diff --git a/src/orbitdata.jl b/src/orbitdata.jl
index ce3c534..5e98f2a 100644
--- a/src/orbitdata.jl
+++ b/src/orbitdata.jl
@@ -186,25 +186,19 @@ function (g::GroupRingElem)(y::GroupRingElem)
     return res
 end
 
-###############################################################################
-#
-#  perm actions
-#
-###############################################################################
-
-function (g::Generic.Perm)(y::GroupRingElem)
+function (g::GroupElem)(y::GroupRingElem)
     RG = parent(y)
     result = zero(RG, eltype(y.coeffs))
 
     for (idx, c) in enumerate(y.coeffs)
-        if c!= zero(eltype(y.coeffs))
+        if !iszero(c)
             result[g(RG.basis[idx])] = c
         end
     end
     return result
 end
 
-function (g::Generic.Perm)(y::GroupRingElem{T, <:SparseVector}) where T
+function (g::GroupElem)(y::GroupRingElem{T, <:SparseVector}) where T
     RG = parent(y)
     index = [RG.basis_dict[g(RG.basis[idx])] for idx in y.coeffs.nzind]
 
@@ -213,6 +207,12 @@ function (g::Generic.Perm)(y::GroupRingElem{T, <:SparseVector}) where T
     return result
 end
 
+###############################################################################
+#
+#  perm && WreathProductElems actions: MatAlgElem
+#
+###############################################################################
+
 function (p::Generic.Perm)(A::MatAlgElem)
     length(p.d) == size(A, 1) == size(A,2) || throw("Can't act via $p on matrix of size $(size(A))")
     result = similar(A)
@@ -224,24 +224,6 @@ function (p::Generic.Perm)(A::MatAlgElem)
     return result
 end
 
-###############################################################################
-#
-#  WreathProductElems action on MatAlgElem
-#
-###############################################################################
-
-function (g::WreathProductElem)(y::GroupRingElem)
-    RG = parent(y)
-    result = zero(RG, eltype(y.coeffs))
-
-    for (idx, c) in enumerate(y.coeffs)
-        if c!= zero(eltype(y.coeffs))
-            result[g(RG.basis[idx])] = c
-        end
-    end
-    return result
-end
-
 function (g::WreathProductElem{N})(A::MatAlgElem) where N
     # @assert N == size(A,1) == size(A,2)
     flips = ntuple(i->(g.n[i].d[1]==1 && g.n[i].d[2]==2 ? 1 : -1), N)
@@ -257,8 +239,6 @@ function (g::WreathProductElem{N})(A::MatAlgElem) where N
             else
                 result[g.p[i], g.p[j]] = -x
             end
-            # result[i, j] = AbstractAlgebra.mul!(x, x, flips[i]*flips[j])
-            # this mul! needs to be separately defined, but is 2x faster
         end
     end
     return result
@@ -266,33 +246,33 @@ end
 
 ###############################################################################
 #
-#  Action of WreathProductElems on AutGroupElem
+# perm && WreathProductElems actions: Automorphism
 #
 ###############################################################################
 
-function AutFG_emb(A::AutGroup, g::WreathProductElem)
+function (g::GroupElem)(a::Automorphism)
+    Ag = parent(a)(g)
+    Ag_inv = inv(Ag)
+    res = append!(Ag, a, Ag_inv)
+    return Groups.freereduce!(res)
+end
+
+(A::AutGroup)(p::Generic.Perm) = A(Groups.AutSymbol(p))
+
+function (A::AutGroup)(g::WreathProductElem)
     isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
     parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
     elt = one(A)
     Id = one(parent(g.n.elts[1]))
-    flips = Groups.AutSymbol[Groups.flip_autsymbol(i) for i in 1:length(g.p.d) if g.n.elts[i] != Id]
-    Groups.r_multiply!(elt, flips, reduced=false)
-    Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
+    for i in 1:length(g.p.d)
+        if g.n.elts[i] != Id
+            push!(elt, Groups.flip(i))
+        end
+    end
+    push!(elt, Groups.AutSymbol(g.p))
     return elt
 end
 
-function (g::WreathProductElem)(a::Groups.Automorphism)
-    A = parent(a)
-    g_emb = AutFG_emb(A,g)
-    res = deepcopy(g_emb)
-    res = Groups.r_multiply!(res, a.symbols, reduced=false)
-    res = Groups.r_multiply!(res, [inv(s) for s in reverse!(g_emb.symbols)])
-    return res
-end
 
-function (p::Generic.Perm)(a::Groups.Automorphism)
-    res = parent(a)(Groups.perm_autsymbol(p))
-    res = Groups.r_multiply!(res, a.symbols, reduced=false)
-    res = Groups.r_multiply!(res, [Groups.perm_autsymbol(inv(p))])
-    return res
-end
+# fallback:
+Base.one(p::Generic.Perm) = Perm(length(p.d))
diff --git a/test/actions.jl b/test/actions.jl
index fd9ce04..8d84a2a 100644
--- a/test/actions.jl
+++ b/test/actions.jl
@@ -3,12 +3,9 @@
     ssgs(M::MatAlgebra, i, j) = (S = [Eij(M, i, j), Eij(M, j, i)];
         S = unique([S; inv.(S)]); S)
 
-    rmul = Groups.rmul_autsymbol
-    lmul = Groups.lmul_autsymbol
-
     function ssgs(A::AutGroup, i, j)
-        rmuls = [rmul(i,j), rmul(j,i)]
-        lmuls = [lmul(i,j), lmul(j,i)]
+        rmuls = [Groups.transvection_R(i,j), Groups.transvection_R(j,i)]
+        lmuls = [Groups.transvection_L(i,j), Groups.transvection_L(j,i)]
         gen_set = A.([rmuls; lmuls])
         return unique([gen_set; inv.(gen_set)])
     end
@@ -33,7 +30,7 @@
         elt2 = E_R[rand(sizes[1]:sizes[2])]
         y = 2RG(elt2) - RG(elt)
 
-        for G in [PermGroup(N), WreathProduct(PermGroup(2), PermGroup(N))]
+        for G in [SymmetricGroup(N), WreathProduct(SymmetricGroup(2), SymmetricGroup(N))]
             @test all(g(one(M)) == one(M) for g in G)
             @test all(rdict[g(m)] <= sizes[1] for g in G for m in S)
             @test all(g(m)*g(n) == g(m*n) for g in G for m in S for n in S)
@@ -50,9 +47,9 @@
             Sij = ssgs(M, i,j)
             Δij= PropertyT.spLaplacian(RG, Sij)
 
-            @test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in PermGroup(N))
+            @test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in SymmetricGroup(N))
 
-            @test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(PermGroup(2), PermGroup(N)))
+            @test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
         end
     end
 end
@@ -79,7 +76,7 @@ end
         elt2 = E_R[rand(sizes[1]:sizes[2])]
         y = 2RG(elt2) - RG(elt)
 
-        for G in [PermGroup(N), WreathProduct(PermGroup(2), PermGroup(N))]
+        for G in [SymmetricGroup(N), WreathProduct(SymmetricGroup(2), SymmetricGroup(N))]
             @test all(g(one(M)) == one(M) for g in G)
             @test all(rdict[g(m)] <= sizes[1] for g in G for m in S)
             @test all(g(m)*g(n) == g(m*n) for g in G for m in S for n in S)
@@ -95,9 +92,9 @@ end
         Sij = ssgs(M, i,j)
         Δij= PropertyT.spLaplacian(RG, Sij)
 
-        @test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in PermGroup(N))
+        @test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in SymmetricGroup(N))
 
-        @test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(PermGroup(2), PermGroup(N)))
+        @test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(SymmetricGroup(2), SymmetricGroup(N)))
     end
 end