Polytopes of high rank for the alternating groups

There is a well-known correspondence between abstract regular polytopes and string C-groups. In this paper, for each d ⩾ 3 , a string C-group with d generators, isomorphic to an alternating group of degree n is constructed (for some n ⩾ 9 ), or equivalently an abstract regular d-polytope, is produced with automorphism group Alt ( n ) . A method that extends the CPR graph of a polytope to a different CPR graph of a larger (or possibly isomorphic) polytope is used to prove that various groups are themselves string C-groups.