diff --git a/src/Groups.jl b/src/Groups.jl
index 03fa1ec..a93c3f3 100644
--- a/src/Groups.jl
+++ b/src/Groups.jl
@@ -23,13 +23,13 @@ include("arithmetic.jl")
 include("findreplace.jl")
 
 module New
-import Groups: AutSymbol, GSymbol, λ, ϱ, RTransvect, LTransvect
 using DataStructures
 
 include("new_types.jl")
 include("new_hashing.jl")
 include("normalform.jl")
 
+include("new_transvections.jl")
 include("gersten_relations.jl")
 include("new_autgroups.jl")
 
diff --git a/src/gersten_relations.jl b/src/gersten_relations.jl
index ea86e6d..79014b8 100644
--- a/src/gersten_relations.jl
+++ b/src/gersten_relations.jl
@@ -1,12 +1,9 @@
 function gersten_alphabet(n::Integer; commutative::Bool = true)
-    indexing = [[i, j] for i in 1:n for j in 1:n if i ≠ j]
-    rmuls = [ϱ(i, j) for (i, j) in indexing]
+    indexing = [(i, j) for i in 1:n for j in 1:n if i ≠ j]
+    S = [ϱ(i, j) for (i, j) in indexing]
 
-    S = if commutative
-        Vector{AutSymbol}(rmuls)
-    else
-        lmuls = [λ(i, j) for (i, j) in indexing]
-        AutSymbol[rmuls; lmuls]
+    if !commutative
+        append!(S, [λ(i, j) for (i, j) in indexing])
     end
 
     return Alphabet(S)
diff --git a/src/new_autgroups.jl b/src/new_autgroups.jl
index a8da29a..aea1766 100644
--- a/src/new_autgroups.jl
+++ b/src/new_autgroups.jl
@@ -89,37 +89,13 @@ domain(f::FPGroupElement{<:AutomorphismGroup}) = deepcopy(parent(f).domain)
 evaluate(f::FPGroupElement{<:AutomorphismGroup{<:FreeGroup}}) =
     evaluate!(domain(f), f)
 
-function evaluate!(t::NTuple{N, T}, f::FPGroupElement{<:AutomorphismGroup{<:FreeGroup}}) where {T<:FPGroupElement, N}
+function evaluate!(
+    t::NTuple{N,T},
+    f::FPGroupElement{<:AutomorphismGroup{<:FreeGroup}},
+) where {T<:FPGroupElement,N}
     A = alphabet(f)
     for idx in word(f)
-        t = evaluate!(t, A[idx])::NTuple{N, T}
+        t = evaluate!(t, A[idx], alphabet(object(parent(f))))::NTuple{N,T}
     end
     return t
 end
-
-function evaluate!(v::NTuple{N, T}, s::AutSymbol) where {N, T}
-    @assert s.pow in (-1, 1)
-    return evaluate!(v, s.fn, isone(s.pow))::NTuple{N, T}
-end
-
-function evaluate!(v, ϱ::RTransvect, flag)
-    if flag
-        append!(New.word(v[ϱ.i]), New.word(v[ϱ.j]   ))
-    else
-        append!(New.word(v[ϱ.i]), New.word(v[ϱ.j]^-1))
-    end
-    _setnormalform!(v[ϱ.i], false)
-    _setvalidhash!(v[ϱ.i], false)
-    return v
-end
-
-function evaluate!(v, λ::LTransvect, flag)
-    if flag
-        prepend!(New.word(v[λ.i]), New.word(v[λ.j]   ))
-    else
-        prepend!(New.word(v[λ.i]), New.word(v[λ.j]^-1))
-    end
-    _setnormalform!(v[λ.i], false)
-    _setvalidhash!(v[λ.i], false)
-    return v
-end
diff --git a/src/new_transvections.jl b/src/new_transvections.jl
new file mode 100644
index 0000000..3dd3832
--- /dev/null
+++ b/src/new_transvections.jl
@@ -0,0 +1,85 @@
+abstract type GSymbol end
+Base.literal_pow(::typeof(^), t::GSymbol, ::Val{-1}) = inv(t)
+
+struct Transvection <: GSymbol
+    id::Symbol
+    ij::UInt8
+    inv::Bool
+
+    function Transvection(id::Symbol, i::Integer, j::Integer, inv=false)
+        @assert id in (:ϱ, :λ)
+        return new(id, _indices(UInt8(i),UInt8(j)), inv)
+    end
+end
+
+ϱ(i, j) = Transvection(:ϱ, i, j)
+λ(i, j) = Transvection(:λ, i, j)
+
+_indices(ij::UInt8) = (ij & 0xf0)>>4, (ij & 0x0f)
+
+function _indices(i::UInt8, j::UInt8)
+    @boundscheck @assert i < typemax(i) ÷ 2
+    @boundscheck @assert j < typemax(j) ÷ 2
+    sizeof
+    return (i << 4) + j
+end
+
+indices(t::Transvection) = Int.(_indices(t.ij))
+
+function Base.getproperty(t::Transvection, s::Symbol)
+    s === :i && return first(indices(t))
+    s === :j && return last(indices(t))
+    return Core.getfield(t, s)
+end
+
+function Base.show(io::IO, t::Transvection)
+    id = if t.id === :ϱ
+        "ϱ"
+    else # if t.id === :λ
+        "λ"
+    end
+    # print(io, id, Groups.subscriptify(t.i), ".", Groups.subscriptify(t.j))
+    print(io, id, "_", t.i, ",", t.j)
+    t.inv && print(io, "^-1")
+end
+
+Base.inv(t::Transvection) = Transvection(t.id, _indices(t.ij)..., !t.inv)
+
+Base.:(==)(t::Transvection, s::Transvection) =
+    t.id === s.id && t.ij == s.ij && t.inv == s.inv
+Base.hash(t::Transvection, h::UInt) = hash(t.id, hash(t.ij, hash(t.inv, h)))
+
+function evaluate!(v::Tuple, t::Transvection, A::Alphabet)
+    i, j = indices(t)
+    @assert i ≤ length(v) && j ≤ length(v)
+
+    @inbounds begin
+        if t.id === :ϱ
+            if !t.inv
+                append!(word(v[i]), word(v[j]))
+            else
+                # append!(word(v[i]), inv(A, word(v[j])))
+                for l in Iterators.reverse(word(v[j]))
+                    push!(word(v[i]), inv(A, l))
+                end
+            end
+        else # if t.id === :λ
+            if !t.inv
+                # prepend!(word(v[i]), word(v[j]))
+                for l in Iterators.reverse(word(v[j]))
+                    pushfirst!(word(v[i]), l)
+                end
+            else
+                # prepend!(word(v[i]), inv(A, word(v[j])))
+                for l in word(v[j])
+                    pushfirst!(word(v[i]), inv(A, l))
+                end
+            end
+        end
+
+        _setnormalform!(v[i], false)
+        _setvalidhash!(v[i], false)
+    end
+
+    return v
+end