// this script checks all genus 3 generating triples
// setup
SetLogFile("genus3check.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(7);
CheckTriple(3,7,[7,7,7],G,x,x,x^5);

print "###case 2";
G<x> := CyclicGroup(7);
CheckTriple(3,7,[7,7,7],G,x,x^2,x^4);

print "###case 3";
G<x> := CyclicGroup(8);
CheckTriple(3,8,[4,8,8],G,x^6,x,x);

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

print "###case 5";
G<x> := CyclicGroup(9);
CheckTriple(3,9,[3,9,9],G,x^3,x^5,x);

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

print "###case 7";
G<x> := CyclicGroup(12);
CheckTriple(3,12,[3,4,12],G,x^8,x^3,x);

print "###case 7";
G<x> := CyclicGroup(12);
CheckTriple(3,12,[3,4,12],G,x^8,x^3,x);

print "###case 8";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2=b,b^2,c^3,c^a=c^2>;
x := a;
y := c;
CheckTriple(3,12,[4,4,6],G,x,x*y^2,x^2*y);
[Order(x),Order(y)];
[y^x,y^(-1)];


print "###case 9";
G<x> := CyclicGroup(14);
CheckTriple(3,14,[2,7,14],G,x^7,x^6,x);

print "###case 10";
G<x,y> := AbelianGroup([8,2]);
CheckTriple(3,16,[2,8,8],G,y,y-x,x);

print "###case 11";
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(3,16,[2,8,8],G,x,x*y^(-1),y);
[Order(x),Order(y)];
[y^x,y^5];

print "###case 12";
G<x,y> := AbelianGroup([4,4]);
CheckTriple(3,16,[4,4,4],G,x,y,-x-y);

print "###case 13";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2=b,b^2,c^2=d,d^2,c^a=c*d>;
x := a;
y := c;
CheckTriple(3,16,[4,4,4],G,x,y,(x*y)^(-1));
[Order(x),Order(y)];
[y^x,y^(-1)];


print "###case 14";
G<x,y> := PolycyclicGroup<x,y|x^3,y^7,y^x=y^2>;
CheckTriple(3,21,[3,3,7],G,x,(y*x)^(-1),y);

print "###case 15";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2,b^2=c,c^2=d,d^3,b^a=b*d,c^a=c*d^2,d^a=d^2>;
x :=a;
y :=b;
CheckTriple(3,24,[2,4,12],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^5];

print "###case 16";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2,b^3,c^2,d^2,c^b=d,d^b=c*d>;
x := a;
y := b;
z := c;
CheckTriple(3,24,[2,6,6],G,z,y*x,(z*y*x)^(-1));
[Order(x),Order(y), Order(z)];
[z,z^y,z^(y^2)];


print "###case 17";
G<x,y> := SpecialLinearGroup(2,3);
CheckTriple(3,24,[3,3,6],G,x,x^y,(x*x^y)^(-1));

print "###case 18";
G:= SymmetricGroup(4);
x := G!(1,2,3,4); y :=G!(1,4,2,3);
x;
y;
CheckTriple(3,24,[3,4,4],G,(x*y)^(-1),x,y);

print "###case 19";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^2,b^2,c^2=d,d^2=e,e^2,c^a=b*c*d,d^a=d*e>;
x := a;
y := b;
z := c;
CheckTriple(3,32,[2,4,8],G,x,x*z,(z)^(-1));
[Order(x),Order(y),Order(z)];
[z^x,y*z^3];


print "###case 20";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^2,b^2,c^2=d,d^2=e,e^2,b^a=b*e,c^a=b*c*d,c^b=c*e>;
x := a;
y := b;
z := c;
CheckTriple(3,32,[2,4,8],G,x,x*z,(z)^(-1));
[Order(x),Order(y),Order(z)];
[y^x,y*z^4];
[z^x,y*z^3];
[z^y,z^5];

print "###case 21";
G<x,y,z,w,v> := PolycyclicGroup<x,y,z,w,v|x^2=v,y^3,z^2=v,w^2=v,v^2,z^y=w,w^y=z*w,w^z=w*v>;
CheckTriple(3,48,[2,3,12],G,x*z,y*z*w,x*y^2*z*w);

print "###case 22";
G<x,y,z>:= DirectProduct(CyclicGroup(2),SymmetricGroup(4));
CheckTriple(3,48,[2,4,6],G,x*z,y^(-1),y*z*x);

print "###case 23";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^3,b^2=d,c^2=e,d^2,e^2,b^a=c,c^a=b*c*d*e,d^a=e,e^a=d*e>;
x := a;
y := b;
z := c;
CheckTriple(3,48,[3,3,4],G,x,(y*x)^(-1),y);
[Order(x),Order(y), Order(z)];
[y^x,z];
[z^x,(y*z)^(-1)];

print "###case 24";
G<a,b,c,d,e,f> := PolycyclicGroup<a,b,c,d,e,f|a^2,b^3,c^2=e,d^2=f,e^2,f^2,b^a=b^2,c^a=d,d^a=c,c^b=d,d^b=c*d*e*f,e^a=f,f^a=e,e^b=f,f^b=e*f>;
x := a;
y := b;
z := c;
w := d;
CheckTriple(3,96,[2,3,8],G,x*y^(-1),y*w,x*z^(-1));
[Order(x),Order(y), Order(z),Order(w)];
[y^x,y^2];
[z^x,w];
[w^x,z];
[z^y,w];
[w^y,(z*w)^(-1)];

print "###case 25";
P := SymmetricGroup(8);
x := P!(1,6)(2,3)(4,5)(7,8);
y := P!(1,8,7)(2,6,4);
z := P!(1,2,3,4,5,6,7);
G := sub<P|[x,y,z]>;
CheckTriple(3,168,[2,3,7],G,x^(-1),y^(-1),z^(-1));


// finish
UnsetLogFile();