Irreducible finite integral matrix groups of degree 8 and 10

Abstract

The lattices of eight- and ten-dimensional Euclidean space with irreducible automorphism group or, equivalently, the conjugacy classes of these groups in$\mathrm {GL}_n(\mathbb {Z})$for$n = 8,10$, are classified in this paper. The number of types is 52 in the case$n = 8$, and 47 in the case$n = 10$. As a consequence of this classification one has 26, resp. 46, conjugacy classes of maximal finite irreducible subgroups of$\mathrm {GL}_8(\mathbb {Z})$, resp.$\mathrm {GL}_{10}(\mathbb {Z})$. In particular, each such group is absolutely irreducible, and therefore each of the maximal finite groups of degree 8 turns up in earlier lists of classifications.