diff --git a/src/orbitdata.jl b/src/orbitdata.jl
index 151b82b..5f2d397 100644
--- a/src/orbitdata.jl
+++ b/src/orbitdata.jl
@@ -58,9 +58,14 @@ function orthSVD(M::AbstractMatrix{T}) where {T<:AbstractFloat}
     return fact.U[:,1:M_rank]
 end
 
-orbit_decomposition(G::Group, E::AbstractVector, rdict=GroupRings.reverse_dict(E)) = orbit_decomposition(collect(G), E, rdict)
+orbit_decomposition(
+    G::Group,
+    E::AbstractVector,
+    rdict = GroupRings.reverse_dict(E);
+    op = ^,
+) = orbit_decomposition(collect(G), E, rdict; op=op)
 
-function orbit_decomposition(elts::AbstractVector{<:GroupElem}, E::AbstractVector, rdict=GroupRings.reverse_dict(E))
+function orbit_decomposition(elts::AbstractVector{<:GroupElem}, E::AbstractVector, rdict=GroupRings.reverse_dict(E); op=^)
 
     tovisit = trues(size(E));
     orbits = Vector{Vector{Int}}()
@@ -71,7 +76,7 @@ function orbit_decomposition(elts::AbstractVector{<:GroupElem}, E::AbstractVecto
         if tovisit[i]
             g = E[i]
             Threads.@threads for j in eachindex(elts)
-                orbit[j] = rdict[elts[j](g)]
+                orbit[j] = rdict[op(g, elts[j])]
             end
             tovisit[orbit] .= false
             push!(orbits, unique(orbit))
@@ -139,7 +144,7 @@ end
 function perm_repr(g::GroupElem, E::Vector, E_dict)
     p = Vector{Int}(undef, length(E))
     for (i,elt) in enumerate(E)
-        p[i] = E_dict[g(elt)]
+        p[i] = E_dict[elt^g]
     end
     return p
 end
@@ -178,29 +183,33 @@ end
 #
 ###############################################################################
 
-function (g::GroupRingElem)(y::GroupRingElem)
+function Base.:^(y::GroupRingElem, g::GroupRingElem, op = ^)
     res = parent(y)()
     for elt in GroupRings.supp(g)
-        res += g[elt]*elt(y)
+        res += g[elt] * ^(y, elt, op)
     end
     return res
 end
 
-function (g::GroupElem)(y::GroupRingElem)
+function Base.:^(y::GroupRingElem, g::GroupElem, op = ^)
     RG = parent(y)
     result = zero(RG, eltype(y.coeffs))
 
     for (idx, c) in enumerate(y.coeffs)
         if !iszero(c)
-            result[g(RG.basis[idx])] = c
+            result[op(RG.basis[idx], g)] = c
         end
     end
     return result
 end
 
-function (g::GroupElem)(y::GroupRingElem{T, <:SparseVector}) where T
+function Base.:^(
+    y::GroupRingElem{T,<:SparseVector},
+    g::GroupElem,
+    op = ^,
+) where {T}
     RG = parent(y)
-    index = [RG.basis_dict[g(RG.basis[idx])] for idx in y.coeffs.nzind]
+    index = [RG.basis_dict[op(RG.basis[idx], g)] for idx in y.coeffs.nzind]
 
     result = GroupRingElem(sparsevec(index, y.coeffs.nzval, y.coeffs.n), RG)
 
@@ -213,28 +222,29 @@ end
 #
 ###############################################################################
 
-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))")
+function Base.:^(A::MatAlgElem, p::Generic.Perm)
+    length(p.d) == size(A, 1) == size(A, 2) ||
+        throw("Can't act via $p on matrix of size $(size(A))")
     result = similar(A)
-    @inbounds for i in 1:size(A, 1)
-        for j in 1:size(A, 2)
-            result[p[i], p[j]] = A[i,j] # action by permuting rows and colums/conjugation
+    @inbounds for i = 1:size(A, 1)
+        for j = 1:size(A, 2)
+            result[p[i], p[j]] = A[i, j] # action by permuting rows and colums/conjugation
         end
     end
     return result
 end
 
-function (g::WreathProductElem{N})(A::MatAlgElem) where N
+function Base.:^(A::MatAlgElem, g::WreathProductElem{N}) 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)
+    flips = ntuple(i -> (g.n[i].d[1] == 1 && g.n[i].d[2] == 2 ? 1 : -1), N)
     result = similar(A)
     R = base_ring(parent(A))
     tmp = R(1)
 
-    @inbounds for i = 1:size(A,1)
-        for j = 1:size(A,2)
+    @inbounds for i = 1:size(A, 1)
+        for j = 1:size(A, 2)
             x = A[i, j]
-            if flips[i]*flips[j] == 1
+            if flips[i] * flips[j] == 1
                 result[g.p[i], g.p[j]] = x
             else
                 result[g.p[i], g.p[j]] = -x
@@ -250,7 +260,7 @@ end
 #
 ###############################################################################
 
-function (g::GroupElem)(a::Automorphism)
+function Base.:^(a::Automorphism, g::GroupElem)
     Ag = parent(a)(g)
     Ag_inv = inv(Ag)
     res = append!(Ag, a, Ag_inv)
@@ -261,10 +271,11 @@ end
 
 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")
+    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]))
-    for i in 1:length(g.p.d)
+    for i = 1:length(g.p.d)
         if g.n.elts[i] != Id
             push!(elt, Groups.flip(i))
         end
diff --git a/src/sqadjop.jl b/src/sqadjop.jl
index 310cf97..a7eaf2a 100644
--- a/src/sqadjop.jl
+++ b/src/sqadjop.jl
@@ -19,7 +19,7 @@ function Sq(RG::GroupRing, N::Integer)
 
     sq = RG()
     for σ in Alt_N
-        GroupRings.addeq!(sq, *(σ(Δ₂), σ(Δ₂), false))
+        GroupRings.addeq!(sq, *(Δ₂^σ, Δ₂^σ, false))
     end
     return sq÷factorial(N-2)
 end
@@ -30,7 +30,7 @@ function Adj(RG::GroupRing, N::Integer)
     Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
 
     Alt_N = [g for g in SymmetricGroup(N) if parity(g) == 0]
-    Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
+    Δ₂s = Dict(σ=>Δ₂^σ for σ in Alt_N)
     adj = RG()
 
     for σ in Alt_N
@@ -52,7 +52,7 @@ function Op(RG::GroupRing, N::Integer)
     Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
 
     Alt_N = [g for g in SymmetricGroup(N) if parity(g) == 0]
-    Δ₂s = Dict(σ=>σ(Δ₂) for σ in Alt_N)
+    Δ₂s = Dict(σ=>Δ₂^σ for σ in Alt_N)
     op = RG()
 
     for σ in Alt_N