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%I #15 May 07 2021 00:52:18
%S 2,12,48,1152,3840,103680,2903040,696729600,1393459200,8360755200,
%T 81749606400,1961990553600,51011754393600,1428329123020800,
%U 42849873690624000,1371195958099968000,46620662575398912000
%N Maximal order of a finite subgroup of the group GL(n,Q).
%C a(n) is also the maximal degree over Q of an algebraic number that has what the authors call conjugate dimension equal n.
%C From _Jianing Song_, May 06 2021: (Start)
%C a(n) is also the maximal order of a finite subgroup of the group GL(n,Z). This is because every finite subgroup of GL(n,Q) is conjugate to a subgroup of GL(n,Z). See the Math Overflow link below.
%C Conjecture: a(n)/2 is the maximal order of a finite subgroup of the group SL(n,Q) (or equivalently, SL(n,Z), since a finite subgroup of SL(n,Q) must be conjugate to a subgroup of SL(n,Z)). This is obviously true for odd n: given a finite subgroup G of SL(n,Q), a subgroup of GL(n,Q) of order 2*|G| can be attained by adjoining -I_n where I_n is the identity matrix. The conjecture is also correct for n = 2, since the finite subgroup of the maximal order of SL(n,Q) is isomorphic to C_6. (End)
%H N. Berry, A. Dubickas, N. D. Elkies, B. Poonen and C. Smyth, <a href="https://arxiv.org/abs/math/0308069">The conjugate dimension of algebraic numbers</a>, arXiv:math/0308069 [math.NT], 2003-2004; Quart. J. Math., 55 (2004), 237-252.
%H Math Overflow, <a href="https://mathoverflow.net/questions/221351/reference-for-a-linear-algebra-result">Reference for a linear algebra result</a>
%F For all n other than the seven exceptional values 2, 4, 6, 7, 8, 9, 10, a(n) = A000165(n) = 2^n * n! and the relevant group is the group of n X n signed permutation matrices.
%Y Cf. A000165.
%K nonn,easy
%O 1,1
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 15 2003
%E More terms from _N. J. A. Sloane_, Dec 20 2006
%E Terms 2, 4, 6, 7, 8, 9, 10 corrected by _M. F. Hasler_, Dec 17 2007
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