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 augmented with a specific symmetry (which isn't present in the polyhedron itself, specifically swapping the endpoints of the edges dividing the squares and octahedra) to form the Mathieu group M24.