%I #16 Jan 31 2014 09:59:55

%S 1,0,1,1,3,0,5,3,8,0,9,3,11,0,3,9,15,0,17,3,5,0,21,5,34,0,35,5,27,0,

%T 29,17,9,0,15,18,35,0,11,9,39,0,41,9,24,0,45,21,76,0,15,11,51,0,27,19,

%U 17,0,57,15,59,0,40,97,33,0,65,15,21,0,69,37,71,0,39,17,45,0,77,34,218,0,81,15,45,0,27,27,87,0,55,21,29,0,51,43,95,0,72,34

%N The number of isomorphism classes of connected, Generalized Alexander quandles of order n.

%C Given a group G and an automorphism f of G define the binary operation * on G by x*y = f(xy^(-1))y. Then (G,*) is a quandle. We call this a Generalized Alexander quandle. If G is abelian then (G,*) is an Alexander quandle (see A193024). (G,*) is connected if the group generated by the right translations of (G,*) is transitive on G.

%H J. Scott Carter, <a href="http://arxiv.org/abs/1002.4429">A Survey of Quandle Ideas</a>, arXiv:1002.4429 [math.GT]

%H W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, <a href="http://arxiv.org/abs/1312.3307">Quandle Colorings of Knots and Applications</a>, arXiv preprint arXiv:1312.3307, 2013

%o (GAP)

%o IsConnected:=function(A)

%o local B,LL;

%o B:=TransposedMat(A);

%o LL:=List(B,x->PermList(x));

%o return IsTransitive(Group(LL),[1..Length(A)]);

%o end;;

%o MakeGAlex:=function(f,g)

%o local e,n,QM,i,j;

%o e:=Elements(g);

%o n:=Length(e);

%o QM:=List([1..n],t->[1..n]);

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

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

%o QM[i][j]:=Position(e,Image(f,e[i]*e[j]^(-1))*e[j]);

%o od;

%o od;

%o return QM;

%o end;;

%o a:=[];;

%o for n in [1..100] do

%o a[n]:=0;

%o N:=NrSmallGroups(n);

%o for u in [1..N] do

%o g:=SmallGroup(n,u);

%o ag:=AutomorphismGroup(g);;

%o eag:=List(ConjugacyClasses(ag),Representative);

%o for t in eag do

%o QM:=MakeGAlex(t,g);

%o if IsConnected(QM) then a[n]:=a[n]+1; fi;

%o od;

%o od;

%o od;;

%o a;

%Y Cf. A193067, A181771.

%Y See also Index to OEIS under quandles.

%K nonn

%O 1,5

%A _W. Edwin Clark_, Aug 04 2013