// AutTest is modified from Dr. S. Allen Broughton's 
// ThetaQuest as part of tileclass.mgm

AutTest := function(G,a,b,ta,tb)
   	aP := {@ <a^i,ta^i> : i in [1..(Order(a)-1)] @};
   	bP := {@ <b^i,tb^i> : i in [1..(Order(b)-1)] @};
	
	W := {@<Id(G),Id(G)>@};
	W1 := W;
	W2 := W;
	P1 := aP;
	P2 := bP;
	while #W lt #G do
	     	W1 := {@ <g[1]*x[1],g[2]*x[2]> : g in P1, x in W1 @};
	     	W2 := {@ <g[1]*x[1],g[2]*x[2]> : g in P2, x in W2 @};
	     	W := W join (W1 join W2);
	    	P3 := P1;
		P1 := P2;
		P2 := P3;
	end while;

   	X   := {@ z[1]:z in W @};
   	thX := {@ z[2]:z in W @};
   	LaX := {@ a*x :x in X @};
   	LbX := {@ b*x :x in X @};
   	LtaX := {@ ta*x :x in X @};
   	LtbX := {@ tb*x :x in X @};

	if not (#X eq #G) then 
		return false; 
	end if;
   	if not (#thX eq #G) then 
		return false; 
	end if;

   	iY  := [Position(X,z):z in thX];
   	iLa := [Position(X,z):z in LaX];
   	iLb := [Position(X,z):z in LbX];
   	iLta := [Position(X,z):z in LtaX];
   	iLtb := [Position(X,z):z in LtbX];

   	S := SymmetricGroup(#G);
   	th := S!iY;
   	La := S!iLa;
   	Lb := S!iLb;
   	Lta := S!iLta;
   	Ltb := S!iLtb;


	if (La^th eq Lta) and (Lb^th eq Ltb) then 
		return true;
	else 
		return false;
      	end if;
end function;



Graph := function(G, l, m, n)
	aleph := [];
	beis := [];

  	K := {@ g : g in G @};
	for p in K do
		if Order(p) eq l then
			for q in K do
				if Order(q) eq m then
					if Order(q^-1*p^-1) eq n then
						if #G eq #sub<G | p, q> then
							aleph := Append(aleph,p);
							beis := Append(beis,q);
						end if;
					end if;
				end if;
			end for;
		end if;
	end for;

	Alpha := [aleph[1]];
	Beta := [beis[1]];

	for w in [2..#aleph] do
		automorphic := false;
		for k in [1..#Alpha] do
			if  automorphic then
				break k;
			end if;
			automorphic := AutTest(G,Alpha[k],Beta[k],aleph[w],beis[w]);
		end for;
		if not automorphic then
			Alpha := Append(Alpha,aleph[w]);
			Beta := Append(Beta,beis[w]);
		end if;
	end for;

	for w in [1..#Alpha] do

		a := Alpha[w];
		b := Beta[w];
		c := b^-1*a^-1;
	
		S, V, E := EmptyGraph(3*#G);
		for k in [1..#G] do	
	 		S := S + {V!k, V!(k + #G)} + {V!k, V!(k + 2*#G)} + {V!(k + #G), V!(k + 2*#G)};
			S := S + {V!k, V!(Position(K, K[k]*a) + #G)} + {V!k, V!(Position(K, K[k]*c^-1) + 2*#G)} + {V!(k + #G), V!(Position(K, K[k]*b) + 2*#G)};
		end for;
		//print "Graph Made";
		ia := a^-1;
		ib := b^-1;
		ic := c^-1;	

		aP := {@ <a^k, ia^k> : k in [1..(l-1)] @};
		bP := {@ <b^k, ib^k> : k in [1..(m-1)] @};
		cP := {@ <c^k, ic^k> : k in [1..(n-1)] @};

		P := {@<Id(G),Id(G)>@};
		P1 := P;
		P2 := P;
		W1 := aP;
		W2 := cP;
		while #P lt #G do
			P1 := {@ <g[1]*x[1],g[2]*x[2]> : g in W1, x in P1 @};
			P2 := {@ <g[1]*x[1],g[2]*x[2]> : g in W2, x in P2 @};
		        P := P join (P1 join P2);
	        	W3 := W1;
			W1 := W2;
			W2 := W3;
		end while;
	
		Q := {@<Id(G),Id(G)>@};
		Q1 := Q;
		Q2 := Q;
		W1 := aP;
		W2 := bP;
		while #Q lt #G do
			Q1 := {@ <g[1]*x[1],g[2]*x[2]> : g in W1, x in Q1 @};
			Q2 := {@ <g[1]*x[1],g[2]*x[2]> : g in W2, x in Q2 @};
	        	Q := Q join (Q1 join Q2);
	        	W3 := W1;
			W1 := W2;
			W2 := W3;
		end while;
	

		R := {@<Id(G),Id(G)>@};
		R1 := R;
		R2 := R;
		W1 := bP;
		W2 := cP;
		while #R lt #G do
			R1 := {@ <g[1]*x[1],g[2]*x[2]> : g in W1, x in R1 @};
			R2 := {@ <g[1]*x[1],g[2]*x[2]> : g in W2, x in R2 @};
		        R := R join (R1 join R2);
		        W3 := W1;
			W1 := W2;
			W2 := W3;
		end while;

		//print "Theta's found!";
		Rp := [];
		Rq := [];
		Rr := [];	

		for p in P do
			if p[1] eq p[2] then
				Rp := Append(Rp,V!(Position(K,p[1])));
			end if;
			if p[1] eq p[2]*ia^-1 then
				Rp := Append(Rp,V!(Position(K,p[1]) + #G));
			end if;
			if p[1] eq p[2]*ic then
				Rp := Append(Rp,V!(Position(K,p[1]) + 2*#G));
			end if;
		end for;
		
		for q in Q do
			if q[1] eq q[2] then
				Rq := Append(Rq,V!(Position(K,q[1]) + #G));
			end if;
			if q[1] eq q[2]*ia then
				Rq := Append(Rq,V!(Position(K,q[1])));
			end if;
			if q[1] eq q[2]*ib^-1 then
				Rq := Append(Rq,(Position(K,q[1]) + 2*#G));
			end if;
		end for;


		for r in R do
			if r[1] eq r[2] then
				Rr := Append(Rr,V!(Position(K,r[1]) + 2*#G));
			end if;	
			if r[1] eq r[2]*ic^-1 then
				Rr := Append(Rr,V!(Position(K,r[1])));
			end if;
			if r[1] eq r[2]*ib then
				Rr := Append(Rr,V!(Position(K,r[1]) + #G));
			end if;
		end for;

		Rp := {r : r in Rp};	
		Rq := {r : r in Rq};
		Rr := {r : r in Rr};	
		
		//print "Fixed points found!";

		//print "a = ", a;
		//print "b = ", b;
		//print "c = ", c;
		
		if IsConnected(S - Rp) then
			//print "The tiling does not seperate along the p reflection.";
		else
			//print "The tiling seperates along the p reflection.";
		end if;	
		if IsConnected(S - Rq) then
			//print "The tiling does not seperate along the q reflection.";
		else
			//print "The tiling seperates along the q reflection.";
		end if;
		if IsConnected(S - Rr) then
			//print "The tiling does not seperate along the r reflection.";
		else
			//print "The tiling seperates along the r reflection.";
		end if;
		
	end for;	


	return <#G,#Alpha,"G">;

	//return "";
end function;