I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are — up to conjugacy in $GL(n,Q)$ — in each dimension the group of signed permutation matrices of order $2^nn!$. This is proven in an article by Walter Feit which appeared as "Orders of finite linear groups" in the Proceedings of the First Jamaican Conference on Group Theory and its Applications in 1996. In this paper also the "exceptional" dimensions are treated.
Even though I tried for a fair bit I couldn't get my hands on the paper. And also was I unable to find the list of exceptional groups someplace else. So does anyone know if I can find this list elsewhere? Or can someone maybe post this list as an answer?
Any help is greatly appreciated.
Best Answer
In fact, I have a copy of a preprint by Feit of this paper. I have not checked the results, but here is what Feit says: the group of signed permutation matrices is of maximal order as a finite subgroup of ${\rm GL}(n,\mathbb{Q})$, except in the following cases (Feit's Theorem A).
$n = 2, W(G_{2})$ of order $12$.
$n = 4, W(F_{4}),$ order $1152$.
$n = 6, W(E_{6}) \times C_{2},$ order $103680.$
$n = 7, W(E_{7}),$ order $2903040.$
$n = 8, W(E_{8}),$ order $696729600$.
$n = 9, W(E_{8}) \times W(A_{1})$, order $1393459200$ (reducible).
$n = 10, W(E_{8}) \times W(G_{2}),$ order $8360755200$ (reducible).