NHC:=PermutationGroup<22|(2, 17)(4, 16)(6, 13)(7, 14)(8, 9)(10, 11)(15, 18)(19, 20),(1, 6)(2, 14)(3, 10)(4, 7)(9, 12)(15, 18)(16, 17)(20, 21), (2, 17)(4, 16)(6, 13)(7, 14)(8, 9)(10, 11)(15, 18)(19, 20), (1, 16, 4)(2, 14, 13)(3, 11, 10)(5, 15, 18)(6, 7, 17)(8, 12, 9), (1, 15, 17)(2, 14, 13)(4, 5, 7)(6, 16, 18)(8, 9, 12)(20, 21, 22), (1, 5)(2, 7)(4, 18)(6, 13)(14, 17)(15, 16)(19, 20)(21, 22)>;



C:=PermutationGroup<22|(1, 4, 16)(2, 13, 14)(3, 10, 11)(5, 18, 15)(6, 17, 7)(8, 9, 12)>;




///the quotient group NHC/C
//

Q:=quo<NHC|C>;




///the symmetric group S4 of degree 4
//

S4:=PermutationGroup<4|(3,4),(2,3),(3,4),Sym(4)!1,(1,3,2),(1,2)(3,4)>;


f:=hom<Q->S4|S4.1,S4.2,S4.3,S4.4,S4.5,S4.6>;




///check that f is bijective
//

if Order(Kernel(f)) eq 1 then
 if Order(Image(f)) eq Order(S4) then
  print("");
  print("f is a group isomorphism Q--->S4");
 end if;
end if;