// this script checks all genus 7 generating triples
// setup
SetLogFile("genus7check.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;

load "smallpermgroups.mgm";

// tests

print "###case 1";
G<x> := CyclicGroup(15);
CheckTriple(7,15,[15,15,15],G,x,x,x^13);

print "###case 2";
G<x> := CyclicGroup(16);
CheckTriple(7,16,[8,16,16],G,x^2,x,x^13);

print "###case 3";
G<x> := CyclicGroup(16);
CheckTriple(7,16,[8,16,16],G,x^2,x^3,x^11);

print "###case 4";
G<x> := CyclicGroup(16);
CheckTriple(7,16,[8,16,16],G,x^2,x^7,x^7);

print "###case 5";
G<x> := CyclicGroup(18);
CheckTriple(7,18,[6,9,18],G,x^3,x^14,x);

print "###case 6";
G<x> := CyclicGroup(18);
CheckTriple(7,18,[6,9,18],G,x^3,x^10,x^5);

print "###case 7";
G<x> := CyclicGroup(20);
CheckTriple(7,20,[4,10,20],G,x^5,x^14,x);

print "###case 8";
G<x> := CyclicGroup(21);
CheckTriple(7,21,[3,21,21],G,x^7,x^13,x);

print "###case 9";
G<x> := CyclicGroup(24);
CheckTriple(7,24,[3,8,24],G,x^8,x^15,x);

print "###case 10";
G<x,y> := AbelianGroup([12,2]);
CheckTriple(7,24,[4,6,12],G,3*x,4*x+y,5*x+y);

print "###case 11";
G<x,y> := SpecialLinearGroup(2,3);
CheckTriple(7,24,[6,6,6],G,x*y^2,y*x*y,(x*y^3*x*y)^(-1));

print "###case 12";
G<x,y> := AbelianGroup([9,3]);
CheckTriple(7,27,[3,9,9],G,y,x,8*x+2*y);

print "###case 13";
G<x,y,z> := SmallGroup(27,4);
CheckTriple(7,27,[3,9,9],G,y,x,x^2*y^2*z^2);

print "###case 14";
G<x> := CyclicGroup(28);
CheckTriple(7,28,[2,28,28],G,x^14,x^13,x);

print "###case 15";
G<a,b,c> := SmallGroup(28,1);
x := a;
y := c;
CheckTriple(7,28,[4,4,14],G,x,x*y,x^2*y^(-1));
[Order(x),Order(y)];
[y^x,y^(-1)];


print "###case 16";
G<x> := CyclicGroup(30);
CheckTriple(7,30,[2,15,30],G,x^15,x^14,x);

print "###case 17";
G<x,y> := AbelianGroup([16,2]);
CheckTriple(7,32,[2,16,16],G,y,x,15*x+y);

print "###case 18";
G<a,b,c,d,e> := SmallGroup(32,17);
x := b;
y := a;
CheckTriple(7,32,[2,16,16],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^9];

print "###case 19";
G<x,y,z,w,v> := SmallGroup(32,10);
CheckTriple(7,32,[4,4,8],G,y,x*w,x*y*v);

print "###case 20";
G<x,y,z,w,v> := SmallGroup(32,11);
CheckTriple(7,32,[4,4,8],G,x*y,y*w,x*v);

print "###case 21";
G<x,y,z,w,v> := SmallGroup(32,13);
CheckTriple(7,32,[4,4,8],G,x,x*y*w,y*z*v);

print "###case 22";
G<x,y,z,w,v> := SmallGroup(32,14);
CheckTriple(7,32,[4,4,8],G,x,x*y*w,y*z);

print "###case 23";
G<a,b,c,d> := SmallGroup(36,6);
x := a;
y := d;
z := b;
CheckTriple(7,36,[3,4,12],G,y*z,x*y,x^(-1)*z^(-1));
[Order(x),Order(y),Order(z)];
[y^x,y^2];
[z^x,z];


print "###case 24";
G<a,b,c> := SmallGroup(42,4);
x := a;
y := b*c;
CheckTriple(7,42,[2,6,21],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^13];

print "###case 25";
G<x,y,z,w,v> := SmallGroup(48,33);
CheckTriple(7,48,[2,6,12],G,x*z,y*z*w*v,x*y^2*z*w*v);

print "###case 26";
G<x,y,z,w,v> := SmallGroup(48,32);
CheckTriple(7,48,[3,4,6],G,y,x*w*v,x*y^2*z);

print "###case 27";
G<a,b,c,d> := SmallGroup(54,3);
x := a*b;
y := c^2;
CheckTriple(7,54,[2,6,9],G,x^3,x*y,y^(-1)*x^2);
[Order(x),Order(y)];
[y^x,y^(-1)];

print "###case 28";
G<a,b,c,d> := SmallGroup(54,6);
x := a*b;
y := c^2;
CheckTriple(7,54,[2,6,9],G,x^3,x*y,y^(-1)*x^2);
[Order(x),Order(y)];
[y^x,y^2];

print "###case 29";
G<a,b,c,d> := SmallGroup(56,4);
x := a;
y  := b*d;
CheckTriple(7,56,[2,4,28],G,x,(y*x)^(-1),y);
[Order(x),Order(y)];
[y^x,y^13];

print "###case 30";
G<x,y,z,w> := SmallGroup(56,11);
CheckTriple(7,56,[2,7,7],G,y,(x*y)^(-1),x);

print "###case 31";
G<x,y,z,w,v,u> := SmallGroup(64,38);
CheckTriple(7,64,[2,4,16],G,y,x,x*y*w);
 
print "###case 32";
G<x,y,z,w,v,u> := SmallGroup(64,41);
CheckTriple(7,64,[2,4,16],G,y,x,x*y*w*u);

print "###case 33";
G<x,y,z,w,v> := SmallGroup(72,25);
CheckTriple(7,72,[3,3,6],G,x,x*y*w,x*y^2*z*w*v);

print "###case 34";
G<x,y,z,w,v,u> := SmallGroup(144,127);
CheckTriple(7,144,[2,3,12],G,x*z,y^2*w*v,x*y*v);

print "###case 35";
P := SymmetricGroup(9);
a := P!(1,8)(2,3)(5,6)(7,9);
b := P!(1,8,5)(2,3,7)(4,9,6);
c := P!(1,6,7,2,9,4,5);
G<x,y> := sub<P|[a,b]>; 
CheckTriple(7,504,[2,3,7],G,x,y,(x*y)^(-1));

// finish
UnsetLogFile();