A364944 Order of Aut^4(C_n) = Aut(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, 1, 6, 6, 1, 8, 8, 8, 1, 1, 8, 12, 1, 2, 336, 8, 6, 1, 12, 12, 8, 8, 384, 144, 8, 384, 12, 12, 1, 384, 4608, 1152, 12, 12, 144, 384, 2, 4, 4608, 12, 8, 1536, 384, 64, 1, 2359296, 336, 144, 12, 12, 4608, 1152, 8, 13824, 1536, 36864, 144, 24

Offset

1,8

Links

Jianing Song, Table of n, a(n) for n = 1..159

Example

For n = 69, we have Aut(C_69) = C_2 X C_22, Aut^2(C_69) = C_10 X S_3, Aut^3(C_69) = C_4 X D_12 and Aut^4(C_69) = SmallGroup(32,27) X S_3, so a(69) = |SmallGroup(32,27) X S_3| = 192.
For n = 972, we have Aut(C_972) = C_2 X C_162, Aut^2(C_972) = C_18 X D_12, Aut^3(C_972) = C_6 X S_3 X S_4 and Aut^4(C_972) = C_2 X C_2 X D_12 X S_4, so a(972) = |C_2 X C_2 X D_12 X S_4| = 1152.
For n = 1029, we have Aut(C_1029) = C_2 X C_294, Aut^2(C_1029) = C_42 X D_12, Aut^3(C_1029) = C_6 X D_12 X S_4 and Aut^4(C_1029) = D_12 X S_4 X SmallGroup(96,227), so a(1029) = |D_12 X S_4 X SmallGroup(96,227)| = 27648.
For n = 1944, we have Aut(C_1944) = C_2 X C_2 X C_162, Aut^2(C_1944) = C_2 X C_18 X PSL(2,7), Aut^3(C_1944) = C_6 X S_3 X PGL(2,7) and Aut^4(C_1944) = C_2 X C_2 X D_12 X PGL(2,7), so a(1944) = |C_2 X C_2 X D_12 X PGL(2,7)| = 16128.

Prog

(GAP) A364944 := function(n)
local G, i, L;
G := CyclicGroup(n);
for i in [1..4] do
G := AutomorphismGroup(G);
if i = 4 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;
# it should be noted that the calculation of Aut^4(C_n) can by extremely lengthy for even small n (for example n = 80)

Crossrefs

Cf. A000010 (order of Aut(C_n)), A258615 (order of Aut^2(C_n)), A364129 (order of Aut^3(C_n)), A364917 (order of Aut^k(C_n) for all sufficiently large k).

Sequence in context: A346013 A344697 A364917 * A320832 A034460 A063919

Adjacent sequences: A364941 A364942 A364943 * A364945 A364946 A364947

Keyword

nonn,hard

Author

Jianing Song, Aug 14 2023

Status

approved