cp's OEIS Frontend

A372011 Orders of finite groups for which there is at least one group G such that |Aut(G)| = |G|.

%I A372011 #21 May 06 2024 23:04:52
%S A372011 1,6,8,12,16,20,24,32,36,40,42,48,54,64,72,80,84,96,108,110,120,126,
%T A372011 128,144,156,160,162,168,192,216,220,240,252,256,272,288,312,320,324,
%U A372011 336,342,378,384,432,440,468,480,486,500,504,506,512,544,550,576,624,640,648,660,672
%N A372011 Orders of finite groups for which there is at least one group G such that |Aut(G)| = |G|.
%C A372011 Contains A341298 as a subsequence.
%C A372011 Every symmetric group S_n of order n! is a member for every n not equal to 2 or 6 since every such S_n is complete.
%F A372011 |Out(G)|<=|G| for every entry.
%e A372011 a(2) = 6 since the symmetric group of order 6 has 6 automorphisms.
%o A372011 (GAP)
%o A372011 for n in [1..32] do
%o A372011     for i in [1..NrSmallGroups(n)] do
%o A372011         if Size(AutomorphismGroup(SmallGroup(n,i))) = n then
%o A372011             Print(n,","," ");
%o A372011             break;
%o A372011         fi;
%o A372011     od;
%o A372011 od;
%Y A372011 Cf. A341298 (orders of complete groups).
%K A372011 nonn
%O A372011 1,2
%A A372011 _Miles Englezou_, Apr 19 2024