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

Author

Jianing Song, Aug 14 2023

Keywords

• nonn
• hard

Examples

			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.
		

Links

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

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).

Programs

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)