diff --git a/src/algorithms/rings.jl b/src/algorithms/rings.jl
index 54da522..55793c4 100644
--- a/src/algorithms/rings.jl
+++ b/src/algorithms/rings.jl
@@ -25,7 +25,7 @@ ConstMiniBitSet(l::AbstractVector{<:Integer}) = ConstMiniBitSet{UInt64}(l)
 push(x::ConstMiniBitSet{T}, i::Integer) where {T} = _constminibitset(x.x | (one(T) << (i % UInt8)))
 Base.union(x::ConstMiniBitSet, y::ConstMiniBitSet) = _constminibitset(x.x | y.x)
 Base.intersect(x::ConstMiniBitSet, y::ConstMiniBitSet) = _constminibitset(x.x & y.x)
-Base.in(x::ConstMiniBitSet{T}, i::Integer) where {T} = ((x.x >> (i % UInt8)) & one(T)) % Bool
+Base.in(i::Integer, x::ConstMiniBitSet{T}) where {T} = ((x.x >> (i % UInt8)) & one(T)) % Bool
 Base.setdiff(x::ConstMiniBitSet, y::ConstMiniBitSet) = _constminibitset(x.x & ~y.x)
 Base.symdiff(x::ConstMiniBitSet, y::ConstMiniBitSet) = _constminibitset(x.x ⊻ y.x)
 Base.length(x::ConstMiniBitSet) = count_ones(x.x)
@@ -67,7 +67,6 @@ struct JunctionNode
 end
 JunctionNode() = JunctionNode(Int[], -one(SmallIntType), ConstMiniBitSet{UInt32}())
 JunctionNode(head::Integer, num, roots::ConstMiniBitSet) = JunctionNode(Int[head], num, roots)
-JunctionNode(num::Integer, ::Nothing) = JunctionNode(Int[], num, ConstMiniBitSet{UInt32}())
 JunctionNode(root::Integer) = JunctionNode(1, zero(SmallIntType), ConstMiniBitSet{UInt32}(root))
 
 function Base.show(io::IO, x::JunctionNode)
@@ -1039,7 +1038,7 @@ Base.length(ras::RingAttributions) = length(ras.attrs)
 Base.eltype(::Type{RingAttributions{D}}) where {D} = RingIncluding{D}
 
 function Base.show(io::IO, ras::RingAttributions)
-    println(io, typeof(ras), "(rings per node:", length.(ras.attrs), ')')
+    println(io, typeof(ras), "(rings per node: ", length.(ras.attrs), ')')
 end
 
 struct RingIncluding{D}
@@ -1053,7 +1052,7 @@ function Base.getindex(ri::RingIncluding{D}, j::Integer) where {D}
     return PeriodicNeighborList{D}(.-ofs, newring)
 end
 Base.length(ri::RingIncluding) = length(ri.ras.attrs[ri.i])
-Base.eltype(::Type{RingIncluding{D}}) where {D} = PeriodicNeighborList
+Base.eltype(::Type{RingIncluding{D}}) where {D} = PeriodicNeighborList{D}
 
 function Base.iterate(r::Union{RingAttributions,RingIncluding}, state=1)
     (state % UInt) - 1 < length(r) ? ((@inbounds r[state]), state+1) : nothing
diff --git a/src/precompile.jl b/src/precompile.jl
index 14d2fe2..505b96f 100644
--- a/src/precompile.jl
+++ b/src/precompile.jl
@@ -33,7 +33,7 @@ function _precompile_()
         @enforce Base.precompile(Tuple{typeof(intersect), CMBitSet{T}, CMBitSet{T}})
         @enforce Base.precompile(Tuple{typeof(setdiff), CMBitSet{T}, CMBitSet{T}})
         @enforce Base.precompile(Tuple{typeof(symdiff), CMBitSet{T}, CMBitSet{T}})
-        @enforce Base.precompile(Tuple{typeof(in), CMBitSet{T}, Int})
+        @enforce Base.precompile(Tuple{typeof(in), Int, CMBitSet{T}})
         @enforce Base.precompile(Tuple{typeof(eltype), Type{CMBitSet{T}}})
         @enforce Base.precompile(Tuple{typeof(length), CMBitSet{T}})
         @enforce Base.precompile(Tuple{typeof(isempty), CMBitSet{T}})
@@ -48,7 +48,6 @@ function _precompile_()
         @enforce Base.precompile(Tuple{Type{PeriodicGraphs.JunctionNode},Int,SmallIntT,CMBitSet{T}})
     end
     @enforce Base.precompile(Tuple{Type{PeriodicGraphs.JunctionNode}})
-    @enforce Base.precompile(Tuple{Type{PeriodicGraphs.JunctionNode},Int,Nothing})
     @enforce Base.precompile(Tuple{Type{PeriodicGraphs.JunctionNode},Int})
     @enforce Base.precompile(Tuple{typeof(PeriodicGraphs.unsafe_incr!),Ptr{SmallIntT}})
     @enforce Base.precompile(Tuple{typeof(PeriodicGraphs.symdiff_cycles), Vector{Int}, Vector{Int}})
diff --git a/test/runtests.jl b/test/runtests.jl
index 004a1c2..4292356 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -2,7 +2,7 @@ using Test
 using Graphs
 using PeriodicGraphs
 import PeriodicGraphs: ofs, vertex_permutation, dimensionality, LoopException, extended_gcd
-import StaticArrays: SVector
+using StaticArrays: SVector, SMatrix
 
 # using Aqua
 # Aqua.test_all(PeriodicGraphs)
@@ -574,6 +574,7 @@ end
     @test isempty(outneighbors(g, PeriodicVertex(5, (1, 0, 1))))
     @test only(inneighbors(g, PeriodicVertex(1, (4, 5, 6)))) == PeriodicVertex(3, (4, 5, 7))
     @test indegree(g, 2) == indegree(g, PeriodicVertex3D(2)) == outdegree(g, PeriodicVertex3D(2)) == degree(g, PeriodicVertex3D(2))
+    @test string(neighbors(g, PeriodicVertex(3, (7, -9, 2)))) == string(expected_neighbors)
 end
 
 @testset "Vertex removal" begin
@@ -669,11 +670,15 @@ end
     @test reduce(+, s .* λ) == d
 end
 
+const cpi = PeriodicGraph("2 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 2 3 0 -1 2 5 1 -1 3 4 1 0 4 5 0 -1")
 const nab = PeriodicGraph("3 1 2 0 -1 0 1 3 0 -1 -1 1 4 -1 0 0 1 5 -1 0 -1 2 3 0 0 -1 2 4 -1 0 0 2 5 -1 0 -1 3 4 0 0 0 3 5 0 0 0 4 5 0 0 0")
 const lta = PeriodicGraph("3 1 3 0 0 0 1 7 0 -1 0 1 9 0 0 -1 1 21 -1 0 -1 2 4 0 0 0 2 8 0 -1 0 2 9 0 0 0 2 21 -1 0 0 3 5 0 0 0 3 10 0 0 -1 3 22 -1 0 -1 4 6 0 0 0 4 10 0 0 0 4 22 -1 0 0 5 6 0 0 0 5 11 0 0 0 5 23 -1 0 0 6 12 0 0 0 6 24 -1 0 0 7 8 0 0 0 7 11 0 0 0 7 23 -1 0 0 8 12 0 0 0 8 24 -1 0 0 9 10 0 0 0 9 16 0 -1 0 10 13 0 0 0 11 12 0 0 0 11 14 0 0 0 12 15 0 0 0 13 14 0 0 1 13 15 0 0 0 13 17 0 0 0 14 16 0 0 -1 14 18 0 0 0 15 16 0 0 0 15 19 0 0 0 16 20 0 0 0 17 18 0 0 1 17 19 0 0 0 17 22 0 0 0 18 20 0 0 -1 18 23 0 0 0 19 20 0 0 0 19 24 0 0 0 20 21 0 1 0 21 22 0 0 0 23 24 0 0 0")
 const mtn = PeriodicGraph("3 1 2 0 0 0 1 8 0 0 -1 1 11 0 -1 0 1 29 -1 0 0 2 7 0 0 0 2 12 0 -1 0 2 30 -1 0 0 3 5 0 0 0 3 7 0 0 0 3 9 0 0 0 3 31 -1 0 0 4 6 0 0 0 4 8 0 0 0 4 9 0 0 0 4 32 -1 0 0 5 10 0 0 -1 5 11 0 0 0 5 33 -1 0 0 6 10 0 0 0 6 12 0 0 0 6 34 -1 0 0 7 13 0 0 -1 7 15 0 0 0 8 13 0 0 0 8 16 0 0 0 9 14 0 0 0 9 17 0 0 0 10 13 0 0 0 10 18 0 0 0 11 14 0 0 0 11 20 0 0 -1 12 14 0 0 0 12 19 0 0 0 13 21 0 0 0 14 22 0 0 0 15 17 0 0 0 15 19 0 -1 0 15 23 0 0 0 16 17 0 0 0 16 20 0 -1 0 16 24 0 0 0 17 25 0 0 0 18 19 0 0 0 18 20 0 0 0 18 26 0 0 0 19 27 0 0 0 20 28 0 0 0 21 23 0 0 1 21 24 0 0 0 21 26 0 0 0 22 25 0 0 0 22 27 0 0 0 22 28 0 0 -1 23 29 0 0 0 23 33 0 -1 0 24 30 0 0 0 24 34 0 -1 0 25 29 0 0 0 25 30 0 0 0 26 31 0 0 0 26 32 0 0 0 27 31 0 0 0 27 33 0 0 0 28 32 0 0 0 28 34 0 0 0 29 32 0 0 -1 30 31 0 0 0 33 34 0 0 -1")
 
-@testset "NAB, LTA and MTN examples" begin
+@testset "cpi, NAB, LTA and MTN examples" begin
+    @test is_well_formed(cpi)
+    @test nv(cpi) == 5 && ne(cpi) == 8
+    @test degree(cpi) == [4; fill(3, 4)]
     @test is_well_formed(nab)
     @test ne(nab) == 2*nv(nab) == 10
     @test all(==(4), degree(nab))
@@ -686,21 +691,36 @@ const mtn = PeriodicGraph("3 1 2 0 0 0 1 8 0 0 -1 1 11 0 -1 0 1 29 -1 0 0 2 7 0
 end
 
 @testset "ConstMiniBitSet" begin
+    CMBitSet = PeriodicGraphs.ConstMiniBitSet
+    @test CMBitSet() == CMBitSet{UInt64}()
+    @test only(CMBitSet(52)) == 52
+    @test CMBitSet([3, 4]) == PeriodicGraphs.push(CMBitSet{UInt64}(4), 3)
     for T in (UInt32, UInt64)
-        CMBitSet = PeriodicGraphs.ConstMiniBitSet{T}
-        c = CMBitSet([2,3,13])
+        c = CMBitSet{T}([2,3,13])
         @test 3 ∈ c
         @test length(c) == 3
-        @test 4 ∈ symdiff(c, CMBitSet(4))
-        @test 4 == only(setdiff(CMBitSet([13, 4]), c))
-        @test isempty(CMBitSet())
+        @test 4 ∈ symdiff(c, CMBitSet{T}(4))
+        @test 4 == only(setdiff(CMBitSet{T}([13, 4]), c))
+        @test intersect(c, CMBitSet{T}([13,3,13])) == CMBitSet{T}([3,13])
+        @test isempty(CMBitSet{T}())
         @test collect(c) == [2, 3, 13]
         @test minimum(c) == 2
         @test maximum(c) == 13
+        @test occursin("[2, 3, 13]", string(c))
     end
 end
 
 @testset "Ring statistics" begin
+    dag, vertexnums = PeriodicGraphs.arcs_list(cpi, 1, 10, nothing, Bool[0;1;1;0;0])
+    @test string(dag[1]) == "JunctionNode([])"
+    rea = PeriodicGraphs.RingsEndingAt(dag, 12)
+    @test_throws MethodError length(rea)
+    @test isempty(rea)
+    dag, vertexnums = PeriodicGraphs.arcs_list(cpi, 1, 10)
+    rea = PeriodicGraphs.RingsEndingAt(dag, 12)
+    @test only(rea) == [5, 1, 4, 10, 12]
+    @test eltype(rea) == Vector{Int}
+
     rings_nab = RingAttributions(nab)
     nab_rotated = nab[[3,1,2,4,5]]
     rings_nab_rotated = RingAttributions(nab_rotated)
@@ -718,6 +738,11 @@ end
     @test sort!(length.(rings_mtn[34])) == [fill(5,5); 6; fill(10, 6)]
     @test sort!(length.(rings_mtn[10])) == [fill(5,6); fill(10, 9)]
     @test sort!(length.(rings_mtn[13])) == [fill(5,6); fill(10, 12)]
+    @test eltype(rings_mtn) == PeriodicGraphs.RingIncluding{3}
+    str_rings_mtn = string(rings_mtn)
+    @test occursin("rings per node: [12", str_rings_mtn)
+    @test occursin("15 rings containing vertex 7", string(rings_mtn[7]))
+    @test eltype(rings_mtn[1]) == PeriodicGraphs.PeriodicNeighborList{3}
 
     rings_lta = RingAttributions(lta)
     @test length(rings_lta.rings) == 113
@@ -735,49 +760,51 @@ end
 end
 
 
+module PeriodicSymmetries
 # copied from PeriodicGraphEmbeddings.jl
+    using PeriodicGraphs, Graphs, StaticArrays
 
-using StaticArrays
+    struct PeriodicSymmetry3D{T} <: PeriodicGraphs.AbstractSymmetry
+        vmap::SubArray{PeriodicVertex3D,1,Matrix{PeriodicVertex3D},Tuple{Base.Slice{Base.OneTo{Int}},Int},true}
+        rotation::SMatrix{3,3,Int,9}
+        translation::SVector{3,T}
+    end
+    Base.getindex(symm::PeriodicSymmetry3D, i::Integer) = symm.vmap[i].v
+    function Base.getindex(symm::PeriodicSymmetry3D, x::PeriodicVertex3D)
+        dst = symm.vmap[x.v]
+        _ofs = muladd(symm.rotation, x.ofs, dst.ofs)
+        PeriodicVertex3D(dst.v, _ofs)
+    end
+    function Base.isequal(x::PeriodicSymmetry3D{T}, y::PeriodicSymmetry3D{T}) where T
+        x.vmap == y.vmap && x.rotation == y.rotation && x.translation == y.translation
+    end
 
-struct PeriodicSymmetry3D{T} <: PeriodicGraphs.AbstractSymmetry
-    vmap::SubArray{PeriodicVertex3D,1,Matrix{PeriodicVertex3D},Tuple{Base.Slice{Base.OneTo{Int}},Int},true}
-    rotation::SMatrix{3,3,Int,9}
-    translation::SVector{3,T}
-end
-Base.getindex(symm::PeriodicSymmetry3D, i::Integer) = symm.vmap[i].v
-function Base.getindex(symm::PeriodicSymmetry3D, x::PeriodicVertex3D)
-    dst = symm.vmap[x.v]
-    _ofs = muladd(symm.rotation, x.ofs, dst.ofs)
-    PeriodicVertex3D(dst.v, _ofs)
-end
-function Base.isequal(x::PeriodicSymmetry3D{T}, y::PeriodicSymmetry3D{T}) where T
-    x.vmap == y.vmap && x.rotation == y.rotation && x.translation == y.translation
-end
+    struct SymmetryGroup3D{T} <: PeriodicGraphs.AbstractSymmetryGroup{PeriodicSymmetry3D{T}}
+        vmaps::Matrix{PeriodicVertex3D}
+        rotations::Vector{SMatrix{3,3,Int,9}}
+        translations::Vector{SVector{3,T}}
+        uniquemap::Vector{Int}
+        uniques::Vector{Int}
+        hasmirror::Bool
+    end
 
-struct SymmetryGroup3D{T} <: PeriodicGraphs.AbstractSymmetryGroup{PeriodicSymmetry3D{T}}
-    vmaps::Matrix{PeriodicVertex3D}
-    rotations::Vector{SMatrix{3,3,Int,9}}
-    translations::Vector{SVector{3,T}}
-    uniquemap::Vector{Int}
-    uniques::Vector{Int}
-    hasmirror::Bool
-end
+    (s::SymmetryGroup3D)(i::Integer) = s.uniques[s.uniquemap[i]]
+    Base.unique(s::SymmetryGroup3D) = s.uniques
+    function Base.getindex(s::SymmetryGroup3D{T}, i::Integer) where {T}
+        PeriodicSymmetry3D{T}((@view s.vmaps[:,i]), s.rotations[i], s.translations[i])
+    end
+    Base.iterate(s::SymmetryGroup3D, state=1) = state > length(s) ? nothing : (s[state], state+1)
+    Base.eltype(::Type{SymmetryGroup3D{T}}) where {T} = PeriodicSymmetry3D{T}
+    Base.length(s::SymmetryGroup3D) = length(s.rotations)
+    function Base.one(s::SymmetryGroup3D{T}) where T
+        n = length(s.uniquemap)
+        PeriodicSymmetry3D{T}((@view reshape(collect(PeriodicVertex3D.(Base.OneTo(n))), n, 1)[:,1]),
+                            one(SMatrix{3,3,Int,9}), zero(SVector{3,T}))
+    end
 
-(s::SymmetryGroup3D)(i::Integer) = s.uniques[s.uniquemap[i]]
-Base.unique(s::SymmetryGroup3D) = s.uniques
-function Base.getindex(s::SymmetryGroup3D{T}, i::Integer) where {T}
-    PeriodicSymmetry3D{T}((@view s.vmaps[:,i]), s.rotations[i], s.translations[i])
-end
-Base.iterate(s::SymmetryGroup3D, state=1) = state > length(s) ? nothing : (s[state], state+1)
-Base.eltype(::Type{SymmetryGroup3D{T}}) where {T} = PeriodicSymmetry3D{T}
-Base.length(s::SymmetryGroup3D) = length(s.rotations)
-function Base.one(s::SymmetryGroup3D{T}) where T
-    n = length(s.uniquemap)
-    PeriodicSymmetry3D{T}((@view reshape(collect(PeriodicVertex3D.(Base.OneTo(n))), n, 1)[:,1]),
-                          one(SMatrix{3,3,Int,9}), zero(SVector{3,T}))
 end
 
-const symmetries_lta = SymmetryGroup3D{Rational{Int32}}(
+const symmetries_lta = PeriodicSymmetries.SymmetryGroup3D{Rational{Int32}}(
     PeriodicVertex3D[(4, (0,0,0)) (18, (0,0,0)) (15, (0,0,0)) (3, (0,0,0)) (2, (0,0,0)) (14, (0,0,0)) (19, (0,0,0)) (4, (0,0,0)) (1, (0,0,0)) (19, (0,0,0)) (14, (0,0,0)) (2, (0,0,0)) (3, (0,0,0)) (15, (0,0,0)) (18, (0,0,0)) (11, (0,0,0)) (24, (0,0,0)) (10, (0,0,0)) (21, (0,0,0)) (12, (0,0,0)) (23, (0,0,0)) (9, (0,0,0)) (22, (0,0,0)) (24, (0,0,0)) (11, (0,0,0)) (22, (0,0,0)) (9, (0,0,0)) (23, (0,0,0)) (12, (0,0,0)) (21, (0,0,0)) (10, (0,0,0)) (16, (0,0,0)) (17, (0,0,0)) (8, (0,0,0)) (5, (0,0,0)) (20, (0,0,0)) (13, (0,0,0)) (7, (0,0,0)) (6, (0,0,0)) (17, (0,0,0)) (16, (0,0,0)) (6, (0,0,0)) (7, (0,0,0)) (13, (0,0,0)) (20, (0,0,0)) (5, (0,0,0)) (8, (0,0,0)); (3, (0,0,0)) (19, (0,0,0)) (14, (0,0,0)) (4, (0,0,0)) (1, (0,0,0)) (15, (0,0,0)) (18, (0,0,0)) (3, (0,0,0)) (2, (0,0,0)) (18, (0,0,0)) (15, (0,0,0)) (1, (0,0,0)) (4, (0,0,0)) (14, (0,0,0)) (19, (0,0,0)) (23, (0,0,0)) (12, (0,0,0)) (22, (0,0,0)) (9, (0,0,0)) (24, (0,0,0)) (11, (0,0,0)) (21, (0,0,0)) (10, (0,0,0)) (12, (0,0,0)) (23, (0,0,0)) (10, (0,0,0)) (21, (0,0,0)) (11, (0,0,0)) (24, (0,0,0)) (9, (0,0,0)) (22, (0,0,0)) (13, (0,1,0)) (20, (0,-1,0)) (6, (0,1,0)) (7, (0,-1,0)) (17, (0,1,0)) (16, (0,-1,0)) (5, (0,1,0)) (8, (0,-1,0)) (20, (0,-1,0)) (13, (0,1,0)) (8, (0,-1,0)) (5, (0,1,0)) (16, (0,-1,0)) (17, (0,1,0)) (7, (0,-1,0)) (6, (0,1,0)); (2, (0,0,0)) (14, (0,0,0)) (19, (0,0,0)) (1, (0,0,0)) (4, (0,0,0)) (18, (0,0,0)) (15, (0,0,0)) (2, (0,0,0)) (3, (0,0,0)) (15, (0,0,0)) (18, (0,0,0)) (4, (0,0,0)) (1, (0,0,0)) (19, (0,0,0)) (14, (0,0,0)) (12, (0,0,0)) (23, (0,0,0)) (9, (0,0,0)) (22, (0,0,0)) (11, (0,0,0)) (24, (0,0,0)) (10, (0,0,0)) (21, (0,0,0)) (23, (0,0,0)) (12, (0,0,0)) (21, (0,0,0)) (10, (0,0,0)) (24, (0,0,0)) (11, (0,0,0)) (22, (0,0,0)) (9, (0,0,0)) (20, (0,0,0)) (13, (0,0,0)) (7, (0,0,0)) (6, (0,0,0)) (16, (0,0,0)) (17, (0,0,0)) (8, (0,0,0)) (5, (0,0,0)) (13, (0,0,0)) (20, (0,0,0)) (5, (0,0,0)) (8, (0,0,0)) (17, (0,0,0)) (16, (0,0,0)) (6, (0,0,0)) (7, (0,0,0)); (1, (0,0,0)) (15, (0,0,0)) (18, (0,0,0)) (2, (0,0,0)) (3, (0,0,0)) (19, (0,0,0)) (14, (0,0,0)) (1, (0,0,0)) (4, (0,0,0)) (14, (0,0,0)) (19, (0,0,0)) (3, (0,0,0)) (2, (0,0,0)) (18, (0,0,0)) (15, (0,0,0)) (24, (0,0,0)) (11, (0,0,0)) (21, (0,0,0)) (10, (0,0,0)) (23, (0,0,0)) (12, (0,0,0)) (22, (0,0,0)) (9, (0,0,0)) (11, (0,0,0)) (24, (0,0,0)) (9, (0,0,0)) (22, (0,0,0)) (12, (0,0,0)) (23, (0,0,0)) (10, (0,0,0)) (21, (0,0,0)) (17, (0,1,0)) (16, (0,-1,0)) (5, (0,1,0)) (8, (0,-1,0)) (13, (0,1,0)) (20, (0,-1,0)) (6, (0,1,0)) (7, (0,-1,0)) (16, (0,-1,0)) (17, (0,1,0)) (7, (0,-1,0)) (6, (0,1,0)) (20, (0,-1,0)) (13, (0,1,0)) (8, (0,-1,0)) (5, (0,1,0)); (8, (0,-1,0)) (11, (0,0,0)) (24, (0,0,0)) (7, (0,-1,0)) (6, (0,0,0)) (23, (0,0,0)) (12, (0,0,0)) (8, (0,-1,0)) (5, (0,0,0)) (12, (0,0,0)) (23, (0,0,0)) (6, (0,0,0)) (7, (0,-1,0)) (24, (0,0,0)) (11, (0,0,0)) (15, (0,0,0)) (18, (0,0,0)) (16, (0,-1,0)) (17, (0,0,0)) (14, (0,0,0)) (19, (0,0,0)) (13, (0,0,0)) (20, (0,-1,0)) (18, (0,0,0)) (15, (0,0,0)) (20, (0,-1,0)) (13, (0,0,0)) (19, (0,0,0)) (14, (0,0,0)) (17, (0,0,0)) (16, (0,-1,0)) (21, (0,1,0)) (10, (0,0,0)) (1, (0,1,0)) (4, (0,0,0)) (9, (0,1,0)) (22, (0,0,0)) (2, (0,1,0)) (3, (0,0,0)) (10, (0,0,0)) (21, (0,1,0)) (3, (0,0,0)) (2, (0,1,0)) (22, (0,0,0)) (9, (0,1,0)) (4, (0,0,0)) (1, (0,1,0)); (7, (0,-1,0)) (12, (0,0,0)) (23, (0,0,0)) (8, (0,-1,0)) (5, (0,0,0)) (24, (0,0,0)) (11, (0,0,0)) (7, (0,-1,0)) (6, (0,0,0)) (11, (0,0,0)) (24, (0,0,0)) (5, (0,0,0)) (8, (0,-1,0)) (23, (0,0,0)) (12, (0,0,0)) (19, (0,0,0)) (14, (0,0,0)) (20, (0,-1,0)) (13, (0,0,0)) (18, (0,0,0)) (15, (0,0,0)) (17, (0,0,0)) (16, (0,-1,0)) (14, (0,0,0)) (19, (0,0,0)) (16, (0,-1,0)) (17, (0,0,0)) (15, (0,0,0)) (18, (0,0,0)) (13, (0,0,0)) (20, (0,-1,0)) (22, (0,1,0)) (9, (0,0,0)) (3, (0,1,0)) (2, (0,0,0)) (10, (0,1,0)) (21, (0,0,0)) (4, (0,1,0)) (1, (0,0,0)) (9, (0,0,0)) (22, (0,1,0)) (1, (0,0,0)) (4, (0,1,0)) (21, (0,0,0)) (10, (0,1,0)) (2, (0,0,0)) (3, (0,1,0)); 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(24, (-1,-1,0)) (7, (0,0,0)) (6, (1,0,0)) (23, (-1,-1,0)) (12, (0,0,0)) (5, (1,0,0)) (8, (0,0,0)) (12, (0,-1,0)) (23, (-1,0,0)) (6, (0,0,0)) (7, (1,0,0)) (24, (-1,0,0)) (11, (0,-1,0)) (8, (1,0,0)) (5, (0,0,0)) (16, (0,0,0)) (17, (0,0,-1)) (14, (0,-1,1)) (19, (0,0,0)) (13, (0,0,-1)) (20, (0,0,0)) (15, (0,0,0)) (18, (0,-1,1)) (20, (0,0,-1)) (13, (0,0,0)) (19, (0,-1,0)) (14, (0,0,1)) (17, (0,0,0)) (16, (0,0,-1)) (18, (0,0,1)) (15, (0,-1,0)) (1, (1,1,1)) (4, (0,0,0)) (9, (0,1,-1)) (22, (-1,0,0)) (2, (0,1,0)) (3, (1,0,1)) (21, (-1,1,0)) (10, (0,0,-1)) (3, (0,0,1)) (2, (1,1,0)) (22, (-1,0,-1)) (9, (0,1,0)) (4, (1,0,0)) (1, (0,1,1)) (10, (0,0,0)) (21, (-1,1,-1)); (23, (-1,-1,0)) (8, (0,0,0)) (5, (1,0,0)) (24, (-1,-1,0)) (11, (0,0,0)) (6, (1,0,0)) (7, (0,0,0)) (11, (0,-1,0)) (24, (-1,0,0)) (5, (0,0,0)) (8, (1,0,0)) (23, (-1,0,0)) (12, (0,-1,0)) (7, (1,0,0)) (6, (0,0,0)) (20, (0,0,0)) (13, (0,0,-1)) (18, (0,-1,1)) (15, (0,0,0)) (17, (0,0,-1)) (16, (0,0,0)) (19, (0,0,0)) (14, (0,-1,1)) (16, (0,0,-1)) (17, (0,0,0)) (15, (0,-1,0)) (18, (0,0,1)) (13, (0,0,0)) (20, (0,0,-1)) (14, (0,0,1)) (19, (0,-1,0)) (3, (1,1,1)) (2, (0,0,0)) (10, (0,1,-1)) (21, (-1,0,0)) (4, (0,1,0)) (1, (1,0,1)) (22, (-1,1,0)) (9, (0,0,-1)) (1, (0,0,1)) (4, (1,1,0)) (21, (-1,0,-1)) (10, (0,1,0)) (2, (1,0,0)) (3, (0,1,1)) (9, (0,0,0)) (22, (-1,1,-1)); 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(18, (-1,-1,0)) (2, (0,1,0)) (3, (1,0,0)) (19, (-1,-1,0)) (14, (0,0,0)) (4, (1,0,0)) (1, (0,1,0)) (14, (0,-1,0)) (19, (-1,0,0)) (3, (0,0,0)) (2, (1,1,0)) (18, (-1,0,0)) (15, (0,-1,0)) (1, (1,1,0)) (4, (0,0,0)) (21, (0,1,0)) (10, (0,0,-1)) (23, (0,-1,1)) (12, (0,0,0)) (22, (0,0,-1)) (9, (0,1,0)) (24, (0,0,0)) (11, (0,-1,1)) (9, (0,1,-1)) (22, (0,0,0)) (12, (0,-1,0)) (23, (0,0,1)) (10, (0,0,0)) (21, (0,1,-1)) (11, (0,0,1)) (24, (0,-1,0)) (5, (1,1,1)) (8, (0,-1,0)) (13, (0,1,-1)) (20, (-1,-1,0)) (6, (0,1,0)) (7, (1,-1,1)) (17, (-1,1,0)) (16, (0,-1,-1)) (7, (0,-1,1)) (6, (1,1,0)) (20, (-1,-1,-1)) (13, (0,1,0)) (8, (1,-1,0)) (5, (0,1,1)) (16, (0,-1,0)) (17, (-1,1,-1)); (17, (-1,-1,-1)) (21, (-1,1,0)) (10, (1,0,-1)) (17, (-1,-1,0)) (16, (0,0,-1)) (10, (1,0,0)) (21, (-1,1,-1)) (13, (0,-1,-1)) (20, (-1,0,0)) (22, (-1,0,-1)) (9, (1,1,0)) (20, (-1,0,-1)) (13, (0,-1,0)) (9, (1,1,-1)) (22, (-1,0,0)) (1, (1,1,1)) (4, (0,0,-1)) (5, (1,-1,1)) (8, (0,0,0)) (4, (1,0,-1)) (1, (0,1,1)) (8, (1,0,0)) (5, (0,-1,1)) (2, (0,1,-1)) (3, (1,0,1)) (6, (0,-1,0)) (7, (1,0,1)) (3, (0,0,1)) (2, (1,1,-1)) (7, (0,0,1)) (6, (1,-1,0)) (11, (1,1,1)) (24, (-1,-1,0)) (15, (0,1,-1)) (18, (-1,-1,1)) (24, (-1,1,0)) (11, (1,-1,1)) (18, (-1,1,1)) (15, (0,-1,-1)) (23, (-1,-1,1)) (12, (1,1,0)) (19, (-1,-1,-1)) (14, (0,1,1)) (12, (1,-1,0)) (23, (-1,1,1)) (14, (0,-1,1)) (19, (-1,1,-1)); (16, (-1,-1,-1)) (10, (0,1,0)) (21, (0,0,-1)) (16, (-1,-1,0)) (17, (0,0,-1)) (21, (0,0,0)) (10, (0,1,-1)) (20, (0,-1,-1)) (13, (-1,0,0)) (9, (0,0,-1)) (22, (0,1,0)) (13, (-1,0,-1)) (20, (0,-1,0)) (22, (0,1,-1)) (9, (0,0,0)) (4, (1,1,0)) (1, (0,0,0)) (8, (1,-1,1)) (5, (0,0,0)) (1, (1,0,0)) (4, (0,1,0)) (5, (1,0,0)) (8, (0,-1,1)) (3, (0,1,0)) (2, (1,0,0)) (7, (0,-1,0)) (6, (1,0,1)) (2, (0,0,0)) (3, (1,1,0)) (6, (0,0,1)) (7, (1,-1,0)) (24, (0,1,1)) (11, (0,-1,0)) (18, (0,1,0)) (15, (-1,-1,0)) (11, (0,1,0)) (24, (0,-1,1)) (15, (-1,1,0)) (18, (0,-1,0)) (12, (0,-1,1)) (23, (0,1,0)) (14, (-1,-1,0)) (19, (0,1,0)) (23, (0,-1,0)) (12, (0,1,1)) (19, (0,-1,0)) (14, (-1,1,0)); (15, (-1,-1,0)) (3, (0,1,0)) (2, (1,0,0)) (14, (-1,-1,0)) (19, (0,0,0)) (1, (1,0,0)) (4, (0,1,0)) (19, (0,-1,0)) (14, (-1,0,0)) (2, (0,0,0)) (3, (1,1,0)) (15, (-1,0,0)) (18, (0,-1,0)) (4, (1,1,0)) (1, (0,0,0)) (10, (0,1,0)) (21, (0,0,-1)) (12, (0,-1,1)) (23, (0,0,0)) (9, (0,0,-1)) (22, (0,1,0)) (11, (0,0,0)) (24, (0,-1,1)) (22, (0,1,-1)) (9, (0,0,0)) (23, (0,-1,0)) (12, (0,0,1)) (21, (0,0,0)) (10, (0,1,-1)) (24, (0,0,1)) (11, (0,-1,0)) (8, (1,0,1)) (5, (0,0,0)) (20, (0,0,-1)) (13, (-1,0,0)) (7, (0,0,0)) (6, (1,0,1)) (16, (-1,0,0)) (17, (0,0,-1)) (6, (0,0,1)) (7, (1,0,0)) (13, (-1,0,-1)) (20, (0,0,0)) (5, (1,0,0)) (8, (0,0,1)) (17, (0,0,0)) (16, (-1,0,-1)); (14, (-1,-1,0)) (4, (0,1,0)) (1, (1,0,0)) (15, (-1,-1,0)) (18, (0,0,0)) (2, (1,0,0)) (3, (0,1,0)) (18, (0,-1,0)) (15, (-1,0,0)) (1, (0,0,0)) (4, (1,1,0)) (14, (-1,0,0)) (19, (0,-1,0)) (3, (1,1,0)) (2, (0,0,0)) (22, (0,1,0)) (9, (0,0,-1)) (24, (0,-1,1)) (11, (0,0,0)) (21, (0,0,-1)) (10, (0,1,0)) (23, (0,0,0)) (12, (0,-1,1)) (10, (0,1,-1)) (21, (0,0,0)) (11, (0,-1,0)) (24, (0,0,1)) (9, (0,0,0)) (22, (0,1,-1)) (12, (0,0,1)) (23, (0,-1,0)) (6, (1,1,1)) (7, (0,-1,0)) (17, (0,1,-1)) (16, (-1,-1,0)) (5, (0,1,0)) (8, (1,-1,1)) (13, (-1,1,0)) (20, (0,-1,-1)) (8, (0,-1,1)) (5, (1,1,0)) (16, (-1,-1,-1)) (17, (0,1,0)) (7, (1,-1,0)) (6, (0,1,1)) (20, (0,-1,0)) (13, (-1,1,-1)); (13, (-1,-1,-1)) (22, (-1,1,0)) (9, (1,0,-1)) (13, (-1,-1,0)) (20, (0,0,-1)) (9, (1,0,0)) (22, (-1,1,-1)) (17, (0,-1,-1)) (16, (-1,0,0)) (21, (-1,0,-1)) (10, (1,1,0)) (16, (-1,0,-1)) (17, (0,-1,0)) (10, (1,1,-1)) (21, (-1,0,0)) (3, (1,1,1)) (2, (0,0,-1)) (6, (1,-1,1)) (7, (0,0,0)) (2, (1,0,-1)) (3, (0,1,1)) (7, (1,0,0)) (6, (0,-1,1)) (4, (0,1,-1)) (1, (1,0,1)) (5, (0,-1,0)) (8, (1,0,1)) (1, (0,0,1)) (4, (1,1,-1)) (8, (0,0,1)) (5, (1,-1,0)) (12, (1,1,1)) (23, (-1,-1,0)) (19, (0,1,-1)) (14, (-1,-1,1)) (23, (-1,1,0)) (12, (1,-1,1)) (14, (-1,1,1)) (19, (0,-1,-1)) (24, (-1,-1,1)) (11, (1,1,0)) (15, (-1,-1,-1)) (18, (0,1,1)) (11, (1,-1,0)) (24, (-1,1,1)) (18, (0,-1,1)) (15, (-1,1,-1)); (10, (-1,0,-1)) (17, (0,1,0)) (16, (0,-1,-1)) (10, (-1,0,0)) (21, (0,0,-1)) (16, (0,-1,0)) (17, (0,1,-1)) (22, (0,0,-1)) (9, (-1,0,0)) (20, (0,-1,-1)) (13, (0,1,0)) (9, (-1,0,-1)) (22, (0,0,0)) (13, (0,1,-1)) (20, (0,-1,0)) (5, (1,1,0)) (8, (0,-1,0)) (4, (1,0,1)) (1, (0,0,0)) (8, (1,-1,0)) (5, (0,1,0)) (1, (1,0,0)) (4, (0,0,1)) (6, (0,1,0)) (7, (1,-1,0)) (3, (0,0,0)) (2, (1,0,1)) (7, (0,-1,0)) (6, (1,1,0)) (2, (0,0,1)) (3, (1,0,0)) (15, (0,1,1)) (18, (0,-1,0)) (24, (0,1,0)) (11, (-1,-1,0)) (18, (0,1,0)) (15, (0,-1,1)) (11, (-1,1,0)) (24, (0,-1,0)) (19, (0,-1,1)) (14, (0,1,0)) (12, (-1,-1,0)) (23, (0,1,0)) (14, (0,-1,0)) (19, (0,1,1)) (23, (0,-1,0)) (12, (-1,1,0)); (9, (-1,0,-1)) (13, (0,1,0)) (20, (0,-1,-1)) (9, (-1,0,0)) (22, (0,0,-1)) (20, (0,-1,0)) (13, (0,1,-1)) (21, (0,0,-1)) (10, (-1,0,0)) (16, (0,-1,-1)) (17, (0,1,0)) (10, (-1,0,-1)) (21, (0,0,0)) (17, (0,1,-1)) (16, (0,-1,0)) (6, (1,1,0)) (7, (0,-1,0)) (2, (1,0,1)) (3, (0,0,0)) (7, (1,-1,0)) (6, (0,1,0)) (3, (1,0,0)) (2, (0,0,1)) (5, (0,1,0)) (8, (1,-1,0)) (1, (0,0,0)) (4, (1,0,1)) (8, (0,-1,0)) (5, (1,1,0)) (4, (0,0,1)) (1, (1,0,0)) (19, (0,1,1)) (14, (0,-1,0)) (23, (0,1,0)) (12, (-1,-1,0)) (14, (0,1,0)) (19, (0,-1,1)) (12, (-1,1,0)) (23, (0,-1,0)) (15, (0,-1,1)) (18, (0,1,0)) (11, (-1,-1,0)) (24, (0,1,0)) (18, (0,-1,0)) (15, (0,1,1)) (24, (0,-1,0)) (11, (-1,1,0)); (12, (-1,-1,0)) (5, (0,1,0)) (8, (1,-1,0)) (11, (-1,-1,0)) (24, (0,0,0)) (7, (1,-1,0)) (6, (0,1,0)) (24, (0,-1,0)) (11, (-1,0,0)) (8, (0,-1,0)) (5, (1,1,0)) (12, (-1,0,0)) (23, (0,-1,0)) (6, (1,1,0)) (7, (0,-1,0)) (13, (0,1,0)) (20, (0,-1,-1)) (15, (0,-1,1)) (18, (0,0,0)) (16, (0,-1,-1)) (17, (0,1,0)) (14, (0,0,0)) (19, (0,-1,1)) (17, (0,1,-1)) (16, (0,-1,0)) (18, (0,-1,0)) (15, (0,0,1)) (20, (0,-1,0)) (13, (0,1,-1)) (19, (0,0,1)) (14, (0,-1,0)) (2, (1,1,1)) (3, (0,0,0)) (21, (0,1,-1)) (10, (-1,0,0)) (1, (0,1,0)) (4, (1,0,1)) (9, (-1,1,0)) (22, (0,0,-1)) (4, (0,0,1)) (1, (1,1,0)) (10, (-1,0,-1)) (21, (0,1,0)) (3, (1,0,0)) (2, (0,1,1)) (22, (0,0,0)) (9, (-1,1,-1)); (11, (-1,-1,0)) (6, (0,1,0)) (7, (1,-1,0)) (12, (-1,-1,0)) (23, (0,0,0)) (8, (1,-1,0)) (5, (0,1,0)) (23, (0,-1,0)) (12, (-1,0,0)) (7, (0,-1,0)) (6, (1,1,0)) (11, (-1,0,0)) (24, (0,-1,0)) (5, (1,1,0)) (8, (0,-1,0)) (17, (0,1,0)) (16, (0,-1,-1)) (19, (0,-1,1)) (14, (0,0,0)) (20, (0,-1,-1)) (13, (0,1,0)) (18, (0,0,0)) (15, (0,-1,1)) (13, (0,1,-1)) (20, (0,-1,0)) (14, (0,-1,0)) (19, (0,0,1)) (16, (0,-1,0)) (17, (0,1,-1)) (15, (0,0,1)) (18, (0,-1,0)) (4, (1,1,1)) (1, (0,0,0)) (22, (0,1,-1)) (9, (-1,0,0)) (3, (0,1,0)) (2, (1,0,1)) (10, (-1,1,0)) (21, (0,0,-1)) (2, (0,0,1)) (3, (1,1,0)) (9, (-1,0,-1)) (22, (0,1,0)) (1, (1,0,0)) (4, (0,1,1)) (21, (0,0,0)) (10, (-1,1,-1))],
     SMatrix{3, 3, Int64, 9}[[-1 0 0; 0 -1 0; 0 0 -1], [0 -1 0; 1 0 0; 0 0 1], [0 1 0; -1 0 0; 0 0 -1], [-1 0 0; 0 -1 0; 0 0 1], [1 0 0; 0 1 0; 0 0 -1], [0 1 0; -1 0 0; 0 0 1], [0 -1 0; 1 0 0; 0 0 -1], [1 0 0; 0 -1 0; 0 0 -1], [-1 0 0; 0 1 0; 0 0 1], [0 -1 0; -1 0 0; 0 0 -1], [0 1 0; 1 0 0; 0 0 1], [-1 0 0; 0 1 0; 0 0 -1], [1 0 0; 0 -1 0; 0 0 1], [0 1 0; 1 0 0; 0 0 -1], [0 -1 0; -1 0 0; 0 0 1], [0 0 1; 1 0 0; 0 1 0], [0 0 -1; -1 0 0; 0 -1 0], [0 0 1; 0 -1 0; 1 0 0], [0 0 -1; 0 1 0; -1 0 0], [0 0 1; -1 0 0; 0 -1 0], [0 0 -1; 1 0 0; 0 1 0], [0 0 1; 0 1 0; -1 0 0], [0 0 -1; 0 -1 0; 1 0 0], [0 0 -1; 1 0 0; 0 -1 0], [0 0 1; -1 0 0; 0 1 0], [0 0 -1; 0 -1 0; -1 0 0], [0 0 1; 0 1 0; 1 0 0], [0 0 -1; -1 0 0; 0 1 0], [0 0 1; 1 0 0; 0 -1 0], [0 0 -1; 0 1 0; 1 0 0], [0 0 1; 0 -1 0; -1 0 0], [0 1 0; 0 0 1; 1 0 0], [0 -1 0; 0 0 -1; -1 0 0], [1 0 0; 0 0 1; 0 -1 0], [-1 0 0; 0 0 -1; 0 1 0], [0 -1 0; 0 0 1; -1 0 0], [0 1 0; 0 0 -1; 1 0 0], [-1 0 0; 0 0 1; 0 1 0], [1 0 0; 0 0 -1; 0 -1 0], [0 -1 0; 0 0 -1; 1 0 0], [0 1 0; 0 0 1; -1 0 0], [-1 0 0; 0 0 -1; 0 -1 0], [1 0 0; 0 0 1; 0 1 0], [0 1 0; 0 0 -1; -1 0 0], [0 -1 0; 0 0 1; 1 0 0], [1 0 0; 0 0 -1; 0 1 0], [-1 0 0; 0 0 1; 0 -1 0]],
     SVector{3, Rational{Int32}}[[0//1, 1//4, 3//4], [5//8, 5//8, 0//1], [3//8, 5//8, 3//4], [0//1, 1//4, 0//1], [0//1, 0//1, 3//4], [3//8, 5//8, 0//1], [5//8, 5//8, 3//4], [0//1, 1//4, 3//4], [0//1, 0//1, 0//1], [5//8, 5//8, 3//4], [3//8, 5//8, 0//1], [0//1, 0//1, 3//4], [0//1, 1//4, 0//1], [3//8, 5//8, 3//4], [5//8, 5//8, 0//1], [1//8, 5//8, 1//4], [7//8, 5//8, 1//2], [1//8, 1//4, 7//8], [7//8, 0//1, 7//8], [1//8, 5//8, 1//2], [7//8, 5//8, 1//4], [1//8, 0//1, 7//8], [7//8, 1//4, 7//8], [7//8, 5//8, 1//2], [1//8, 5//8, 1//4], [7//8, 1//4, 7//8], [1//8, 0//1, 7//8], [7//8, 5//8, 1//4], [1//8, 5//8, 1//2], [7//8, 0//1, 7//8], [1//8, 1//4, 7//8], [3//8, 3//4, 7//8], [5//8, 1//2, 7//8], [0//1, 3//4, 1//2], [0//1, 1//2, 1//4], [5//8, 3//4, 7//8], [3//8, 1//2, 7//8], [0//1, 3//4, 1//4], [0//1, 1//2, 1//2], [5//8, 1//2, 7//8], [3//8, 3//4, 7//8], [0//1, 1//2, 1//2], [0//1, 3//4, 1//4], [3//8, 1//2, 7//8], [5//8, 3//4, 7//8], [0//1, 1//2, 1//4], [0//1, 3//4, 1//2]],
@@ -812,13 +839,17 @@ end
         @test canonicalize_ri(r1) == canonicalize_ri(r2)
     end
     withid = PeriodicGraphs.IncludingIdentity(symmetries_lta)
-    @test eltype(withid) == PeriodicSymmetry3D{Rational{Int32}}
+    @test eltype(withid) == PeriodicSymmetries.PeriodicSymmetry3D{Rational{Int32}}
     @test length(withid) == 48
-    id = first(withid)
-    @test isequal(id, one(symmetries_lta))
+    id, id2 = withid
+    @test isequal(id, withid[1]) && isequal(id, one(symmetries_lta))
+    @test isequal(id2, withid[2]) && isequal(id2, first(symmetries_lta))
     trivsymmgroup = PeriodicGraphs.IncludingIdentity(NoSymmetryGroup(lta))
     @test unique(trivsymmgroup) == 1:nv(lta)
+    @test trivsymmgroup(13) == 13
     @test length(trivsymmgroup) == 1
+    @test PeriodicGraphs.IncludingIdentity(trivsymmgroup) === trivsymmgroup
+    @test_throws ErrorException one(trivsymmgroup)
     triv = only(trivsymmgroup)
     @test triv[5] == 5
     @test triv[PeriodicVertex3D(14)] == PeriodicVertex3D(14)