// this script checks all genus 4 generating triples
// setup
SetLogFile("genus4check.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(9);
CheckTriple(4,9,[9,9,9],G,x,x,x^7);

print "###case 2";
G<x> := CyclicGroup(10);
CheckTriple(4,10,[5,10,10],G,x^2,x,x^7);

print "###case 3";
G<x> := CyclicGroup(10);
CheckTriple(4,10,[5,10,10],G,x^8,x,x);

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

print "###case 5";
G<x> := CyclicGroup(12);
CheckTriple(4,12,[4,6,12],G,x^3,x^2,x^7);

print "###case 6";
G<x,y> := AbelianGroup([6,2]);
CheckTriple(4,12,[6,6,6],G,x,x+y,4*x+y);

print "###case 7";
G<x> := CyclicGroup(15);
CheckTriple(4,15,[3,5,15],G,x^5,x^3,x^7);

print "###case 8";
G<x> := CyclicGroup(16);
CheckTriple(4,16,[2,16,16],G,x^8,x^3,x^5);

print "###case 9";
G<x,y,z,w> := SmallGroup(16,9);
CheckTriple(4,16,[4,4,8],G,y,x*z*w,x*y*z);

print "###case 10";
G<x> := CyclicGroup(18);
CheckTriple(4,18,[2,9,18],G,x^9,x^2,x^7);

print "###case 11";
G<x,y> := AbelianGroup([6,3]);
CheckTriple(4,18,[3,6,6],G,y,x+y,5*x+y);

print "###case 12";
G<x,y,z> := DirectProduct(SymmetricGroup(3),CyclicGroup(3));
CheckTriple(4,18,[3,6,6],G,x*z,y*z,y*x^2*z);

print "###case 13";
G<x,y,z> := DirectProduct(SymmetricGroup(3),CyclicGroup(3));
CheckTriple(4,18,[3,6,6],G,x,y*z^2,y*x^2*z);


print "###case 14";
G<x,y> := AbelianGroup([10,2]);
CheckTriple(4,20,[2,10,10],G,y,x+y,9*x);


print "###case 15";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2=b,b^2,c^5,c^a=c^4>;
x := a;
y := c;
CheckTriple(4,20,[4,4,5],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^(-1)];

print "###case 16";
G<a,b,c> := PolycyclicGroup<a,b,c|a^2=b,b^2,c^5,c^a=c^2,c^b=c^4>;
x := a;
y := c;
CheckTriple(4,20,[4,4,5],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^2];

print "###case 17";
G<a,b,c,d> := PolycyclicGroup<a,b,c,d|a^2,b^2=c,c^2=d,d^3,b^a=b*c*d>;
x := a;
y := b;
CheckTriple(4,24,[2,6,12],G,x,x*y^(-1),y);
[Order(a),Order(y)];
[y^x,y^7];

print "###case 18";
G<x,y,z,w> := SmallGroup(24,3);
CheckTriple(4,24,[3,4,6],G,x,y*w,(x*y*w)^(-1));

print "###case 19";
G<a,b,c,d,e> := PolycyclicGroup<a,b,c,d,e|a^2,b^2=c,c^2=d,d^2=e,e^2,b^a=b*c*d,c^a=c*d*e,d^a=d*e>;
x := a;
y := b;
CheckTriple(4,32,[2,4,16],G,x,x*y^(-1),y);
[Order(x),Order(y)];
[y^x,y^7];

print "###case 20";
G<x,y,z,w> := SmallGroup(36,10);
CheckTriple(4,36,[2,6,6],G,x*y,y*z*w^2,x*z^2*w^2);

print "###case 21";
G<x,y,z,w> := SmallGroup(36,12);
CheckTriple(4,36,[2,6,6],G,x,y*z*w^2,x*y*z^2*w^2);

print "###case 22";
G<x,y,z,w> := SmallGroup(36,11);
CheckTriple(4,36,[3,3,6],G,x,x^2*y*z*w,y^2*z*w);

print "###case 23";
G<a,b,c,d> := SmallGroup(36,9);
x := a;
y := c;
z := d;
CheckTriple(4,36,[3,4,4],G,y,(x*y)^(-1),x);
[Order(x),Order(y),Order(z)];
[y^x,y*z^2];
[z^x,y^2*z^2];

print "###case 24";
G<x,y,z,w> := SmallGroup(40,8);
CheckTriple(4,40,[2,4,10],G,x,x*y*z*w,y*z*w^4);

print "###case 25";
G := AlternatingGroup(5);
x := G!(1,2)(3,4);
y := G!(1,2,3,4,5);
CheckTriple(4,60,[2,5,5],G,x,x*y^2,y^3);

print "###case 26";
G<x,y,z,w,v> := SmallGroup(72,42);
CheckTriple(4,72,[2,3,12],G,x,y*z^2*w*v,x*y^2*z^2*v);

print "###case 27";
G<x,y,z,w,v> := SmallGroup(72,40);
CheckTriple(4,72,[2,4,6],G,y,x*y*z*w^2*v^2,x*w^2*v);


print "###case 28";
G := SymmetricGroup(5);
x := G!(1,2);
y := G!(1,2,3,4,5);
CheckTriple(4,120,[2,4,5],G,x,(y*x)^(-1),y);


// finish
UnsetLogFile();