Weakly regular polytope

An object is weakly regular if its symmetry acts transitively on elements of each rank. For example a polyhedron is weakly regular if its symmetry is isohedral, isotoxal, and isogonal. All regular polytopes are weakly regular, but there are non-regular polytopes that are weakly regular.

Weakly regular polyhedra
A weakly regular polyhedron is isotoxal, isogonal, and isotopic. The finite, non-skew polyhedra that are weakly regular are all regular polyhedra. However relaxing these requirements allows polyhedra that are weakly regular but not regular:

The rhombic tiling is a weakly regular, but not regular, Euclidean tiling.

The quotient of the square tiling by (2,1) and (-1,2) is a weakly regular, but not regular, finite abstract polyhedron. It can also be embedded as a skew polyhedron in 4-dimensional space.

There are also four polytope compounds which are transitive on all their elements, but not regular. These are the regular compounds, excluding the stella octangula which is regular.