diff --git a/experimental/LieAlgebras/src/LieAlgebras.jl b/experimental/LieAlgebras/src/LieAlgebras.jl
index d6cd094e36c6..e866dfbd8e64 100644
--- a/experimental/LieAlgebras/src/LieAlgebras.jl
+++ b/experimental/LieAlgebras/src/LieAlgebras.jl
@@ -74,6 +74,7 @@ import ..Oscar:
   ngens,
   order,
   parent_type,
+  permutation_group,
   rank,
   root,
   roots,
diff --git a/experimental/LieAlgebras/src/WeylGroup.jl b/experimental/LieAlgebras/src/WeylGroup.jl
index 821fe43e161e..e4d0fd98e021 100644
--- a/experimental/LieAlgebras/src/WeylGroup.jl
+++ b/experimental/LieAlgebras/src/WeylGroup.jl
@@ -474,6 +474,59 @@ function isomorphism(::Type{FPGroup}, W::WeylGroup; set_properties::Bool=true)
   return MapFromFunc(W, G, iso, isoinv)
 end
 
+function permutation_group(W::WeylGroup; set_properties::Bool=true)
+  return codomain(isomorphism(PermGroup, W; set_properties))
+end
+
+function isomorphism(::Type{PermGroup}, W::WeylGroup; set_properties::Bool=true)
+  @req is_finite(W) "Weyl group is not finite"
+  R = root_system(W)
+  type, ordering = root_system_type_with_ordering(R)
+
+  if length(type) != 1
+    error("Not implemented (yet)")
+  end
+  if !issorted(ordering)
+    error("Not implemented (yet)")
+  end
+  coxeter_type, n = only(type)
+  if coxeter_type == :A
+    G = symmetric_group(n + 1)
+
+    iso = function (w::WeylGroupElem)
+      reduce(*, [cperm(G, [i, i + 1]) for i in word(w)]; init=cperm(G))
+    end
+
+    isoinv = function (p::PermGroupElem)
+      word = UInt8[]
+      for cycle in cycles(p)
+        transpositions = [
+          sort([c, cycle[i + 1]]) for (i, c) in enumerate(cycle) if i < length(cycle)
+        ]
+        for t in transpositions
+          word = reduce(
+            vcat,
+            [
+              [i for i in t[1]:(t[2] - 1)],
+              [i for i in reverse(t[1]:(t[2] - 2))],
+              word,
+            ],
+          )
+        end
+      end
+      return W(word)
+    end
+  else
+    error("Not implemented (yet)")
+  end
+
+  if set_properties
+    set_order(G, order(W))
+  end
+
+  return MapFromFunc(W, G, iso, isoinv)
+end
+
 ###############################################################################
 # ReducedExpressionIterator
 
diff --git a/experimental/LieAlgebras/test/WeylGroup-test.jl b/experimental/LieAlgebras/test/WeylGroup-test.jl
index 76178206bce4..e78ad3d73ad5 100644
--- a/experimental/LieAlgebras/test/WeylGroup-test.jl
+++ b/experimental/LieAlgebras/test/WeylGroup-test.jl
@@ -52,6 +52,7 @@ include(
 
   @testset "WeylGroup Group conformace test for $(Wname)" for (Wname, W) in [
     ("A1", weyl_group(:A, 1)),
+    ("A5", weyl_group(:A, 5)),
     ("B4", weyl_group(root_system(:B, 4))),
     ("D5", weyl_group(cartan_matrix(:D, 5))),
     ("F4+G2", weyl_group((:F, 4), (:G, 2))),
@@ -111,9 +112,55 @@ include(
             if is_finite(W) # remove once rand(W) is implemented for infinite groups
               w = rand(W)
               @test w == inv(iso)(iso(w))
+              v = rand(W)
+              @test iso(v * w) == iso(v) * iso(w)
+              @test v * w == inv(iso)(iso(v) * iso(w))
             end
             g = rand_pseudo(G)
             @test g == iso(inv(iso)(g))
+            h = rand_pseudo(G)
+            @test inv(iso)(h * g) == inv(iso)(h) * inv(iso)(g)
+            @test h * g == iso(inv(iso)(h) * inv(iso)(g))
+          end
+        end
+      end
+    end
+
+    if has_root_system_type(root_system(W))
+      type, ordering = root_system_type_with_ordering(root_system(W))
+      if length(type) == 1 && issorted(ordering) && only(type)[1] == :A # only implemented for A_n (yet)
+        @testset "isomorphism(PermGroup, ::WeylGroup; set_properties=$set_properties)" for set_properties in
+                                                                                           [
+          false, true
+        ]
+          G = permutation_group(W; set_properties)
+          if (is_finite(W) && ngens(W) < 6) || set_properties #= for sane runtime =#
+            @test is_finite(G) == is_finite(W)
+            is_finite(W) && @test order(G) == order(W)
+          end
+
+          iso = isomorphism(PermGroup, W; set_properties)
+          @test W == domain(iso)
+          G = codomain(iso)
+          if (is_finite(W) && ngens(W) < 6) || set_properties #= for sane runtime =#
+            @test is_finite(G) == is_finite(W)
+            is_finite(W) && @test order(G) == order(W)
+            if ngens(W) < 10 #= for sane runtime =#
+              for _ in 1:5
+                if is_finite(W) # remove once rand(W) is implemented for infinite groups
+                  w = rand(W)
+                  @test w == inv(iso)(iso(w))
+                  v = rand(W)
+                  @test iso(v * w) == iso(v) * iso(w)
+                  @test v * w == inv(iso)(iso(v) * iso(w))
+                end
+                g = rand_pseudo(G)
+                @test g == iso(inv(iso)(g))
+                h = rand_pseudo(G)
+                @test inv(iso)(h * g) == inv(iso)(h) * inv(iso)(g)
+                @test h * g == iso(inv(iso)(h) * inv(iso)(g))
+              end
+            end
           end
         end
       end