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Numbers m such that there are precisely 15 groups of order m.
20

%I #54 May 13 2023 23:51:12

%S 24,54,81,84,136,220,228,250,260,328,340,372,513,516,580,584,620,625,

%T 686,712,740,776,804,884,891,904,948,999,1060,1096,1236,1375,1377,

%U 1420,1460,1508,1524,1544,1668,1780,1812,1863,1864,1911,1924,1928,1940,1956,1971,1972,2056,2132,2180

%N Numbers m such that there are precisely 15 groups of order m.

%H Muniru A Asiru, <a href="/A294156/b294156.txt">Table of n, a(n) for n = 1..1061</a>

%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/">The Small Groups Library</a>

%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">Small Even Order Groups</a> [<a href="https://web.archive.org/web/20171109093930/http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">archived copy</a>]

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F A294156 = { m | A000001(m) = 15 }. - _M. F. Hasler_, Oct 24 2017

%e For m = 24, the 15 groups of order 24 are C3 : C8, C24, SL(2,3), C3 : Q8, C4 x S3, D24, C2 x (C3 : C4), (C6 x C2) : C2, C12 x C2, C3 x D8, C3 x Q8, S4, C2 x A4, C2 x C2 x S3, C6 x C2 x C2 and for n = 54 the 15 groups of order 54 are D54, C54, C3 x D18, C9 x S3, ((C3 x C3) : C3) : C2, (C9 : C3) : C2, (C9 x C3) : C2, ((C3 x C3) : C3) : C2, C18 x C3, C2 x ((C3 x C3) : C3), C2 x (C9 : C3), C3 x C3 x S3, C3 x ((C3 x C3) : C2), (C3 x C3 x C3) : C2, C6 x C3 x C3 where C, D, Q, S, A and SL mean Cyclic, Dihedral, Quaternion, Symmetric, Alternating and Special Linear group. The symbols x and : mean direct and semi-direct products respectively.

%t Select[ Range@2000, FiniteGroupCount@# == 15 &] (* _Robert G. Wilson v_, Oct 24 2017 *)

%o (GAP) A294156 := Filtered([1..2015], n -> NumberSmallGroups(n) = 15);

%Y Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), this sequence (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

%K nonn

%O 1,1

%A _Muniru A Asiru_, Oct 24 2017