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In recent years, much work has been done on the classication of abstract regular polytopes by their local and global topological type. Ab- stract regular polytopes are combinatorial structures which generalize the well- known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the cor- responding classication of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks 5 and 6. For large classes of such polytopes, their groups are explicitly identied using twist- ing operations on quotients of Coxeter groups. In particular, this leads to new classication results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-gures.

Twisted Groups and Locally Toroidal Regular Polytopes