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