A263447 -id:A263447 - cp's OEIS frontend

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.

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.

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 ]

cp's OEIS Frontend

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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