A001228 -id:A001228 - cp's OEIS frontend

A174671 Divisors of the order of the Monster group, sorted into decreasing order.

808017424794512875886459904961710757005754368000000000, 404008712397256437943229952480855378502877184000000000, 269339141598170958628819968320570252335251456000000000, 202004356198628218971614976240427689251438592000000000, 161603484958902575177291980992342151401150873600000000

Offset: 1

Reinhard Zumkeller, Apr 02 2010

Let Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131;
a(n) = Mnr / A174670(n);
the sequence is finite with A174601(26) = 424488960 terms;
a(n) = A174670(424488960 - n + 1).

			a(1) = Mnr;
a(424488960) = 1, the last term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
Index entries for sequences related to divisors of numbers

A174848 Squarefree kernels of orders of sporadic simple groups.

Original entry on oeis.org

330, 330, 43890, 2310, 210, 53130, 2310, 9690, 53130, 2310, 3570, 79170, 30030, 1360590, 53130, 53130, 30030, 43890, 177521190, 1607970, 11741730, 690690, 75992317170, 340510170, 325046311590, 1618964990108856390

Offset: 1

Reinhard Zumkeller, Apr 02 2010

a(n) = A007947(A001228(n)).

			a(1) = 2*3*5*11;
a(2) = 2*3*5*11;
a(3) = 2*3*5*7*11*19;
a(4) = 2*3*5*7*11 = 11#;
a(5) = 2*3*5*7 = 7#;
a(6) = 2*3*5*7*11*23;
a(7) = 2*3*5*7*11 = 11#;
a(8) = 2*3*5*17*19;
a(9) = 2*3*5*7*11*23;
a(10) = 2*3*5*7*11 = 11#;
a(11) = 2*3*5*7*17;
a(12) = 2*3*5*7*13*29;
a(13) = 2*3*5*7*11*13 = 13#;
a(14) = 2*3*5*7*11*19*31;
a(15) = 2*3*5*7*11*23;
a(16) = 2*3*5*7*11*23;
a(17) = 2*3*5*7*11*13 = 13#;
a(18) = 2*3*5*7*11*19;
a(19) = 2*3*5*7*11*31*37*67;
a(20) = 2*3*5*7*13*19*31;
a(21) = 2*3*5*7*11*13*17*23;
a(22) = 2*3*5*7*11*13*23;
a(23) = 2*3*5*7*11*23*29*31*37*43;
a(24) = 2*3*5*7*11*13*17*23*29;
a(25) = 2*3*5*7*11*13*17*19*23*31*47;
a(26) = PROD(A002267(k): 1<=k<=15) = 2*3*5*7*11*13*17*19*23*29*31*41*47*59*71.

Eric Weisstein's World of Mathematics, Sporadic Group

Cf. A174670.

A001051 Number of subgroups of order n in orthogonal group O(3).

Original entry on oeis.org

1, 3, 1, 5, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 10, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8

Offset: 1

J. H. Conway

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Orthogonal Group.

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

a[2] = 3; a[4] = 5; a[12] = 8; a[24] = 10; a[48] = a[60] = a[120] = 8; a[n_] := Switch[Mod[n, 4], 0, 7, 1, 1, 2, 5, 3, 1]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Oct 15 2013 *)

A001051(n) = if((12==n)||(48==n)||(60==n)||(120==n),8,if(24==n,10,if((4==n)||(2==n),1+n,[1,5,1,7][1+((n-1)%4)]))); \\ Antti Karttunen, Jan 15 2019

Has period 1 5 1 7 except that a(2) = 3, a(4) = 5, a(12) = 8, a(24) = 10, a(48) = a(60) = a(120) = 8.

Data section extended up to a(120) by Antti Karttunen, Jan 15 2019

A051881 Number of subgroups of order n in special orthogonal group SO(3).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

J. H. Conway

			The groups are "nn", of order n; "22n", of order 2n; "332", "432", "532" of orders 12,24,60.

Antti Karttunen, Table of n, a(n) for n = 1..12000

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

a[2] = 1; a[12|24|60] = 3; a[n_] := 2-Mod[n, 2]; Array[a, 105] (* Jean-François Alcover, Nov 12 2015 *)

a(n)=if(n==2||n==12||n==24||n==60, if(n>2,3,1), if(n%2,1,2)) \\ Charles R Greathouse IV, Nov 10 2015

def a(n):
    if n == 2:
        return 1
    elif n in {12, 24, 60}:
        return 3
    else:
        return 2 - n % 2 # Paul Muljadi, Oct 21 2024

Has period 1, 2 except for a(2) = 1, a(12) = a(24) = a(60) = 3.

More terms from James Sellers and David W. Wilson, Dec 16 1999

A119648 Orders for which there is more than one simple group.

Original entry on oeis.org

20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480

Offset: 1

N. J. A. Sloane, Jul 29 2006

nonn

All such orders are composite numbers (since there is only one group of any prime order).
Orders which are repeated in A109379.
Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010
a(1) = 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8 (see A137863). - Bernard Schott, May 18 2020

			From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start)
a(1)=|A_8|=8!/2=20160,
a(2)=|C_3(3)|=4585351680,
a(3)=|C_3(5)|=228501000000000, and
a(4)=|C_4(3)|=65784756654489600. (End)

See A001034 for references and other links.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. See also. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
W. Kimmerle et al., Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups, Proc. London Math. Soc., (3) 60 (1990), 89-122. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
David A. Madore, Orders of non-Abelian simple groups
Wikipedia, Classification of finite simple groups [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
Index entries for sequences related to groups

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition), A137863 (orders of simple groups which are non-cyclic and non-alternating).

sp(n, q) 1/2 q^n^2.(q^(2.i) - 1, i, 1, n) [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] [This line contained some nonascii characters which were unreadable]

For n>1, a(n) is obtained as (1/2) q^(m^2)Prod(q^(2i)-1, i=1..m) for appropriate m>2 and q equal to a power of some odd prime number. [Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

A137863 Orders of simple groups which are non-cyclic and non-alternating.

Original entry on oeis.org

168, 504, 660, 1092, 2448, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348, 175560, 178920, 194472, 246480, 262080

Offset: 1

Artur Jasinski, Feb 16 2008

nonn

From Bernard Schott, Apr 26 2020: (Start)
About a(16) = 20160; 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but, 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8.
Indeed, 20160 is the smallest order for which there exist two nonisomorphic simple groups and it is the order of this group PSL_3(4) that was missing in the data. The first proof that there exist two nonisomorphic simple groups of this order was given by the American mathematician Ida May Schottenfels (1900) [see the link]. (End)

			From _Bernard Schott_, Apr 27 2020: (Start)
Two particular examples:
a(1) = 168 is the order of the smallest non-cyclic and non-alternating simple group, this Lie group is the projective special linear group PSL_2(7) that is isomorphic to the general linear group GL_3(2).
a(12) = 7920 is the order of the smallest sporadic group (A001228), the Mathieu group M_11. (End)

L. E. Dickson, Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.

David Madore, Orders of non abelian simple groups
Ida May Schottenfels, Two non isomorphic simple groups of the same order 20160, Annals of Mathematics, Second Series, Vol. 1, No. 1/4 (1900), pp. 147-152.

Cf. A001034, A001710, A005180, A109379.

Subsequence: A001228 (sporadic groups).

More terms from R. J. Mathar, Apr 23 2009
a(16) = 20160 inserted by Bernard Schott, Apr 26 2020
Incorrect formula and programs removed by R. J. Mathar, Apr 27 2020
Terms checked by Bernard Schott, Apr 26 2020

A261717 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups of which the n-th largest sporadic simple group is a subquotient.

Original entry on oeis.org

19, 12, 2, 16, 4, 10, 7, 1, 5, 7, 3, 1, 3, 1, 3, 4, 5, 3, 1, 3, 4, 2, 1, 2, 2, 1

Offset: 1

Sven Simon, Charles R Greathouse IV, Franklin T. Adams-Watters, Oct 24 2015

The sum of all elements of the sequence is the same as that of A263447.

			The sporadic group Fi_23 is a subquotient of the sporadic groups Fi_23 (itself), Fi_24, the Baby Monster and the Monster, so a(21) = 4.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. Page 238.

Wikipedia, Sporadic group

Cf. A001228, A263447.

A263447 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups which are subquotients of the n-th largest sporadic simple group.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 3, 1, 5, 3, 1, 1, 4, 3, 7, 6, 4, 5, 4, 1, 6, 12, 6, 9, 12, 20

Offset: 1

Sven Simon and Charles R Greathouse IV, Oct 18 2015

A group is a subquotient of itself, so a(n) >= 1.
It is well-known that a(26) = 20, the so-called "happy family". Trivially a(1) = 1 and a(2) = 2 since M_11 is a subquotient of M_12.
The sequence was generated from the diagram of subquotient relations on page 238 of the ATLAS, together with the update that J_1 is not involved in M (which replaces the single question mark in the table with a plus sign).

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. See page 238.

Wikipedia, Sporadic group

Cf. A001228, A261717 (another version).

Terms confirmed by N. J. A. Sloane, Oct 19 2015

A330585 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the sporadic finite simple groups.

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 12180, 14880, 20160, 20160, 25920, 29120, 32736, 58800, 62400, 95040, 102660, 126000, 175560, 178920, 181440, 265680, 372000, 443520, 604800

Offset: 1

Hal M. Switkay, Dec 18 2019

By the classification theorem for finite simple groups, there are exactly 26 sporadic finite simple groups, whose orders form A001228. The online ATLAS includes lists of the maximal subgroups of these groups, and entries for their simple subquotients.

Subsequence of A083207. - Ivan N. Ianakiev, Jan 02 2020

			This list includes the orders of all non-cyclic simple groups of order less than 9828. L2(27), of order 9828, does not appear as a subquotient of any of the sporadic finite simple groups.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Hal M. Switkay, Table of n, a(n) for n = 1..82
Ivan N. Ianakiev, Subsequence of A083207, Proof
David A. Madore, Orders of non-abelian simple groups
R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Cf. A109379, A001228, A083207.

A119631 Orders of non-Abelian simple groups of rank at least four.

Original entry on oeis.org

60, 360, 2520, 7920, 20160, 95040, 175560, 181440, 443520, 604800, 1814400, 9999360, 10200960, 13685760, 17971200, 19958400, 44352000, 50232960, 174182400, 197406720, 211341312, 239500800, 244823040, 898128000

Offset: 1

N. J. A. Sloane, Jun 10 2006

nonn

This includes all the sporadic simple groups (A001228) except one (the Monster).

David A. Madore, Table of n, a(n) for n = 1..346 [taken from link below]
David A. Madore, More terms
Index entries for sequences related to groups

Cf. A001034, A119630.

cp's OEIS Frontend

A174671 Divisors of the order of the Monster group, sorted into decreasing order.

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A174848 Squarefree kernels of orders of sporadic simple groups.

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A001051 Number of subgroups of order n in orthogonal group O(3).

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A051881 Number of subgroups of order n in special orthogonal group SO(3).

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A119648 Orders for which there is more than one simple group.

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A137863 Orders of simple groups which are non-cyclic and non-alternating.

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A261717 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups of which the n-th largest sporadic simple group is a subquotient.

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A263447 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups which are subquotients of the n-th largest sporadic simple group.

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