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

1,1

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

Table of n, a(n) for n=1..68.

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

Prog

(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;;

Crossrefs

Cf. A001034, A005180, A060793.

Sequence in context: A275083 A098114 A135805 * A118058 A269037 A144524

Adjacent sequences: A327909 A327910 A327911 * A327913 A327914 A327915

Keyword

nonn

Author

Sébastien Palcoux, Sep 29 2019

Status

approved