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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.

%I A321224 #31 Sep 21 2019 14:56:34
%S A321224 266,506,759,1045,1288,1463,3795
%N 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.
%C A321224 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 A321224 There are no other sporadic numbers less than 4096 (see computation below).
%C A321224 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 A321224 Derek Holt suggested another sequence where we also allow the extensions of the sporadic simple groups.
%D A321224 The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
%H A321224 S. Palcoux, <a href="https://mathoverflow.net/q/336691/34538">The sporadic numbers</a> (version: 2019-07-22), MathOverflow.
%o A321224 (GAP)
%o A321224 IsSporadic:=function(G)
%o A321224    if not IsSimple(G) then
%o A321224       return false;
%o A321224    else
%o A321224       return IsomorphismTypeInfoFiniteSimpleGroup(G).series="Spor";
%o A321224    fi;
%o A321224 end;;
%o A321224 SporadicNumbers:=function(b1,b2)
%o A321224    local L,i,n,a,j,G;
%o A321224    L:=[];
%o A321224    for i in [b1..b2] do
%o A321224       n:=NrPrimitiveGroups(i);
%o A321224       if n>2 then
%o A321224          a:=0;
%o A321224          for j in [1..n] do
%o A321224             G:=PrimitiveGroup(i,j);
%o A321224             if not G=SymmetricGroup(i) and not G=AlternatingGroup(i) and not IsSporadic(G) then
%o A321224                a:=1;
%o A321224                break;
%o A321224             fi;
%o A321224          od;
%o A321224          if a=0 then
%o A321224             Add(L,i);
%o A321224          fi;
%o A321224       fi;
%o A321224    od;
%o A321224    return L;
%o A321224 end;;
%o A321224 SporadicNumbers(1,4095);
%o A321224 # gives: [ 266, 506, 759, 1045, 1288, 1463, 3795 ]
%Y A321224 Cf. A102842, A001228, A174601, A174848, A261717, A263447, A121236.
%K A321224 nonn,fini,more
%O A321224 1,1
%A A321224 _Sébastien Palcoux_, Aug 27 2019