Low-Dimensional Lattices. II. Subgroups of GL(n,Z)

The maximal finite irreducible groups of n x n integers for n = 4, 5, ..., 9, 11, 13, 17, 19, 23 were determined by Dade, Ryskov, Bulow, Plesken & Pohst and Plesken, as the automorphism groups of certain quadratic forms. This paper presents a geometric description of the corresponding n-dimensional lattices, and gives coordinates which display their symmetries and minimal vectors. Some very interesting lattices appear.