Moonshine paths for 3A and 6A nodes of the extended E8-diagram

We continue the program to make a moonshine path between a node of the extended E8-diagram and the Monster. Our theory is a concrete model expressing some of the mysterious connections identified by John McKay, George Glauberman and Simon Norton. In this article, we treat the 3A- and 6A-nodes. We determine the orbits of triples (x,y,z) in the Monster where z∈2B, x,y∈2A∩C(z) and xy∈3A∪6A. Such x, y correspond to a rootless EE8-pair in the Leech lattice. For the 3A and 6A cases, we shall say something about the “half Weyl groups”, which are proposed in the Glauberman–Norton theory. Most work in this article is with lattices, due to their connection with dihedral subgroups of the Monster. These lattices are M+N, where M,N is the relevant pair of EE8-sublattices, and their annihilators in the Leech lattice. The isometry groups of these four lattices are analyzed.