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

1,1

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

Table of n, a(n) for n=1..7.

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

Prog

(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 ]

Crossrefs

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

Sequence in context: A091676 A061662 A348657 * A028528 A260134 A242320

Adjacent sequences: A321221 A321222 A321223 * A321225 A321226 A321227

Keyword

nonn,fini,more

Author

Sébastien Palcoux, Aug 27 2019

Status

approved