x:=(24,21,20,13,19,5,23,3,18,16,6,9,22,14,4,2,17,8,15,12,7,11,10);;
y:=(4,8,9,12,10)(7,6,3,5,11)(15,19,20,23,21)(18,17,14,16,22);;
m23 := Group( x,y);
Size(m23);
H := Subgroup(m23, [ x*y, x^6*y^2 ] );;
Size( H );
Index( G, H );

F2 := FreeGroup("x1","x2");
x1 := F2.1; x2 := F2.2;
G := F2 / [(x1*x2)^2,(x1^5*x2^7)^4];
PG := PresentationFpGroup( G );
Pm23 := PresentationViaCosetTable( m23, F2, [x1*x2,x1^6*x2^2]);