C2 x S4 x S3
r0 := (2,4)(5,8)(6,11)(10,14)(12,17)(13,18);
r1 := (1,2)(3,6)(5,9)(7,10)(8,11)(12,15)(13,16);
r2 := (3,7)(5,10)(6,12)(8,14)(9,15)(11,17);
r3 := (1,3)(2,5)(4,8)(6,13)(9,16)(11,18);
F := FreeGroup(4);
r0 := F.1;; r1 := F.2;; r2 := F.3;; r3 := F.4;;
G := F/[ r1^2, r2^2, r0^2, r3^2, (r0*r3)^2, (r0*r2)^2, (r3*r2)^3, r3*r2*r1*r3*r2*r3*r1*r2, (r1*r0*r1*r2)^2, (r2*r1)^4, (r0*r1)^6 ];;
type 0: 6 elements
type 1: 24 elements
type 2: 6 elements
type 3: 6 elements