diff --git a/src/abelianize.jl b/src/abelianize.jl
index b62595b..e059feb 100644
--- a/src/abelianize.jl
+++ b/src/abelianize.jl
@@ -21,3 +21,49 @@ function _abelianize(
     end
 end
 
+function _abelianize(
+    i::Integer,
+    source::AutomorphismGroup{<:Groups.SurfaceGroup},
+    target::MatrixGroups.SpecialLinearGroup{N, T}) where {N, T}
+    n = ngens(Groups.object(source))
+    @assert n == N
+    g = alphabet(source)[i].autFn_word
+    result = one(target)
+    for l in word(g)
+        append!(word(result), _abelianize(l, parent(g), target))
+    end
+
+    return word(result)
+end
+
+function Groups._abelianize(
+    i::Integer,
+    source::AutomorphismGroup{<:Groups.SurfaceGroup},
+    target::MatrixGroups.SymplecticGroup{N, T}
+) where {N, T}
+    @assert iseven(N)
+    As = alphabet(source)
+    At = alphabet(target)
+
+    SlN = let genus = Groups.genus(Groups.object(source))
+        @assert 2genus == N
+        MatrixGroups.SpecialLinearGroup{2genus}(T)
+    end
+
+    ab = Groups.Homomorphism(Groups._abelianize, source, SlN, check=false)
+
+    matrix_spn_map = let S = gens(target)
+        Dict(MatrixGroups.matrix_repr(g)=> word(g) for g in union(S, inv.(S)))
+    end
+
+    # renumeration:
+    # (f1, f2, f3, f4, f5, f6) = (a₁, a₂, a₃, b₁, b₂, b₃) →
+    # → (b₃, a₃, b₂, a₂, b₁, a₁)
+    # hence p = [6, 4, 2, 5, 3, 1]
+    p = [reverse(2:2:N); reverse(1:2:N)]
+
+    g = source([i])
+    Mg = MatrixGroups.matrix_repr(ab(g))[p,p]
+
+    return matrix_spn_map[Mg]
+end
diff --git a/src/matrix_groups/Spn.jl b/src/matrix_groups/Spn.jl
index 66e658c..630a56a 100644
--- a/src/matrix_groups/Spn.jl
+++ b/src/matrix_groups/Spn.jl
@@ -28,19 +28,22 @@ function Base.show(
     ::MIME"text/plain",
     sp::Groups.AbstractFPGroupElement{<:SymplecticGroup{N}}
 ) where {N}
+    normalform!(sp)
     print(io, "$N×$N Symplectic matrix: ")
     KnuthBendix.print_repr(io, word(sp), alphabet(sp))
     println(io)
     Base.print_array(io, matrix_repr(sp))
 end
 
+_offdiag_idcs(n) = ((i,j) for i in 1:n for j in 1:n if i ≠ j)
+
 function symplectic_gens(N, T=Int8)
     iseven(N) || throw(ArgumentError("N needs to be even!"))
     n = N÷2
 
-    a_ijs = [ElementarySymplectic{N}(:A, i,j, one(T)) for (i,j) in offdiagonal_indexing(n)]
+    a_ijs = [ElementarySymplectic{N}(:A, i,j, one(T)) for (i,j) in _offdiag_idcs(n)]
     b_is =  [ElementarySymplectic{N}(:B, n+i,i, one(T)) for i in 1:n]
-    c_ijs = [ElementarySymplectic{N}(:B, n+i,j, one(T)) for (i,j) in offdiagonal_indexing(n)]
+    c_ijs = [ElementarySymplectic{N}(:B, n+i,j, one(T)) for (i,j) in _offdiag_idcs(n)]
 
     S = [a_ijs; b_is; c_ijs]
 
diff --git a/test/homomorphisms.jl b/test/homomorphisms.jl
index 8419b73..97f4736 100644
--- a/test/homomorphisms.jl
+++ b/test/homomorphisms.jl
@@ -2,8 +2,12 @@ function test_homomorphism(hom)
     F = hom.source
     @test isone(hom(one(F)))
     @test all(inv(hom(g)) == hom(inv(g)) for g in gens(F))
-    @test all(isone(hom(g)*hom(inv(g))) for g in gens(F))
-    @test all(hom(g*h) == hom(g)*hom(h) for g in gens(F) for h in gens(F))
+    @test all(isone(hom(g) * hom(inv(g))) for g in gens(F))
+    @test all(hom(g * h) == hom(g) * hom(h) for g in gens(F) for h in gens(F))
+    @test all(
+        hom(inv(g * h)) == inv(hom(g * h)) == hom(inv(h)) * hom(inv(g)) for
+        g in gens(F) for h in gens(F)
+    )
 end
 
 @testset "Homomorphisms" begin
@@ -46,4 +50,22 @@ end
 
         test_homomorphism(hom)
     end
+
+    @testset "Correctness of autπ₁Σ → SpN" begin
+
+        GENUS = 3
+        π₁Σ = Groups.SurfaceGroup(GENUS, 0)
+        autπ₁Σ = AutomorphismGroup(π₁Σ)
+
+        SpN = MatrixGroups.SymplecticGroup{2GENUS}(Int8)
+
+        hom = Groups.Homomorphism(
+            Groups._abelianize,
+            autπ₁Σ,
+            SpN,
+            check = false,
+        )
+
+        test_homomorphism(hom)
+    end
 end
diff --git a/test/matrix_groups.jl b/test/matrix_groups.jl
index fb13d6f..92555b2 100644
--- a/test/matrix_groups.jl
+++ b/test/matrix_groups.jl
@@ -33,4 +33,16 @@ using Groups.MatrixGroups
             end
         end
     end
+
+    @testset "Sp(6, ℤ)" begin
+        Sp6 = MatrixGroups.SymplecticGroup{6}(Int8)
+
+        @testset "GroupsCore conformance" begin
+            test_Group_interface(Sp6)
+            g = Sp6(rand(1:length(alphabet(Sp6)), 10))
+            h = Sp6(rand(1:length(alphabet(Sp6)), 10))
+
+            test_GroupElement_interface(g, h)
+        end
+    end
 end