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A364944 Order of Aut^4(C_n) = Aut(Aut(Aut(Aut(C_n)))), where C_n is the cyclic group of order n.

%I A364944 #22 Aug 22 2023 21:19:21
%S A364944 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,
%T A364944 384,144,8,384,12,12,1,384,4608,1152,12,12,144,384,2,4,4608,12,8,1536,
%U A364944 384,64,1,2359296,336,144,12,12,4608,1152,8,13824,1536,36864,144,24
%N A364944 Order of Aut^4(C_n) = Aut(Aut(Aut(Aut(C_n)))), where C_n is the cyclic group of order n.
%H A364944 Jianing Song, <a href="/A364944/b364944.txt">Table of n, a(n) for n = 1..159</a>
%e A364944 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.
%e A364944 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.
%e A364944 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.
%e A364944 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.
%o A364944 (GAP) A364944 := function(n)
%o A364944 local G, i, L;
%o A364944 G := CyclicGroup(n);
%o A364944 for i in [1..4] do
%o A364944 G := AutomorphismGroup(G);
%o A364944 if i = 4 then return Size(G); fi;
%o A364944 L := DirectFactorsOfGroup(G);
%o A364944 if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
%o A364944 L := List(L, x->IdGroup(x));
%o A364944 G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
%o A364944 fi; od; end;
%o A364944 # it should be noted that the calculation of Aut^4(C_n) can by extremely lengthy for even small n (for example n = 80)
%Y A364944 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).
%K A364944 nonn,hard
%O A364944 1,8
%A A364944 _Jianing Song_, Aug 14 2023