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, 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, 8, 16, 16, 16, 32, 32, 32, 32, 48, 64, 96, 192, 192, 20160, 16, 6, 12, 48, 54, 432, 18, 8, 20, 24, 40, 40, 12, 42, 10, 110, 22, 8, 16, 16, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 144, 336

Offset: 1

Author

Benoit Jubin, Apr 06 2008, Apr 15 2008

Keywords

• nonn
• tabf

Comments

The length of the n-th row is A000001(n).

The largest value of the n-th row is A059773(n).

The number phi(n) = A000010(n) appears in the n-th row.

Examples

			The table begins as follows:
   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
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)^* = Z_2 and the symmetric group S_3.
		

Links

• D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

Crossrefs

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

Programs

GAP

# GAP 4
Print("\n") ;
for o in [ 1 .. 33 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        H := AutomorphismGroup(g) ;
        ho := Order(H) ;
        Add(og,ho) ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013