Grünbaum polyhedron

The Grünbaum polyhedron is a realization of the Fricke-Klein map in 3-dimensional Euclidean space with the maximum possible symmetry. The Fricke-Klein map is regular, and thus the Grünbaum polyhedron is regular under its automorphism group, however it is not regular under its symmetry group.

Construction
The Grünbaum polyhedron can be constructed by blending two non-identical snub cubes with coincident vertices. The inner snub cube is the same as the outer snub cube with each of its square faces rotated about its own center a quarter turn. If a uniform snub cube is used the inner snub cube will self-intersect. Typically a non-uniform version is used to avoid this.