A321224 - OEIS

%I #31 Sep 21 2019 14:56:34

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

%N 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.

%C 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).

%C There are no other sporadic numbers less than 4096 (see computation below).

%C 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).

%C Derek Holt suggested another sequence where we also allow the extensions of the sporadic simple groups.

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

%H S. Palcoux, <a href="https://mathoverflow.net/q/336691/34538">The sporadic numbers</a> (version: 2019-07-22), MathOverflow.

%o (GAP)

%o IsSporadic:=function(G)

%o    if not IsSimple(G) then

%o       return false;

%o    else

%o       return IsomorphismTypeInfoFiniteSimpleGroup(G).series="Spor";

%o    fi;

%o end;;

%o SporadicNumbers:=function(b1,b2)

%o    local L,i,n,a,j,G;

%o    L:=[];

%o    for i in [b1..b2] do

%o       n:=NrPrimitiveGroups(i);

%o       if n>2 then

%o          a:=0;

%o          for j in [1..n] do

%o             G:=PrimitiveGroup(i,j);

%o             if not G=SymmetricGroup(i) and not G=AlternatingGroup(i) and not IsSporadic(G) then

%o                a:=1;

%o                break;

%o             fi;

%o          od;

%o          if a=0 then

%o             Add(L,i);

%o          fi;

%o       fi;

%o    od;

%o    return L;

%o end;;

%o SporadicNumbers(1,4095);

%o # gives: [ 266, 506, 759, 1045, 1288, 1463, 3795 ]

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

%K nonn,fini,more

%O 1,1

%A _Sébastien Palcoux_, Aug 27 2019