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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.

%I A137316 #25 Feb 28 2024 13:26:54
%S A137316 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 A137316 8,16,16,16,32,32,32,32,48,64,96,192,192,20160,16,6,12,48,54,432,18,8,
%U A137316 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 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.
%C A137316 The length of the n-th row is A000001(n).
%C A137316 The largest value of the n-th row is A059773(n).
%C A137316 The number phi(n) = A000010(n) appears in the n-th row.
%H A137316 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 A137316 The table begins as follows:
%e A137316    1
%e A137316    1
%e A137316    2
%e A137316    2   6
%e A137316    4
%e A137316    2   6
%e A137316    6
%e A137316    4   8   8  24 168
%e A137316    6  48
%e A137316    4  20
%e A137316   10
%e A137316    4  12  12  12  24
%e A137316   12
%e A137316    6  42
%e A137316 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.
%o A137316 (GAP) # GAP 4
%o A137316 Print("\n") ;
%o A137316 for o in [ 1 .. 33 ] do
%o A137316     n := NumberSmallGroups(o) ;
%o A137316     og := [] ;
%o A137316     for i in [1 .. n] do
%o A137316         g := SmallGroup(o,i) ;
%o A137316         H := AutomorphismGroup(g) ;
%o A137316         ho := Order(H) ;
%o A137316         Add(og,ho) ;
%o A137316     od;
%o A137316     Sort(og) ;
%o A137316     Print(og) ;
%o A137316     Print("\n") ;
%o A137316 od; # _R. J. Mathar_, Jul 13 2013
%Y A137316 Cf. A064767, A060249, A060817, A062771, A060249, A002618, A061350.
%K A137316 nonn,tabf
%O A137316 1,3
%A A137316 _Benoit Jubin_, Apr 06 2008, Apr 15 2008