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.

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

Offset: 1

Author

Sébastien Palcoux, Aug 27 2019

Keywords

• nonn
• fini
• more

Comments

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.

References

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

Links

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

Crossrefs

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

Programs

GAP

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 ]