A001228 -id:A001228 - cp's OEIS frontend

A360389 The orders of 4-transitive permutation groups.

24, 120, 360, 720, 2520, 5040, 7920, 20160, 40320, 95040, 181440, 362880, 1814400, 3628800, 10200960, 19958400, 39916800, 239500800, 244823040, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000

Offset: 1

Hal M. Switkay, Feb 05 2023

nonn

The 4-transitive permutation groups are either: 1) symmetric groups of degree k for k >= 4, with order k! = A000142(k); 2) alternating groups of degree k for k >= 6, with order k!/2 = A001710(k); or 3) Mathieu groups of degree 11, 12, 23, or 24, with order A001228(k), where k = 1, 2, 6, or 9 respectively.

Hal M. Switkay, Table of n, a(n) for n = 1..44
Shreeram S. Abhyankar, Galois Theory on the Line in Non-Zero Characteristic, Bulletin of the AMS, 27 (1992), 68-133.

Cf. A000142, A001228, A001710.

A321224 Sporadic numbers: n is defined to be sporadic if the set of groups G not in {A_n, S_n} and having a core-free maximal subgroup of index n is nonempty and contains only sporadic simple groups.

Original entry on oeis.org

266, 506, 759, 1045, 1288, 1463, 3795

Offset: 1

Sébastien Palcoux, Aug 27 2019

A finite group G has a core-free maximal subgroup H of index n if and only if it is a primitive permutation group of degree n (acting on the set G/H of cosets).
There are no other sporadic numbers less than 4096 (see computation below).
According to Derek Holt, the next sporadic number is 4180, and the last one should be 492693551703971265784426771318116315247411200000000 (coming from the maximal subgroup 41:40 of the Monster, and assuming that L_2(13) is not maximal).
Derek Holt suggested another sequence where we also allow the extensions of the sporadic simple groups.

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.

S. Palcoux, The sporadic numbers (version: 2019-07-22), MathOverflow.

Cf. A102842, A001228, A174601, A174848, A261717, A263447, A121236.

IsSporadic:=function(G)
   if not IsSimple(G) then
      return false;
   else
      return IsomorphismTypeInfoFiniteSimpleGroup(G).series="Spor";
   fi;
end;;
SporadicNumbers:=function(b1,b2)
   local L,i,n,a,j,G;
   L:=[];
   for i in [b1..b2] do
      n:=NrPrimitiveGroups(i);
      if n>2 then
         a:=0;
         for j in [1..n] do
            G:=PrimitiveGroup(i,j);
            if not G=SymmetricGroup(i) and not G=AlternatingGroup(i) and not IsSporadic(G) then
               a:=1;
               break;
            fi;
         od;
         if a=0 then
            Add(L,i);
         fi;
      fi;
   od;
   return L;
end;;
SporadicNumbers(1,4095);
# gives: [ 266, 506, 759, 1045, 1288, 1463, 3795 ]

A329191 The prime divisors of the orders of the sporadic finite simple groups.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 67, 71

Offset: 1

Hal M. Switkay, Nov 07 2019

This list is complete according to the classification theorem for finite simple groups.
This list includes all primes < 72 except 53 and 61, which do not divide the order of any sporadic finite simple group.
All entries on this list divide the order of the Monster, except 37, 43, and 67.

			The first term is necessarily 2, by the Feit-Thompson theorem.

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].

R. A. Wilson et al., ATLAS of Finite Group Representations, version 3, Sporadic groups

Cf. A001228, A005180, A174670.

A379755 Orders k of groups where at least one group has a simple automorphism group.

Original entry on oeis.org

3, 4, 6, 8, 16, 32, 64, 128, 256, 512, 1024

Offset: 1

Miles Englezou, Jan 01 2025

A group G that has a simple automorphism group Aut(G) is either abelian or simple and complete (that is, the center Z(G) = 1 and G = Aut(G)). Proof: since the group of inner automorphisms Inn(G) is a normal subgroup of Aut(G), if Aut(G) is simple then Inn(G) = 1 or Aut(G). When Inn(G) = 1, G is abelian. That G is simple and complete when Inn(G) = Aut(G) can be proved by considering that the homomorphism f: G -> Aut(G), with Inn(G) as image and Z(G) as kernel, is surjective, and that a surjective homomorphism preserves normal subgroups. As Aut(G) is simple, therefore G must also be simple, otherwise any normal N in G would correspond to a normal f(N) in Aut(G). Since G is nonabelian, Z(G) = 1, and Aut(G) = Inn(G) = G/Z(G) = G. Hence, when Aut(G) is simple and Inn(G) = Aut(G), G is simple and complete.

As the order of any simple complete group is a term, the orders of various sporadic simple groups are also terms, including 7920, 10200960, and 244823040, corresponding to the Mathieu groups M_11, M_23, and M_24, and also 175560, corresponding to the Janko group J_1.

			3 is a term since Aut(C_3) = C_2, which is simple.
8 is a term since Aut(C_2 x C_2 x C_2) = PSL(3,2), which is simple.
7920 is a term since Aut(M_11) = M_11, which is simple. (M_11 is one of the Mathieu groups).

Cf. A000079, A001228.

A:=[];
for n in [1..100] do
    if n>=8 and Unique(Factors(n))=[2] then
        A:=Concatenation(A,[n]);
        continue;
    fi;
    for i in [1..NrSmallGroups(n)] do
        G:=SmallGroup(n,i);
        if IsAbelian(G)=false and Order(Centre(G))>1 then
            continue;
        fi;
        Aut:=AutomorphismGroup(G);
        if IsSimpleGroup(Aut)=true then
            A:=Concatenation(A,[n]);
            break;
        fi;
    od;
od;
Print(A);

2^m is a term for m >= 3. This is because every elementary abelian group of order 2^m has the projective special linear group PSL(m,2) as an automorphism group, which is simple for every such m.

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A360389 The orders of 4-transitive permutation groups.

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A321224 Sporadic numbers: n is defined to be sporadic if the set of groups G not in {A_n, S_n} and having a core-free maximal subgroup of index n is nonempty and contains only sporadic simple groups.

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A329191 The prime divisors of the orders of the sporadic finite simple groups.

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A379755 Orders k of groups where at least one group has a simple automorphism group.

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