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 G L n ( Z ) \mathrm {GL}_n(\mathbb {Z}) for n = 8 , 10 n = 8,10 , are classified in this paper. The number of types is 52 in the case n = 8 n = 8 , and 47 in the case n = 10 n = 10 . As a consequence of this classification one has 26, resp. 46, conjugacy classes of maximal finite irreducible subgroups of G L 8 ( Z ) \mathrm {GL}_8(\mathbb {Z}) , resp. G L 10 ( Z ) \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.