Two rank six geometries for the Higman–Sims sporadic group Original Research Article

We construct two rank six geometries that are firm and residually connected and on which the Higman–Sims sporadic group HS acts flag-transitively. Both of these geometries have five rank five residues isomorphic to geometries of the Mathieu group image. They continue chains of geometries obtained for image, image, image and image. Moreover, they are the first rank six geometries constructed for HS.