cube.

It has the highest genus of any known quasi-convex Stewart toroid, that being 46.

Symmetry
Each of the Z4 tunnels can be oriented in two ways. All together, they can be made part of several different axial symmetries that are subsymmetries of dodecahedral symmetry. But the orientation of the rod must also be considered relative to the symmetry of all the Z4 tunnels. It is possible to build a version of this toroid with chiral triangular pyramidal symmetry, but most constructions of it will have identity symmetry.