# The Regular Map S3:{3,7}, the Small Cubicuboctahedron, and the Mathieu Group M24

This page, which is mostly derived from How to Make the Mathieu Group M24, describes connections between the regular map S3:{3,7} (the dual of the Klein map), the small cubicuboctahedron, and the sporadic simple group M24.

Its rotational symmetry group is PSL(2,7), or the isomorphic PSL(3,2).

It has 24 vertices, 56 triangular faces, and 84 edges. We plan to remove 36 of its edges, thereby reducing its faces by 36, so as to leave it with 24 vertices, 20 faces, and 48 edges. Colour eight single faces pink, six pairs of faces orange, and six strings of six faces green, as shown to the right. Note that the pink faces form two distinct sets, shown by different shades of pink.

We have reduced the rotational symmetry group to A4, a subgroup of PSL(2,7). Now remove the edges that separate faces of the same colour, as shown to the right. We can regard the result as having eight triangular faces, six square faces, and six octagonal faces.

If we ignore fact that the faces are not quite the right shape in the diagram, we find that by removing the distinction betwen the two sets of pink faces, we have increased the rotational symmetry group to S4, also a subgroup of PSL(2,7).

What we have now has the same set of faces as the small cubicuboctahedron. The figure to the right can be immersed in three-space as a small cubicuboctahedron, with rotational symmetry group S4.

### The Mathieu Group M24 The Mathieu group M24 is the largest of the Mathieu groups, with 244,823,040 elements. It is a sporadic simple group, and is 5-transitive.

M24 can be regarded as a permutation group on the 24 vertices of the figures shown on this page. It is then generated by two permutations:

• A permutation of order 7, being a rotation of the original S3:{3,7} that fixes a set of three antipodal vertices. This generator is an element of its rotational symmetry group, PSL(2,7).
• A permutation of order 2, that interchanges the pairs of vertices joined by thick lines in the diagram to the right. These thick lines are those edges of the original S3:{3,7} that coincide with diagonals of the square orange faces or with major diagonals of the green octagonal faces.