Trivial group

The trivial group also known as the zero group, is the simplest possible group. It contains 1 element, its identity. Polytopes whose symmetry group are isomorphic to the trivial group are called asymmetrical.

Symmetries isomorphic to the trivial group

 * Pointic symmetry
 * I
 * I×I
 * I×I×I
 * I×I×I×I

Notable asymmetrical polytopes
Most classes of polytopes that are objects of study (regular, uniform, noble, etc.) are symmetrical. However, interesting questions arise from asymmetrical polytopes as well, such as asking what the smallest asymmetrical polytopes are in a given category using some measure such as facet count.

By facet count, the smallest known asymmetrical abstract polytope (apart from the trivial examples of the nullitope and the point) is a tetrahedron of rank 3 with 1 digon, 2 triangles, and 1 quadrilateral as faces. The triangles share an edge.

Asymmetrical polyhedra with all regular faces are believed to have a minimum possible face count of 9. An example is a square pyramid blended with a pentagonal pyramid, angled so that the apex of each pyramid is adjacent to the base of the other.

All Johnson solids are symmetrical, but there are asymmetrical Blind polytopes, all of which are special cuts of the hexacosichoron.