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
KEYWORD
nonn,hard
AUTHOR
Jianing Song, Aug 14 2023
STATUS
approved