

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, 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
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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), 231238.


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)^x = Z_2 and the symmetric group S_3.


PROG

(GAP4) 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: A092384 A061915 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



