A327912 Orders of perfect non-simple groups.

120, 336, 720, 960, 1080, 1320, 1344, 1920, 2160, 2184, 2688, 3000, 3600, 3840, 4860, 4896, 5040, 5376, 5760, 6840, 7200, 7500, 7560, 7680, 9720, 10080, 10752, 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

Offset: 1

Author

Sébastien Palcoux, Sep 29 2019

Keywords

• nonn

Comments

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.

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.

References

• The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
• D.F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.

Links

• The GAP Group, GAP Data Library "Perfect Groups".

Crossrefs

Cf. A001034, A005180, A060793.

Programs

GAP

OrderPerfectNonSimple:=function(n1,n2)
   local it,S,G,L,o,No,i,c;
   it:=SimpleGroupsIterator(n1,n2);
   S:=[];
   for G in it do
      Add(S,Order(G));
   od;
   L:=[];
   for o in [n1..n2] do
      c:=0;
      for i in S do
         if i=o then
            c:=c+1;
         fi;
      od;
      No:=NumberPerfectGroups(o);
      if No>c then
         Add(L,o);
         if c>0 then
            Print([o,c,No]);
         fi;
      fi;
   od;
   return L;
end;;