// this script checks all genus 5 generating triples
// setup
SetLogFile("genus5check.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(11);
CheckTriple(5,11,[11,11,11],G,x,x,x^9);

print "###case 2";
G<x> := CyclicGroup(11);
CheckTriple(5,11,[11,11,11],G,x,x^2,x^8);

print "###case 3";
G<x> := CyclicGroup(12);
CheckTriple(5,12,[6,12,12],G,x^2,x^5,x^5);

print "###case 4";
G<x> := CyclicGroup(15);
CheckTriple(5,15,[3,15,15],G,x^5,x^2,x^8);

print "###case 5";
G<x,y> := AbelianGroup([8,2]);
CheckTriple(5,16,[4,8,8],G,2*x+y,x,5*x+y);

print "###case 6";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2,b^2=c,c^2=d,d^2,b^a=b*d>;
x := a;
y := b;
CheckTriple(5,16,[4,8,8],G,x*y^2,x*y,y^5);
[Order(x),Order(y)];
[y^x,y^5];


print "###case 7";
G<x> := CyclicGroup(20);
CheckTriple(5,20,[2,20,20],G,x^10,x,x^9);

print "###case 8";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2=b,b^2,c^5=1,c^a=c^4>;
x := a;
y := c;
CheckTriple(5,20,[4,4,10],G,x,x*y,x^2*y^4);
[Order(x),Order(y)];
[y^x,y^(-1)];

print "###case 9";
G<x> := CyclicGroup(22);
CheckTriple(5,22,[2,11,22],G,x^11,x^2,x^9);

print "###case 10";
G<x,y> := AbelianGroup([12,2]);
CheckTriple(5,24,[2,12,12],G,6*x+y,x,5*x+y);

print "###case 11";
G<x,y,z,w> := PolycyclicGroup<x,y,z,w|x^3,y^2,z^2,w^2,z^x=w,w^x=z*w>;
CheckTriple(5,24,[3,6,6],G,x,x*y*z*w,x*y*z);

print "###case 12";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2=b,b^2,c^2,d^3,d^a=d^2>;
x := a;
y := c*d;
CheckTriple(5,24,[4,4,6],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^(-1)];

print "###case 13";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2,b^3,c^5,c^a=c^4>;
x := a*b;
y := c;
CheckTriple(5,30,[2,6,15],G,x^3,x*y,x^2*y^(-1));
[Order(x),Order(y)];
[y^x,y^(-1)];

print "###case 14";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^2=b,b^2=c,c^2,d^2,e^2,d^a=d*e>;
x := a;
y := d;
z := e;
CheckTriple(5,32,[2,8,8],G,y,(x*y)^(-1),x);
[Order(x),Order(y), Order(z)];
[y^x,y*z];

print "###case 15";
G<x,y,z,w,v> := SmallGroup(32,7);
CheckTriple(5,32,[2,8,8],G,y,x*v,x*y*w*v);

print "###case 16";
G<a,b,c,d,e> := SmallGroup(32,2);
x := a;
y := b;
z := c;
CheckTriple(5,32,[4,4,4],G,y,x*y^2,x^3*y);
[Order(x),Order(y),Order(z)];
[y^x,y*z];

print "###case 17";
G<a,b,c,d,e> := SmallGroup(32,6);
x := a;
y := b;
z := c;
w := e;
CheckTriple(5,32,[4,4,4],G,x,x*y,(x^2*y)^(-1));
[Order(x),Order(y), Order(z),Order(w)];
[y^x,y*z];
[z^x,z*w];
 
print "###case 18";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2, b^2=c,c^2,d^5,d^a=d^4>;
x := a;
y := b*d;
CheckTriple(5,40,[2,4,20],G,x,x*y^(-1),y);
[Order(x),Order(y)];
[y^x,y^9];

print "###case 19";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^2,b^2=c,c^2=1,d^3,e^2,b^a=b*e,d^a=d^2>;
x := a;
y := b*d;
z := e;
CheckTriple(5,48,[2,4,12],G,x,(y*z*x)^(-1),y*z);
[Order(x),Order(y), Order(z)];
[y^x,y^5*z];
[z^x,z];


print "###case 20";
G<x,y,z,w,v> := SmallGroup(48,30);
CheckTriple(5,48,[3,4,4],G,z,x*w*v,x*y*z^2*v);

print "###case 21";
G<x,y,z,w,v> := SmallGroup(48,30);
CheckTriple(5,48,[3,4,4],G,z,x*z*w,x*y*v);

print "###case 22";
G := AlternatingGroup(5);
x := G!(1,2,3);
y := G!(3,4,5);
CheckTriple(5,60,[3,3,5],G,x,y,(x*y)^(-1));


print "###case 23";
G<x,y,z,w,v,u> := SmallGroup(64,8);
CheckTriple(5,64,[2,4,8],G,y,x*z*u,x*y*z*w);

print "###case 24";
G<a,b,c,d,e,f> := PolycyclicGroup<a,b,c,d,e,f|a^2=b,b^2,c^2,d^2,e^2,f^2,c^a=c*d,d^a=d*e,e^a=e*f,c^b=c*e,d^b=d*f>;
x := a;
y := c;
z := d;
w:= e;
v := f;
CheckTriple(5,64,[2,4,8],G,y,x,(y*x)^(-1));
[Order(x),Order(y), Order(z),Order(w),Order(v)];
[y^x,y*z];
[z^x,z*w];
[w^x,w*v];
[v^x,v];

print "###case 25";
G<x,y,z,w,v> := SmallGroup(80,49);
CheckTriple(5,80,[2,5,5],G,y,x*y*w,x^4*z*w*v);

print "###case 26";
G<x,y,z,w,v,u> := PolycyclicGroup<x,y,z,w,v,u|x^2,y^3,z^2,w^2,v^2,u^2,y^x=y^2,z^x=z*w,v^x=u,u^x=v,v^y=u,u^y=v*u>;
CheckTriple(5,96,[2,4,6],G,x,x*y*z*v,(y*z*v)^(-1));

print "###case 27";
G<x,y,z,w,v,u> := SmallGroup(96,3);
CheckTriple(5,96,[3,3,4],G,x,x^2*z*u,(z*u)^(-1));

print "###case 28";
P := SymmetricGroup(7);
G<x,y,z> := sub<P |[P!(1,2)(3,4),P!(1,2,3,4,5),P!(6,7)]>;
CheckTriple(5,120,[2,3,10],G,x*z,(y*x)^(-1), y*z);

print "###case 29";
G<x,y,z,w,v,u> := SmallGroup(160,234);
CheckTriple(5,160,[2,4,5],G,x,x*y^4*w*v,y*w*u);

print "###case 30";
G<x,y,z,w,v,u,t> := SmallGroup(192,181);
CheckTriple(5,192,[2,3,8],G,x,y*z*v*t,x*y*z*w*v);



// finish
UnsetLogFile();