A364129 Order of Aut^3(C_n) = Aut(Aut(Aut(C_n))), where C_n is the cyclic group of order n.

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 6, 6, 1, 8, 8, 8, 1, 2, 8, 12, 2, 4, 336, 8, 6, 2, 12, 12, 8, 8, 64, 24, 8, 64, 12, 12, 2, 64, 1152, 192, 12, 12, 24, 64, 4, 10, 1152, 12, 8, 768, 64, 16, 2, 128, 336, 24, 12, 12, 1152, 192, 8, 576, 768, 768, 24, 24, 768, 48, 64, 16, 336, 336, 12, 128, 24, 192, 64, 16, 6144

Offset

1,8

Links

Jianing Song, Table of n, a(n) for n = 1..200

Jianing Song, Structure and SmallGroupId of Aut^3(C_n) for n <= 100

Example

a(24) = 336 since Aut(C_24) = C_2 X C_2 X C_2, Aut^2(C_24) = PSL(2,7) and Aut(Aut(Aut(C_24))) = PGL(2,7).
a(32) = 64 since Aut(C_32) = C_2 X C_8, Aut^2(C_32) = C_2 X D_8 and Aut^3(C_32) = SmallGroup(64,138).
a(40) = 1152 since Aut(C_40) = C_2 X C_2 X C_4, Aut^2(C_40) = SmallGroup(192,1493) and Aut^3(C_40) = C_2 X SmallGroup(576,8654).

Prog

(GAP) A364129 := function(n)
local G, i, L;
G := CyclicGroup(n);
for i in [1..3] do
G := AutomorphismGroup(G);
if i = 3 then return Size(G); fi;
L := DirectFactorsOfGroup(G);
if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
L := List(L, x->IdGroup(x));
G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
fi; od; end;

Crossrefs

Cf. A000010 (order of Aut(C_n)), A258615 (order of Aut^2(C_n)), A364944 (order of Aut^4(C_n)), A364917 (order of Aut^k(C_n) for all sufficiently large k).

Sequence in context: A197420 A128423 A345628 * A336179 A350034 A133419

Adjacent sequences: A364126 A364127 A364128 * A364130 A364131 A364132

Keyword

nonn,hard

Author

Jianing Song, Aug 13 2023

Status

approved