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%I #41 Aug 18 2023 08:25:57
%S 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,
%T 64,24,8,64,12,12,2,64,1152,192,12,12,24,64,4,10,1152,12,8,768,64,16,
%U 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
%N Order of Aut^3(C_n) = Aut(Aut(Aut(C_n))), where C_n is the cyclic group of order n.
%H Jianing Song, <a href="/A364129/b364129.txt">Table of n, a(n) for n = 1..200</a>
%H Jianing Song, <a href="/A364129/a364129.txt">Structure and SmallGroupId of Aut^3(C_n) for n <= 100</a>
%e 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).
%e 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).
%e 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).
%o (GAP) A364129 := function(n)
%o local G, i, L;
%o G := CyclicGroup(n);
%o for i in [1..3] do
%o G := AutomorphismGroup(G);
%o if i = 3 then return Size(G); fi;
%o L := DirectFactorsOfGroup(G);
%o if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
%o L := List(L, x->IdGroup(x));
%o G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
%o fi; od; end;
%Y 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).
%K nonn,hard
%O 1,8
%A _Jianing Song_, Aug 13 2023
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