diff --git a/Project.toml b/Project.toml
index cd77d3f..7e007b9 100644
--- a/Project.toml
+++ b/Project.toml
@@ -15,13 +15,15 @@ AbstractAlgebra = "0.15, 0.16"
 GroupsCore = "^0.3"
 KnuthBendix = "^0.2.1"
 OrderedCollections = "1"
+PermutationGroups = "^0.3"
 ThreadsX = "^0.1.0"
 julia = "1.3, 1.4, 1.5, 1.6"
 
 [extras]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
 BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
+PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Test", "BenchmarkTools", "AbstractAlgebra"]
+test = ["Test", "BenchmarkTools", "AbstractAlgebra", "PermutationGroups"]
diff --git a/test/AutSigma_41.jl b/test/AutSigma_41.jl
index 11c3e6a..f4f92d2 100644
--- a/test/AutSigma_41.jl
+++ b/test/AutSigma_41.jl
@@ -1,12 +1,21 @@
 using PermutationGroups
+using Groups.KnuthBendix
 
 @testset "Wajnryb presentation for Σ₄₁" begin
 
     genus = 4
 
-    G = New.SpecialAutomorphismGroup(New.FreeGroup(2genus))
+    G = SpecialAutomorphismGroup(FreeGroup(2genus))
 
-    T = let G = G; (Tas, Tαs, Tes) = New.mcg_twists(genus)
+    # symplectic pairing goes like this:
+    # in the free Group:
+    # f1 ↔ f5
+    # f2 ↔ f6
+    # f3 ↔ f7
+    # f4 ↔ f8
+
+    T = let G = G
+        (Tas, Tαs, Tes) = Groups.mcg_twists(G)
         Ta = G.(Tas)
         Tα = G.(Tαs)
         Tes = G.(Tes)
@@ -24,24 +33,58 @@ using PermutationGroups
     a8 = T[8]^-1 # Tα₄
 
     b0 = T[2]^-1 # Ta₂
-    a0 = (a1*a2*a3)^4*b0^-1 # from the 3-chain relation
-    X = a4*a5*a3*a4 # auxillary, not present in the Primer
-    b1 = X^-1*a0*X
+    a0 = (a1 * a2 * a3)^4 * b0^-1 # from the 3-chain relation
+    X = a4 * a5 * a3 * a4 # auxillary, not present in the Primer
+    b1 = X^-1 * a0 * X
     b2 = T[10]^-1 # Te₁₃
 
-    As = T[[1,5,9,6,12,7,14,8]] # the inverses of the elements a
+    As = T[[1, 5, 9, 6, 12, 7, 14, 8]] # the inverses of the elements a
+
+
+    @testset "preserving relator" begin
+        F = Groups.object(G)
+
+        R = prod(commutator(gens(F, i), gens(F, i+genus)) for i in 1:genus)
+
+        ## TODO: how to evaluate automorphisms properly??!!!
+
+        for g in T
+            w = one(word(g))
+
+            dg = Groups.domain(g)
+            gens_idcs = first.(word.(dg))
+            img = evaluate(g)
+            A = alphabet(first(dg))
+
+            ltrs_map = Vector{eltype(dg)}(undef, length(KnuthBendix.letters(A)))
+
+            for i in 1:length(KnuthBendix.letters(A))
+                if i in gens_idcs
+                    ltrs_map[i] = img[findfirst(==(i), gens_idcs)]
+                else
+                    ltrs_map[i] = inv(img[findfirst(==(inv(A, i)), gens_idcs)])
+                end
+            end
+
+            for l in word(R)
+                append!(w, word(ltrs_map[l]))
+            end
+
+            @test F(w) == R
+        end
+    end
 
     @testset "commutation relations" begin
         for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for commutativity. Also, a0 is as defined before.
             for (j, aj) in enumerate(As)
-                if abs(i-j) > 1
-                    @test ai*aj == aj*ai
+                if abs(i - j) > 1
+                    @test ai * aj == aj * ai
                 elseif i ≠ j
-                    @test ai*aj != aj*ai
+                    @test ai * aj != aj * ai
                 end
             end
             if i != 4
-                @test a0*ai == ai*a0
+                @test a0 * ai == ai * a0
             end
         end
     end
@@ -49,145 +92,186 @@ using PermutationGroups
     @testset "braid relations" begin
         for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for braid relations.
             for (j, aj) in enumerate(As)
-                if abs(i-j) == 1
-                    @test ai*aj*ai == aj*ai*aj
+                if abs(i - j) == 1
+                    @test ai * aj * ai == aj * ai * aj
                 end
             end
         end
-        @test a0*a4*a0 == a4*a0*a4 # here, a0 and a4 are as before
+        @test a0 * a4 * a0 == a4 * a0 * a4 # here, a0 and a4 are as before
     end
 
     @testset "Lantern relation" begin
 
         @testset "b2 definition" begin
-                @test b2 == (a2*a3*a1*a2)^-1*b1*(a2*a3*a1*a2)
+            @test b2 == (a2 * a3 * a1 * a2)^-1 * b1 * (a2 * a3 * a1 * a2)
 
             # some additional tests, checking what explicitly happens to the generators of the π₁ under b2
-            d = New.domain(b2)
-            im = New.evaluate(b2)
-            z = im[7]*d[7]^-1
+            d = Groups.domain(b2)
+            im = evaluate(b2)
+            z = im[7] * d[7]^-1
 
             @test im[1] == d[1]
-            @test im[2] == z*d[2]*z^-1
-            @test im[3] == z*d[3]*z^-1
+            @test im[2] == z * d[2] * z^-1
+            @test im[3] == z * d[3] * z^-1
             @test im[4] == d[4]
 
-            @test im[5] == d[5]*z^-1
-            @test im[6] == z*d[6]*z^-1
-            @test im[7] == z*d[7]
+            @test im[5] == d[5] * z^-1
+            @test im[6] == z * d[6] * z^-1
+            @test im[7] == z * d[7]
             @test im[8] == d[8]
 
         end
 
         @testset "b2: commutation relations" begin
-            @test b2*a1 == a1*b2
-            @test b2*a2 != a2*b2
-            @test b2*a3 == a3*b2
-            @test b2*a4 == a4*b2
-            @test b2*a5 == a5*b2
-            @test b2*a6 != a6*b2
+            @test b2 * a1 == a1 * b2
+            @test b2 * a2 != a2 * b2
+            @test b2 * a3 == a3 * b2
+            @test b2 * a4 == a4 * b2
+            @test b2 * a5 == a5 * b2
+            @test b2 * a6 != a6 * b2
         end
 
         @testset "b2: braid relations" begin
-            @test a2*b2*a2 == b2*a2*b2
-            @test a6*b2*a6 == b2*a6*b2
+            @test a2 * b2 * a2 == b2 * a2 * b2
+            @test a6 * b2 * a6 == b2 * a6 * b2
         end
 
         @testset "lantern" begin
-            u = (a6*a5)^-1*b1*(a6*a5)
-            x = (a6*a5*a4*a3*a2*u*a1^-1*a2^-1*a3^-1*a4^-1) # yet another auxillary
+            u = (a6 * a5)^-1 * b1 * (a6 * a5)
+            x = (a6 * a5 * a4 * a3 * a2 * u * a1^-1 * a2^-1 * a3^-1 * a4^-1) # yet another auxillary
             # x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
-            @time New.evaluate(x)
-            b3 = x*a0*x^-1
-            @time New.evaluate(b3)
-            @test a0*b2*b1 == a1*a3*a5*b3
+            @time evaluate(x)
+            b3 = x * a0 * x^-1
+            @time evaluate(b3)
+            @test a0 * b2 * b1 == a1 * a3 * a5 * b3
         end
     end
 
 
     @testset "Te₁₂ definition" begin
         G = parent(first(T))
-        F₈ = New.object(G)
-        (a, b, c, d, α, β, γ, δ) = gens(F₈)
+        F₈ = Groups.object(G)
+        (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
 
-        A = KnuthBendix.alphabet(G)
+        A = alphabet(G)
 
-        λ = [i == j ? one(G) : G([A[New.λ(i,j)]]) for i in 1:8, j in 1:8]
-        ϱ = [i == j ? one(G) : G([A[New.ϱ(i,j)]]) for i in 1:8, j in 1:8]
+        λ = [i == j ? one(G) : G([A[Groups.λ(i, j)]]) for i in 1:8, j in 1:8]
+        ϱ = [i == j ? one(G) : G([A[Groups.ϱ(i, j)]]) for i in 1:8, j in 1:8]
 
         g = one(G)
-        # @show g
-        # @show g(Groups.domain(G))
 
         # β ↦ α*β
-        g *= λ[6,5]
-        @test New.evaluate(g)[6] == α*β
+        g *= λ[6, 5]
+        @test evaluate(g)[6] == α * β
 
         # α ↦ a*α*b^-1
-        g *= λ[5,1]*inv(ϱ[5,2])
-        @test New.evaluate(g)[5] == a*α*b^-1
+        g *= λ[5, 1] * inv(ϱ[5, 2])
+        @test evaluate(g)[5] == a * α * b^-1
 
         # β ↦ b*α^-1*a^-1*α*β
-        g *= inv(λ[6,5])
-        @test New.evaluate(g)[6] == b*α^-1*a^-1*α*β
+        g *= inv(λ[6, 5])
+        @test evaluate(g)[6] == b * α^-1 * a^-1 * α * β
 
         # b ↦ α
-        g *= λ[2,5]*inv(λ[2,1]);
-        @test New.evaluate(g)[2] == α
+        g *= λ[2, 5] * inv(λ[2, 1])
+        @test evaluate(g)[2] == α
 
         # b ↦ b*α^-1*a^-1*α
-        g *= inv(λ[2,5]);
-        @test New.evaluate(g)[2] == b*α^-1*a^-1*α
+        g *= inv(λ[2, 5])
+        @test evaluate(g)[2] == b * α^-1 * a^-1 * α
 
         # b ↦ b*α^-1*a^-1*α*b*α^-1
-        g *= inv(ϱ[2,5])*ϱ[2,1];
-        @test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1
+        g *= inv(ϱ[2, 5]) * ϱ[2, 1]
+        @test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1
 
         # b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
-        g *= ϱ[2,5];
-        @test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
+        g *= ϱ[2, 5]
+        @test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1 * a * α * b^-1
 
-        x = b*α^-1*a^-1*α
-        @test New.evaluate(g) == # (a, b,        c, d, α,      β,   γ, δ)
-                                   (a, x*b*x^-1, c, d, α*x^-1, x*β, γ, δ)
+        x = b * α^-1 * a^-1 * α
+        @test evaluate(g) ==
+              (a, x * b * x^-1, c, d, α * x^-1, x * β, γ, δ)
+            # (a, b,            c, d, α,        β,     γ, δ)
         @test g == T[9]
     end
 
-    Base.conj(t::New.Transvection, p::Perm) =
-        New.Transvection(t.id, t.i^p, t.j^p, t.inv)
+    Base.conj(t::Groups.Transvection, p::Perm) =
+        Groups.Transvection(t.id, t.i^p, t.j^p, t.inv)
 
-    function Base.conj(elt::New.FPGroupElement, p::Perm)
+    function Base.conj(elt::FPGroupElement, p::Perm)
         G = parent(elt)
-        A = New.alphabet(elt)
-        return G([A[conj(A[idx], p)] for idx in New.word(elt)])
+        A = alphabet(elt)
+        return G([A[conj(A[idx], p)] for idx in word(elt)])
     end
 
     @testset "Te₂₃ definition" begin
         Te₁₂, Te₂₃ = T[9], T[12]
         G = parent(Te₁₂)
-        F₈ = New.object(G)
-        (a, b, c, d, α, β, γ, δ) = gens(F₈)
+        F₈ = Groups.object(G)
+        (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
 
-        img_Te₂₃ = New.evaluate(Te₂₃)
-        y = c*β^-1*b^-1*β
-        @test img_Te₂₃ == (a, b, y*c*y^-1, d, α, β*y^-1, y*γ, δ)
+        img_Te₂₃ = evaluate(Te₂₃)
+        y = c * β^-1 * b^-1 * β
+        @test img_Te₂₃ == (a, b, y * c * y^-1, d, α, β * y^-1, y * γ, δ)
 
         σ = perm"(1,2,3)(5,6,7)(8)"
         Te₂₃_σ = conj(Te₁₂, σ)
-        # @test New.word(Te₂₃_σ) == New.word(Te₂₃)
+        # @test word(Te₂₃_σ) == word(Te₂₃)
 
-        @test New.evaluate(Te₂₃_σ) == New.evaluate(Te₂₃)
+        @test evaluate(Te₂₃_σ) == evaluate(Te₂₃)
         @test Te₂₃ == Te₂₃_σ
     end
 
     @testset "Te₃₄ definition" begin
         Te₁₂, Te₃₄ = T[9], T[14]
         G = parent(Te₁₂)
-        F₈ = New.object(G)
+        F₈ = Groups.object(G)
         (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
 
         σ = perm"(1,3)(2,4)(5,7)(6,8)"
         Te₃₄_σ = conj(Te₁₂, σ)
         @test Te₃₄ == Te₃₄_σ
     end
+
+    @testset "hyperelliptic τ is central" begin
+
+        A = alphabet(G)
+        λ = Groups.ΡΛ(:λ, A, 2genus)
+        ϱ = Groups.ΡΛ(:ϱ, A, 2genus)
+
+        import Groups: Ta, Tα, Te
+
+        halftwists = map(1:genus-1) do i
+            j = i + 1
+            x = Ta(λ, j) * inv(A, Ta(λ, i)) * Tα(λ, j) * Te(λ, ϱ, i, j)
+            δ = x * Tα(λ, i) * inv(A, x)
+            c =
+                inv(A, Ta(λ, j)) *
+                Te(λ, ϱ, i, j) *
+                Tα(λ, i)^2 *
+                inv(A, δ) *
+                inv(A, Ta(λ, j)) *
+                Ta(λ, i) *
+                δ
+            z =
+                Te(λ, ϱ, j, i) *
+                inv(A, Ta(λ, i)) *
+                Tα(λ, i) *
+                Ta(λ, i) *
+                inv(A, Te(λ, ϱ, j, i))
+
+            G(Ta(λ, i) * inv(A, Ta(λ, j) * Tα(λ, j))^6 * (Ta(λ, j) * Tα(λ, j) * z)^4 * c)
+        end
+
+        τ = (G(Ta(λ, 1) * Tα(λ, 1))^6) * prod(halftwists, init = one(G))
+
+        # τ^genus is trivial but only in autπ₁Σ₄
+        # here we check its centrality
+
+
+
+
+        τᵍ = τ^genus
+        @test_broken all(a * τᵍ == τᵍ * a for a in Groups.gens(G))
+    end
 end
diff --git a/test/runtests.jl b/test/runtests.jl
index e69bce8..2e41ab8 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -26,6 +26,7 @@ include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
     include("fp_groups.jl")
 
     include("AutFn.jl")
+    include("AutSigma_41.jl")
 
     # if !haskey(ENV, "CI")
     #    include("benchmarks.jl")