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A137316 Array read by rows: T(n,k) is the number of automorphisms of the k^th group of order n, where the ordering is such that the rows are nondecreasing. 1

%I

%S 1,1,2,2,6,4,2,6,6,4,8,8,24,168,6,48,4,20,10,4,12,12,12,24,12,6,42,8,

%T 8,16,16,16,32,32,32,32,48,64,96,192,192,20160,16,6,12,48,54,432,18,8,

%U 20,24,40,40,12,42,10,110,22,8,16,16,24,24,24,24,24,24,48,48,48,48,144,336

%N Array read by rows: T(n,k) is the number of automorphisms of the k^th group of order n, where the ordering is such that the rows are nondecreasing.

%C The length of the n^th row is A000001(n).

%C The largest value of the n^th row is A059773(n).

%C The number phi(n) = A000010(n) appears in the n^th row.

%H D. MacHale and R. Sheehy, <a href="http://www.jstor.org/stable/40656888">Finite groups with few automorphisms</a>, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

%e The table begins as follows:

%e 1

%e 1

%e 2

%e 2 6

%e 4

%e 2 6

%e 6

%e 4 8 8 24 168

%e 6 48

%e 4 20

%e 10

%e 4 12 12 12 24

%e 12

%e 6 42

%e The first row with two numbers corresponds to the two groups of order 4, the cyclic group Z_4 and the Klein group Z_2 x Z_2, whose automorphism groups are respectively the group (Z_4)^x = Z_2 and the symmetric group S_3.

%o (GAP4) Print("\n") ;

%o for o in [ 1 .. 33 ] do

%o n := NumberSmallGroups(o) ;

%o og := [] ;

%o for i in [1 .. n] do

%o g := SmallGroup(o,i) ;

%o H := AutomorphismGroup(g) ;

%o ho := Order(H) ;

%o Add(og,ho) ;

%o od;

%o Sort(og) ;

%o Print(og) ;

%o Print("\n") ;

%o od; # _R. J. Mathar_, Jul 13 2013

%Y Cf. A064767, A060249, A060817, A062771, A060249, A002618, A061350.

%K nonn,tabf

%O 1,3

%A _Benoit Jubin_, Apr 06 2008, Apr 15 2008

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)