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

1,3

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.

Links

Table of n, a(n) for n=1..74.

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A061915 A366898 A138565 * A064851 A305353 A134458

Adjacent sequences: A137313 A137314 A137315 * A137317 A137318 A137319

Keyword

nonn,tabf

Author

Benoit Jubin, Apr 06 2008, Apr 15 2008

Status

approved