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Number of isomorphism classes of finite groups of order 11*2^n.
2

%I #19 Oct 19 2024 22:24:44

%S 1,2,4,12,42,195,1387,19324,1083472

%N Number of isomorphism classes of finite groups of order 11*2^n.

%C This appears to be the smallest possible number of groups of order q*2^n for an odd number q.

%C Apparently, a(n) is also the number of isomorphism classes of finite groups of order 19*2^n and, more generally, of order p*2^n for primes p such that p is congruent to 3 modulo 4 and p+1 is not a power of 2.

%C Comment from _Miles Englezou_, Sep 26 2024: (Start)

%C The comment above which starts "Apparently, ... power of 2." is not true. (For a proof see the Miles Englezou link). However, it is true that a(0) to a(8) are the smallest possible number of groups of order q*2^n for an odd number q, and this can be generalized in the way stated below. (For further details see the Miles Englezou link).

%C A correct generalization of the 9 terms:

%C The number of groups of order q*2^n is the least possible for prime q such that q == 3 (mod 4) and where the least 2^m such that 2^m == 1 (mod q) is greater than 2^n. Or put another way, if A014664(A080148(n)) > n, then for q = A000040(A080148(n)) the number of groups of order q*2^n is the least possible. (End)

%D J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 206.

%H John H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</a>.

%H Miles Englezou, <a href="/A139669/a139669.pdf">Proofs</a>

%F a(n) = A000001(11*2^n). - _Max Alekseyev_, Apr 26 2010

%e a(2) is the number of groups of order 11*2^2=44, which is 4 and also the number of groups of order 19*2^2=76, 23*2^2=92, etc.

%p A139669 := n -> GroupTheory[NumGroups](11*2^n);

%Y Cf. A000001, A002145, A014664, A080148.

%K hard,more,nonn

%O 0,2

%A Anthony D. Elmendorf (aelmendo(AT)calumet.purdue.edu), Jun 12 2008

%E a(8) from _Max Alekseyev_, Dec 24 2014