diff --git a/test/AutSigma3.jl b/test/AutSigma3.jl
new file mode 100644
index 0000000..475fb57
--- /dev/null
+++ b/test/AutSigma3.jl
@@ -0,0 +1,75 @@
+@testset "Aut(Σ₃.₀)" begin
+    genus = 3
+
+    π₁Σ = Groups.SurfaceGroup(genus, 0)
+
+    Groups.PermRightAut(p::Perm) = Groups.PermRightAut(p.d)
+    # Groups.PermLeftAut(p::Perm) = Groups.PermLeftAut(p.d)
+    autπ₁Σ = let autπ₁Σ = AutomorphismGroup(π₁Σ)
+        pauts = let p = perm"(1,3,5)(2,4,6)"
+            [Groups.PermRightAut(p^i) for i in 0:2]
+        end
+        T = eltype(KnuthBendix.letters(alphabet(autπ₁Σ)))
+        S = eltype(pauts)
+
+        A = Alphabet(Union{T,S}[KnuthBendix.letters(alphabet(autπ₁Σ)); pauts])
+
+        autG = AutomorphismGroup(
+            π₁Σ,
+            autπ₁Σ.gens,
+            A,
+            ntuple(i->inv(gens(π₁Σ, i)), 2Groups.genus(π₁Σ))
+        )
+
+        autG
+    end
+
+    Al = alphabet(autπ₁Σ)
+    S = [gens(autπ₁Σ); inv.(gens(autπ₁Σ))]
+
+    sautFn = let ltrs = KnuthBendix.letters(Al)
+        parent(first(ltrs).autFn_word)
+    end
+
+    τ = Groups.rotation_element(sautFn)
+
+    @testset "Twists" begin
+        A = KnuthBendix.alphabet(sautFn)
+        λ = Groups.ΡΛ(:λ, A, 2genus)
+        ϱ = Groups.ΡΛ(:ϱ, A, 2genus)
+        @test sautFn(Groups.Te_diagonal(λ, ϱ, 1)) ==
+            conj(sautFn(Groups.Te_diagonal(λ, ϱ, 2)), τ)
+
+        @test sautFn(Groups.Te_diagonal(λ, ϱ, 3)) == sautFn(Groups.Te(λ, ϱ, 3, 1))
+    end
+
+    z = let d = Groups.domain(τ)
+        Groups.evaluate(τ^genus)
+    end
+
+    @test π₁Σ.(word.(z)) == Groups.domain(first(S))
+    d = Groups.domain(first(S))
+    p = perm"(1,3,5)(2,4,6)"
+    @test Groups.evaluate!(deepcopy(d), τ) == d^inv(p)
+    @test Groups.evaluate!(deepcopy(d), τ^2) == d^p
+
+    E, sizes = Groups.wlmetric_ball(S, radius=3)
+    @test sizes == [49, 1813, 62971]
+    B2 = @view E[1:sizes[2]]
+
+    σ = autπ₁Σ(Word([Al[Groups.PermRightAut(p)]]))
+
+    @test conj(S[7], σ) == S[10]
+    @test conj(S[7], σ^2) == S[11]
+    @test conj(S[9], σ) == S[12]
+    @test conj(S[9], σ^2) == S[8]
+
+    @test conj(S[1], σ) == S[4]
+    @test conj(S[1], σ^2) == S[5]
+    @test conj(S[3], σ) == S[6]
+    @test conj(S[3], σ^2) == S[2]
+
+    B2ᶜ = [conj(b, σ) for b in B2]
+    @test B2ᶜ != B2
+    @test Set(B2ᶜ) == Set(B2)
+end
diff --git a/test/AutSigma_41.jl b/test/AutSigma_41.jl
index 16e3a2f..7a9e0d1 100644
--- a/test/AutSigma_41.jl
+++ b/test/AutSigma_41.jl
@@ -5,9 +5,10 @@ using Groups.KnuthBendix
 
     genus = 4
 
-    G = SpecialAutomorphismGroup(FreeGroup(2genus))
+    Fn = FreeGroup(2genus)
+    G = SpecialAutomorphismGroup(Fn)
 
-    T = Groups.mcg_twists(autF)
+    T = Groups.mcg_twists(G)
 
     # symplectic pairing in the free Group goes like this:
     # f1 ↔ f5
@@ -41,7 +42,6 @@ using Groups.KnuthBendix
 
     As = T[[1, 5, 9, 6, 12, 7, 14, 8]] # the inverses of the elements a
 
-
     @testset "preserving relator" begin
         F = Groups.object(G)
 
@@ -123,7 +123,12 @@ using Groups.KnuthBendix
             # x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
             @time evaluate(x)
             b3 = x * a0 * x^-1
-            @time evaluate(b3)
+            b3im = @time evaluate(b3)
+            b3cim = @time let g = b3
+                f = Groups.compiled(g)
+                f(Groups.domain(g))
+            end
+            @test b3im == b3cim
             @test a0 * b2 * b1 == a1 * a3 * a5 * b3
         end
     end
@@ -170,46 +175,13 @@ using Groups.KnuthBendix
 
     @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)
-
-        # τ^genus is trivial but only in autπ₁Σ₄
-        # here we check its centrality
-
+        τ = Groups.rotation_element(G)
         τᵍ = τ^genus
 
         symplectic_gens = let genus = genus, G = G
             π₁Σ = Groups.SurfaceGroup(genus, 0)
-            autπ₁Σ = AutomorphismGroup(π₁Σ)
-            letters = alphabet(autπ₁Σ).letters
-            G.(word(l.autFn_word) for l in letters)
+            s_twists = Groups.symplectic_twists(π₁Σ)
+            G.(word(t.autFn_word) for t in s_twists)
         end
 
         @test all(sg * τᵍ == τᵍ * sg for sg in symplectic_gens)
diff --git a/test/runtests.jl b/test/runtests.jl
index 2e41ab8..c0334c4 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -27,6 +27,7 @@ include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
 
     include("AutFn.jl")
     include("AutSigma_41.jl")
+    include("AutSigma3.jl")
 
     # if !haskey(ENV, "CI")
     #    include("benchmarks.jl")