// The triangle group presentations G_2,3,n and their binary 
// covers from 7.6. The groups have the stated orders

F2<x,y> := FreeGroup(2);
GG := [ quo< F2 | x^2, y^3, (x*y)^r > : r in [2..5]];
[Order(G) : G in GG];
BGG := [ quo< F2 | x^4, y^3 = x^2, (x*y)^r = x^2 > : r in [2..5]];
[Order(G) : G in BGG];