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Irreducible finite integral matrix groups of degree $ 8$ and $ 10$

Author: Bernd Souvignier
Journal: Math. Comp. 63 (1994), 335-350
MSC: Primary 20H15; Secondary 11E12, 20C10, 20C40
MathSciNet review: 1213836
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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.

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Keywords: Integral matrix groups, Bravais groups, integral lattices in Euclidean space
Article copyright: © Copyright 1994 American Mathematical Society

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