Locally toroidal polytopes of rank 6 and sporadic groups

Abstract

Abstract. We augment the list of finite universal locally toroidal regular polytopes of type {3, 3, 4, 3, 3} due to P. McMullen and E. Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y -shaped presentation for the sporadic simple group Fi 22 , and admits S 4 × O + 8 (2):S 3 as the automorphism group. We also discuss further extensions of its quotients in the context of Y -shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3, 3, 4, 3, 3} are related to Y -shaped presentations of orthogonal groups over F 2 . Mixing construction is used in a number of places to describe covers and 2-covers.