// this script checks all genus 6 generating triples
// setup
SetLogFile("genus6check.log":Overwrite := true);

CheckTriple := function(sigma,g,ords,G,a,b,c);
print sigma,g,ords;
print G, [a,b,c];
l := ords[1];
m := ords[2];
n := ords[3];
RH := (2*sigma-2)/g -(1-1/l-1/m-1/n);
return([ RH eq 0, g eq #G, 
         #G eq #sub<G|[a,b,c]>, 
         l eq Order(a),m eq Order(b), n eq Order(c), a*b*c eq Identity(G)]); 
end function;


// tests

print "###case 1";
G<x> := CyclicGroup(13);
CheckTriple(6,13,[13,13,13],G,x,x,x^11);

print "###case 2";
G<x> := CyclicGroup(13);
CheckTriple(6,13,[13,13,13],G,x,x^2,x^10);

print "###case 3";
G<x> := CyclicGroup(13);
CheckTriple(6,13,[13,13,13],G,x,x^3,x^9);

print "###case 4";
G<x> := CyclicGroup(14);
CheckTriple(6,14,[7,14,14],G,x^12,x,x);

print "###case 5";
G<x> := CyclicGroup(14);
CheckTriple(6,14,[7,14,14],G,x^10,x,x^3);

print "###case 6";
G<x> := CyclicGroup(14);
CheckTriple(6,14,[7,14,14],G,x^2,x,x^11);

print "###case 7";
G<x> := CyclicGroup(15);
CheckTriple(6,15,[5,15,15],G,x^12,x^2,x);

print "###case 8";
G<x> := CyclicGroup(15);
CheckTriple(6,15,[5,15,15],G,x^3,x,x^11);

print "###case 9";
G<x> := CyclicGroup(16);
CheckTriple(6,16,[4,16,16],G,x^12,x3,x);


print "###case 10";
G<x> := CyclicGroup(18);
CheckTriple(6,18,[3,18,18],G,x^6,x,x^11);

print "###case 11";
G<x> := CyclicGroup(20);
CheckTriple(6,20,[4,5,20],G,x^5,x^4,x^11);

print "###case 12";
G<x> := CyclicGroup(21);
CheckTriple(6,21,[3,7,21],G,x^7,x^3,x^11);

print "###case 13";
G<x> := CyclicGroup(24);
CheckTriple(6,24,[2,24,24],G,x^12,x,x^11);

print "###case 14";
G<a,b,c,d> := SmallGroup(24,1);
x := a;
y := d;
CheckTriple(6,24,[3,8,8],G,y,(x*y)^(-1),x);
[Order(x),Order(y)];
[y^x,y^(-1)];

print "###case 15";
G<x,y,z,w> := SmallGroup(24,4);
CheckTriple(6,24,[4,4,12],G,x,x*y*z*w,y*z*w^2);

print "###case 16";
G<x,y,z,w> := SmallGroup(24,3);
CheckTriple(6,24,[4,6,6],G,y,x*y*z*w,x^2*y*z);

print "###case 17";
G<x,y,z,w> := PolycyclicGroup<x,y,z,w|x^2,y^2,z^2,w^3,y^x=y*z>;
CheckTriple(6,24,[4,6,6],G,x*y*z,x*z*w^2,y*z*w);

print "###case 18";
G<x,y> := AbelianGroup([5,5]);
CheckTriple(6,25,[5,5,5],G,x,y,-x-y);

print "###case 19";
G<x> := CyclicGroup(26);
CheckTriple(6,26,[2,13,26],G,x^13,x^2,x^11);

print "###case 20";
G<x,y> := AbelianGroup([14,2]);
CheckTriple(6,28,[2,14,14],G,y,-x+y,x);

print "###case 21";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2=b,b^2,c^7,c^a=c^6>;
x := a;
y := c;
CheckTriple(6,28,[4,4,7],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^(-1)];


print "###case 22";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2,b^3,c^5,b^a=b^2>;
x := a;
y := b*c;
CheckTriple(6,30,[2,10,15],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^11];

print "###case 23";
G<a,b,c,d> := SmallGroup(36,3);
x := a;
y := c;
z := d;
CheckTriple(6,36,[2,9,9],G,y,(x*y)^(-1),x);
[Order(x),Order(y),Order(z)];
[y^x,z];
[z^x,y*z];

print "###case 24";
G<x,y> := SmallGroup(39,1);
CheckTriple(6,39,[3,3,13],G,x,(y*x)^(-1),y);

print "###case 25";
G<a,b,c,d,e> := SmallGroup(48,6);
x := a;
y := b*e;
CheckTriple(6,48,[2,4,24],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^11];

print "###case 26";
G<x,y,z,w,v> := SmallGroup(48,15);
CheckTriple(6,48,[2,6,8],G,x,y*v^2,x*y*z*v^2);

print "###case 27";
G<x,y,z,w,v> := SmallGroup(48,29);
CheckTriple(6,48,[2,6,8],G,x,y^2*z*w*v,x*y^2*w*v);

print "###case 28";
G<x,y,z> := SmallGroup(50,3);
CheckTriple(6,50,[2,5,10],G,x,y^3*z^3,x*y^2*z^3);

print "###case 29";
G<x,y,z,w> := SmallGroup(56,7);
CheckTriple(6,56,[2,4,14],G,x,x*y*w^2,y*w^5);

print "###case 30";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^2,b^3=c,c^3,d^2,e^2,b^a=b^2*c^2,c^a=c^2,d^a=e,e^a=d,d^b=e,e^b=d*e>;
x := a;
y := b;
z := d;
w := e;
CheckTriple(6,72,[2,4,9],G,x,(y*z*x)^(-1),y*z);
[Order(x),Order(y),Order(z),Order(w)];
[y^x,y^(-1)];
[z^x,w];
[w^x,z];
[z^y,w];
[w^y,z*w];

print "###case 31";
G<x,y,z> := SmallGroup(75,2);
CheckTriple(6,75,[3,3,5],G,x,x^2*y^4,y);

print "###case 32";
G := SymmetricGroup(5);
a := G!(1, 2)(3, 4);
b := G!(2, 4, 5, 3);
c := (a*b)^(-1);
CheckTriple(6,120,[2,4,6],G,a,b,c);

print "###case 33";
G<x,y,z,w> := SmallGroup(150,5); 
CheckTriple(6,150,[2,3,10],G,x,y^2*z^4*w^3,x*y^2*w);


// finish
UnsetLogFile();