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A327912 Orders of perfect non-simple groups.

%I A327912 #12 Sep 30 2019 09:11:11
%S A327912 120,336,720,960,1080,1320,1344,1920,2160,2184,2688,3000,3600,3840,
%T A327912 4860,4896,5040,5376,5760,6840,7200,7500,7560,7680,9720,10080,10752,
%U A327912 11520,12144,14400,14520,14580,15000,15120,15360,15600,16464,17280,19656,20160,21504,21600,23040,24360,28224,29160,29760,30240,30720,32256,34560,37500,39600,40320,43008,43200,43320,43740,46080,48000,50616,51840,56448,57600,57624,58240,58320,60480
%N A327912 Orders of perfect non-simple groups.
%C A327912 The smallest number n such that there is a simple group and a non-simple perfect group of order n is 20160. So this sequence is A060793 minus A001034 (as sets) for the orders less than 20160. The next known such exceptions are 181440, 262080, 443520 and 604800.
%C A327912 The perfect groups of order 61440, 122880, 172032, 245760, 344064, 491520, 688128, 983040 have not completely been determined yet. Then GAP neither provides the number of these groups nor the groups themselves.
%D A327912 The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
%D A327912 D.F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.
%H A327912 The GAP Group, <a href="https://www.gap-system.org/Datalib/perfect.html">GAP Data Library "Perfect Groups"</a>.
%o A327912 (GAP)
%o A327912 OrderPerfectNonSimple:=function(n1,n2)
%o A327912    local it,S,G,L,o,No,i,c;
%o A327912    it:=SimpleGroupsIterator(n1,n2);
%o A327912    S:=[];
%o A327912    for G in it do
%o A327912       Add(S,Order(G));
%o A327912    od;
%o A327912    L:=[];
%o A327912    for o in [n1..n2] do
%o A327912       c:=0;
%o A327912       for i in S do
%o A327912          if i=o then
%o A327912             c:=c+1;
%o A327912          fi;
%o A327912       od;
%o A327912       No:=NumberPerfectGroups(o);
%o A327912       if No>c then
%o A327912          Add(L,o);
%o A327912          if c>0 then
%o A327912             Print([o,c,No]);
%o A327912          fi;
%o A327912       fi;
%o A327912    od;
%o A327912    return L;
%o A327912 end;;
%Y A327912 Cf. A001034, A005180, A060793.
%K A327912 nonn
%O A327912 1,1
%A A327912 _Sébastien Palcoux_, Sep 29 2019