C2 x S4 x S3
r0 := (2,4)(5,7)(6,8)(9,11)(14,16)(17,18);
r1 := (1,2)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,17);
r2 := (6,9)(8,11)(10,13);
r3 := (1,3)(2,5)(4,7)(6,8)(9,11)(12,15)(14,17)(16,18);
F := FreeGroup(4);
r0 := F.1;; r1 := F.2;; r2 := F.3;; r3 := F.4;;
G := F/[ r2^2, r1^2, r0^2, r3^2, (r3*r2)^2, (r0*r3)^2, (r0*r2)^2, (r3*r1)^3, (r1*r0*r1*r2)^2, (r1*r2)^4, r3*r1*r0*r1*r3*(r0*r1)^2*r0 ];;
type 0: 6 elements
type 1: 36 elements
type 2: 8 elements
type 3: 6 elements