The automorphism group of the Klein quartic can be augmented (by a symmetry which is not realized by a symmetry of the polyhedron, namely "exchanging the two endpoints of the edges that bisect the squares and octahedra) to yield the Mathieu group M24.
The automorphism group of the Klein quartic can be expanded by a symmetry that is not inherent in the polyhedron's symmetries (specifically, exchanging the endpoints of the edges bisecting the squares and octahedra), resulting in the Mathieu group M24.