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

1, 6, 8, 12, 16, 20, 24, 32, 36, 40, 42, 48, 54, 64, 72, 80, 84, 96, 108, 110, 120, 126, 128, 144, 156, 160, 162, 168, 192, 216, 220, 240, 252, 256, 272, 288, 312, 320, 324, 336, 342, 378, 384, 432, 440, 468, 480, 486, 500, 504, 506, 512, 544, 550, 576, 624, 640, 648, 660, 672

Offset: 1

Author

Miles Englezou, Apr 19 2024

Keywords

• nonn

Comments

Contains A341298 as a subsequence.

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.

Examples

			a(2) = 6 since the symmetric group of order 6 has 6 automorphisms.
		

Crossrefs

Cf. A341298 (orders of complete groups).

Programs

GAP

for n in [1..32] do
    for i in [1..NrSmallGroups(n)] do
        if Size(AutomorphismGroup(SmallGroup(n,i))) = n then
            Print(n,","," ");
            break;
        fi;
    od;
od;

Formula

|Out(G)|<=|G| for every entry.