@testset "WreathProducts" begin
   S_3 = SymmetricGroup(3)
   S_2 = SymmetricGroup(2)
   b = perm"(1,2,3)"
   a = perm"(1,2)"

   @testset "Constructors" begin
      @test Groups.WreathProduct(S_2, S_3) isa AbstractAlgebra.Group
      B3 = Groups.WreathProduct(S_2, S_3)
      @test B3 isa Groups.WreathProduct
      @test B3 isa WreathProduct{3, Generic.SymmetricGroup{Int}, Generic.SymmetricGroup{Int}}

      aa = Groups.DirectPowerGroupElem((a^0 ,a, a^2))

      @test Groups.WreathProductElem(aa, b) isa AbstractAlgebra.GroupElem
      x = Groups.WreathProductElem(aa, b)
      @test x isa Groups.WreathProductElem
      @test x isa
         Groups.WreathProductElem{3, Generic.Perm{Int}, Generic.Perm{Int}}

      @test B3.N == Groups.DirectPowerGroup(S_2, 3)
      @test B3.P == S_3

      @test B3(aa, b) == Groups.WreathProductElem(aa, b)
      w = B3(aa, b)
      @test B3(w) == w
      @test B3(b) == Groups.WreathProductElem(one(B3.N), b)
      @test B3(aa) == Groups.WreathProductElem(aa, one(S_3))

      @test B3((a^0 ,a, a^2), b) isa WreathProductElem

      @test B3((a^0 ,a, a^2), b) == B3(aa, b)
   end

   @testset "Types" begin
      B3 = Groups.WreathProduct(S_2, S_3)

      @test elem_type(B3) == Groups.WreathProductElem{3, Generic.Perm{Int}, Generic.Perm{Int}}

      @test parent_type(typeof(one(B3))) == Groups.WreathProduct{3, parent_type(typeof(one(B3.N.group))), Generic.SymmetricGroup{Int}}

      @test parent(one(B3)) == Groups.WreathProduct(S_2,S_3)
      @test parent(one(B3)) == B3
   end

   @testset "Basic operations on WreathProductElem" begin
      aa = Groups.DirectPowerGroupElem((a^0 ,a, a^2))
      B3 = Groups.WreathProduct(S_2, S_3)
      g = B3(aa, b)

      @test g.p == b
      @test g.n == DirectPowerGroupElem(aa.elts)

      h = deepcopy(g)
      @test h == g
      @test !(g === h)

      g = B3(Groups.DirectPowerGroupElem((a ,a, a^2)), g.p)

      @test g.n[1] == parent(g.n[1])(a)
      @test g != h

      @test hash(g) != hash(h)
   end

   @testset "Group arithmetic" begin
      B4 = Groups.WreathProduct(SymmetricGroup(3), SymmetricGroup(4))

      id, a, b = perm"(3)", perm"(1,2)(3)", perm"(1,2,3)"

      x = B4((id,a,b,id), perm"(1,2,3)(4)")
      @test inv(x) == B4((inv(b),id, a,id), perm"(1,3,2)(4)")

      y = B4((a,id,a,b), perm"(1,4)(2,3)")
      @test inv(y) == B4((inv(b), a,id, a), perm"(1,4)(2,3)")

      @test x*y == B4((id,id,b*a,b), perm"(1,3,4)(2)")
      @test y*x == B4(( a, b, id,b), perm"(1,4,2)(3)")

      @test inv(x)*y == B4((inv(b)*a,a,a,b), perm"(1,2,4)(3)")
      @test y*inv(x) == B4((a,a,a,id), perm"(1,4,3)(2)")

      @test (x*y)^6 == ((x*y)^2)^3
   end

   @testset "Iteration" begin
      Wr = WreathProduct(SymmetricGroup(2),SymmetricGroup(4))

      elts = collect(Wr)
      @test elts isa Vector{Groups.WreathProductElem{4, Generic.Perm{Int}, Generic.Perm{Int}}}
      @test order(Wr) == 2^4*factorial(4)

      @test length(elts) == order(Wr)
      @test all((g*inv(g) == one(Wr) for g in elts))
      @test all(inv(g*h) == inv(h)*inv(g) for g in elts for h in elts)
   end

end