Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 497-515.

Classification of Dodecahedral Space Forms

István Prok*

TU Budapest, Department of Geometry H–1521 Budapest XI, Egry J. u. 1, H.II.22, Hungary e-mail: geometry@ccmail.bme.hu

1. Introduction

Considering the regular dodecahedron tilings in the spaces of constant curvature we look for the fixed point free discrete transformation groups that act simply transitively on the tiles. The isometries of a group, mapping a distinguished tile onto its neighbours, identify the faces and determine a dodecahedral space form, moreover, these transformations generate this fundamental group. In order to find all these groups, in this paper we apply a polyhedron algorithm, which was developed theoretically by E. Molnár [5, 7] and implemented to computer by the author [6, 7, 9, 10]. Because of the large number of cases we modify the general algorithm and apply it for finding all the non-equivariant face identifications of a regular polyhedron, and for choosing those ones that generate fixed point free groups, if the number of polyhedra around the edges is given. In this way we obtain the fundamental groups of dodecahedral space forms given by generators and relations. We classify these non-equivariant forms according to the obvious orientability and to the first homology groups. When the vertices are ideal points we shall take into consideration the so-called cusp structure refining our classification. The fundamental groups in different classes will be non-isomorphic, but we shall leave the general problem of isomorphism within the classes open for later papers. The isomorphism problem has already been solved for hyperbolic octahedron spaces in [9], recently. So this paper gives a complete list of regular dodecahedral space forms, but in some small classes the difference of the manifolds (up to isometry) cannot be guaranteed yet. From the results we emphasize that we improved the list of compact dodecahedron spaces by L. A. Best in [1], and we extended the investigations for non-orientable space forms, too.

* Supported by the Hungarian Nat. Found. Sci. Research (OTKA) grant no. T. 7351 (1993).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1998 Heldermann Verlag

2. Regular dodecahedron tilings

In the spherical space S^{3} and in the hyperbolic space H^{3} we consider the regular face-to-face dodecahedron tilings D^{(n)} in which there are n dodecahedra around each edge. The dihedral angles of a dodecahedron D^{(n)} in D^{(n)} are \frac{2π}{n}. Let k^{(n)} denote the number of dodecahedra around the vertices. There are four tilings of this kind. These are as follows.

The tiling D^{(3)} is realized in S^{3} (as the projection of the regular 120 cell onto the surface of the 4-dimensional sphere from its centre). The symmetry group Sym D^{(3)} is the Coxeter group of diagram

◯{5 \atop −}◯{3 \atop −}◯{3 \atop −}◯.

The number of dodecahedra around the vertices is k^{(3)} = 4; the neigbourhood of any vertex is tetrahedronlike.

The tiling D^{(4)} appears in H^{3}. The diagram of its symmetry group Sym D^{(4)} is

◯{5 \atop −}◯{3 \atop −}◯{4 \atop −}◯,

and the neighbourhood of the vertices is octahedronlike, k^{(4)} = 8.

The tiling D^{(5)} also appears in H^{3}. The diagram of its symmetry group Sym D^{(5)} is

◯{5 \atop −}◯{3 \atop −}◯{5 \atop −}◯,

and the neighbourhood of the vertices is icosahedronlike, k^{(5)} = 20.

Finally, the tiling D^{(6)} of H^{3} contains noncompact dodecahedra whose vertices are ideal points lying on the absolute, but the tiles have finite volume. (See [8] for hexagonal bipyramids and [11] for cubes.) The tiling has the symmetry group Sym D^{(6)} of Coxeter diagram

◯{5 \atop −}◯{3 \atop −}◯{6 \atop −}◯.

Since the vertices are ideal points (so called parabolic vertices or ends), all its edges concurrent to such an ideal vertex belong to a parabolic bundle. A horosphere centred at this vertex is orthogonal to this bundle and intersects each tile around the vertex in a horospherical triangle. Since a horosphere is isometric to the Euclidean plane and the angles are congruent to the dihedral angles of D^{(6)} (which is \frac{2π}{6} ), the triangles are regular. Thus, the neighbourhood of any vertex corresponds to the regular triangle tiling {3, 6} of the Euclidean plane, k^{(6)} = ∞.

3. How to generate fixed point free groups acting on a tiling

Considering a tiling D^{(n)} we look for the fixed point free discrete isometry groups that act transitively on the tiles. Let G^{(n)} be a group of this kind, and let us fix a dodecahedron D^{(n)} in D^{(n)}. Since fixed points cannot be inside D^{(n)}, the action of G^{(n)} on D^{(n)} must be simply transitive, so D^{(n)} is a fundamental domain of G^{(n)}. Similarly, because fixed points cannot be inside the faces of D^{(n)} either, each face must be identified with another face by the transformation and its inverse that map D^{(n)} onto its neighbours along these faces. In this way G^{(n)} determines a pairing of the 12 faces of D^{(n)} so that each of the 6 pairs contains different faces. Moreover, an isometry and its inverse that identify the paired faces are attached to each pair. Conversely, the face pairing with the identifying transformations generate the group G^{(n)} geometrically (in general, this generator system is not independent as the following relations may yield).

4. Defining relations by the Poincaré algorithm

We consider a dodecahedron tiling D := D^{(n)} and fix a tile D := D^{(n)} in it. We give a face pairing of D and attach an isometry g and its inverse g^{−1} to each pair (g, g^{−1}) (where g ≠ g^{−1}), so that g maps g^{−1} onto g and D onto its neighbour D^{g} along g. Then g^{−1} maps g onto g^{−1} and D onto D^{g^{−1}} along g^{−1}. Let F denote the set of these identifying transformations (together with their inverses).

We look for the defining relations of the group G generated by F and apply the so-called Poincaré algorithm. (We do not know yet whether G is a fixed point free space group.) We choose a directed edge e_{1} of D and try to form the transformations (as products of the generators) that map D onto its neigbours along e_{1} one after the other, surrounding the edge.

There are two faces along e_{1} in D, let g_{1} be one of them. The isometry g_{1} ∈ F maps D = D^{1} onto D^{g_{1}} along g_{1} and maps the edge e_{1}^{g_{1}^{−1}} =: e_{2} of the face g_{1}^{−1} onto e_{1}.

Now we consider the face g_{2}^{g_{1}} of D^{g_{1}} along the directed edge e_{2}^{g_{1}} = e_{1}, where g_{2}^{g_{1}} ≠ (g_{1}^{−1})^{g_{1}} ≡ g_{1}. The isometry g_{1}^{−1}g_{2}g_{1} (g_{2} ∈ F) maps D^{g_{1}} onto (D^{g_{1}})^{g_{1}^{−1}g_{2}g_{1}} = D^{g_{2}g_{1}} along g_{2}^{g_{1}}. So the transformation g_{2}g_{1} maps D^{1} onto its neighbour D^{g_{2}g_{1}} along the edge

e_{1} and it maps the edge e_{1}^{g_{1}^{−1}g_{2}^{−1}} =: e_{3} of D^{1} onto e_{1}.

We continue this procedure considering the face g_{3}^{g_{2}g_{1}} along the directed edge e_{3}^{g_{2}g_{1}} = e_{1} in D^{g_{2}g_{1}}, where g_{3}^{g_{2}g_{1}} ≠ (g_{2}^{−1})^{g_{2}g_{1}} ≡ g_{2}^{g_{1}}. We apply the isometry g_{1}^{−1}g_{2}^{−1}g_{3}g_{2}g_{1} (g_{3} ∈ F), which maps D^{g_{2}g_{1}} onto its neighbour D^{g_{3}g_{2}g_{1}} along g_{3}^{g_{2}g_{1}}. Mapping D^{1} onto its

neighbour D^{g_{3}g_{2}g_{1}} along e_{1}, the transformation g_{3}g_{2}g_{1} maps the edge e_{1}^{g_{1}^{−1}g_{2}^{−1}g_{3}^{−1}} =: e_{4} of D^{1} onto e_{1}.

Finally, the isometry

g_{1}^{−1}g_{2}^{−1} ⋯ g_{l−1}^{−1}g_{l}g_{l−1} ⋯ g_{2}g_{1} (g_{1}, …, g_{l} ∈ F)

maps D^{g_{l−1}···g_{2}g_{1}} onto D^{g_{l}g_{l−1}···g_{2}g_{1}} along g_{l}^{g_{l−1}···g_{2}g_{1}}. The transformation g_{l} ⋯ g_{2}g_{1}

maps D^{1} onto its neighbour D^{g_{l}···g_{2}g_{1}} along e_{1}, mapping e_{1}^{g_{1}^{−1}g_{2}^{−1}···g_{l}^{−1}} =: e_{l+1} of D^{1} onto e_{1}. Thus, the directed edge e_{l+1} is identical with e_{1}, moreover the face g_{l+1}^{g_{l}···g_{2}g_{1}}, which is different from (g_{l}^{−1})^{g_{l}···g_{2}g_{1}} ≡ g_{l}^{g_{l−1}···g_{2}g_{1}}, along e_{l+1}^{g_{l}···g_{2}g_{1}} = e_{1}^{g_{l}···g_{2}g_{1}} is identical with g_{1}^{g_{l}···g_{2}g_{1}} in D^{g_{l}···g_{2}g_{1}}.

During this algorithm we put together a collection of l dodecahedra D = D^{1}, D^{g_{1}}, …, D^{g_{l−1}···g_{2}g_{1}}, and we get a cycle {e_{1}, e_{2}, …, e_{l}} of directed edges, moreover, a product (a so-called cycle transformation) r := g_{l} ⋯ g_{2}g_{1} of the generators, which is metrically a rotation about e_{1} as g_{1} ↦ g_{1}^{g_{l}···g_{2}g_{1}} indicates (see [5] for more details).

Let us take into consideration that the generated group G should be a fixed point free space group. Then a proper rotation axis, whose all points are fixed, cannot appear. Moreover, oppositely directed edges cannot occur in the cycle, because if an edge and its image occur in opposite directions (then each edge does likewise), its midpoint would be fix (as inversion centre). So in order that G be a fixed point free space group, it is necessary that l should be equal to n (the number of dodecahedra around the edges in the tiling), and the cycle {e_{1}, e_{2}, …, e_{n}} should contain different edges. Thus, we get the defining relation r = g_{l} ⋯ g_{2}g_{1} = 1.

In order to get the other edge cycles defining new relations, we choose a next edge e ∉ {e_{1}, e_{2}, …, e_{l}}, and starting with e we apply the previous method until each edge has occurred in a cycle. Since each cycle has to contain exactly l = n different edges, n has to be a divisor of 30 (the number of edges of a dodecahedron), and the number of cycles is 30/n. This criterion excludes the existence of fixed point free G acting on D^{(4)} (i.e. n cannot be equal to 4).

5. Fixed points at proper vertices

Similarly to the previous method we could describe also the tilings around the vertices determining the vertex stabilizer subgroups of G (see Section 8, moreover [5]). Since we look for only fixed point free groups, these stabilizers must be trivial. This means that the transformations, mapping D onto its neighbour around a vertex in D = D^{(n)}, cannot map this vertex onto itself, namely, exactly k^{(n)} different vertex domains form the local tiling around each vertex. If we add this necessary condition to the previous ones from Section 4, we can exclude the existence of fixed points.

Let the faces be given by their vertex cycles:

g = V_{1} V_{2} V_{3} V_{4} V_{5}
g^{−1} = W_{1}W_{2}W_{3}W_{4}W_{5}
⋮

Now we consider a face pairing (g, g^{−1}), … with the identifying isometries (g, g^{−1}), …, and describe how the isometries map the vertices:

g : W_{1}W_{2}W_{3}W_{4}W_{5} ↦ V_{i_{1}} V_{i_{2}} V_{i_{3}} V_{i_{4}} V_{i_{5}}
g^{−1} : V_{1} V_{2} V_{3} V_{4} V_{5} ↦ W_{j_{1}}W_{j_{2}}W_{j_{3}}W_{j_{4}}W_{j_{5}}
⋮

Starting with the set V = {W_{1}} we look for W_{1} on the left side of the previous mappings, and add the corresponding vertex from the right side to V. So we get V = {W_{1}, V_{i_{1}}, …} at the end of this step. Then, starting with the second element V_{i_{1}} of V, we add its images to V, and so on while we can find a new vertex as the image of any element of V. In this way, finally, V contains the class of vertices whose image domains surround W_{1} combinatorially. Taking the metric conditions into consideration, the cardinality of V must be equal to k^{(n)}. Then we consider a vertex V ∉ V and look for its equivalent vertices, and so on. We assign to each vertex its class. Since each class has to contain k^{(n)} vertices, k^{(n)} must be a divisor of 20 (the number of vertices of the dodecahedron). This criterion excludes the tiling D^{(4)} again.

Remarks. In the case of D^{(6)} we do not have to examine the problem of fixed points at the vertices because they are ideal points lying on the absolute. Moreover, it would be enough to guarantee the correct number in each edge class, if we did not look for the defining relations, too.

6. Finding non-equivariant generator systems

First we look for all the non-equivariant pairings of the 12 faces. A face pairing π is a permutation of the faces formed by 6 disjoint transpositions. We consider the elements s_{1}, s_{2}, …, s_{120} of Sym D (the symmetry group of the dodecahedron) and create the face permutations σ_{1}, σ_{2}, …, σ_{120} determined by them. (These permutations can be computed independently of the space in which the regular dodecahedron D is given.) We say that two pairings π_{1} and π_{2} are equivariant if a symmetry s_{i} transforms π_{1} onto π_{2}, namely π_{2} = σ_{i}^{−1}π_{1}σ_{i}.

Enumerating all the face pairings, we collect the non-equivariant ones into a set Π, which is the empty set in the beginning. We consider the next pairing π in turn and check whether the set of conjugacies

{σ_{1}^{−1}πσ_{1}, σ_{2}^{−1}πσ_{2}, …, σ_{120}^{−1}πσ_{120}}

and Π are disjoint. If so, then we put π into Π (Π := Π∪{π}). Having run the programme of this algorithm on our computer, we obtained 147 non-equivariant face pairings among the 10 395 ones.

Now we choose a tiling D^{(n)} from D^{(3)}, D^{(5)} and D^{(6)} (fixing also the space). We consider the face pairings π_{1}, π_{2}, …, π_{147} ∈ Π one after the other and equip each pairing π_{j} (j ∈ {1, 2, …, 147}) with identifying transformations, attaching an isometry and its inverse to each pair (as we have seen above), in all possible manner. We enumerate these systems of transformations. Since two pentagons can be identified in 10 different ways, the 6 face pairs involve a million cases.

We collect the non-equivariant systems that generate fixed point free discrete transformation groups acting simply transitively on D^{(n)} into a set F_{j}^{(n)}, which is the empty set in the beginning. We consider the next system F = {g_{1}, g_{2}, …, g_{12}} (containing the identifying isometries) in turn. First we form the edge cycles and the equivalence classes of vertices, to decide whether the generated group is suitable, as we have seen in Section 4 and 5. If so, then we create the set

{s_{1}^{−1}Fs_{1}, s_{2}^{−1}Fs_{2}, …, s_{120}^{−1}Fs_{120}}

of conjugacies by the symmetries, and check whether this set and F_{j}^{(n)} are disjoint, in other words we examine if F is equivariant with an earlier generator system in F_{j}^{(n)}. If not, then we put F into F_{j}^{(n)} (F_{j}^{(n)} := F_{j}^{(n)} ∪ {F}).

Enumerating each pairing π_{j} of Π, we get the sets F_{1}^{(n)}, F_{2}^{(n)}, …, F_{147}^{(n)}, and their union

F^{(n)} := ⋃_{j=1}^{147} F_{j}^{(n)}

contains the generator systems of the desired groups. Having examined the 147 million cases by computer, we got these systems. As a result we obtained the numbers: |F^{(3)}| = 2, |F^{(5)}| = 12 and |F^{(6)}| = 78.

Remark. We mention that our algorithm works combinatorially [10]. We substitute the corresponding automorphisms of the incidence structure for the symmetries and the corresponding incidence preserving isomorphisms between faces for the identifying isometries. Since we have assumed that the dodecahedron is regular, the correspondence between the combinatorial mappings and the isometries is one-to-one.

7. Classification of the groups

We enumerate the generator systems contained by F^{(n)}. Applying the Poincaré algorithm we obtain the defining relations in each case. Let us consider a system from F^{(n)}, its identifying isometries generate a fixed point free space group G^{(n)}. Thus, the identification of faces determines a space form S^{3}/G^{(n)} (n = 3) or H^{3}/G^{(n)} (n = 5, 6). We can compute its first homology group H_{1}^{(n)}, which is the Abelianization (the commutator factor group) of the fundamental group G^{(n)}:

H_{1}^{(n)} = G^{(n)}/[G^{(n)}, G^{(n)}].

If the first homology groups of two manifolds are different, then the fundamental groups cannot be isomorphic. First, we include two identifications (or generator systems) from F^{(n)} into the same class in the course of our classification if their first homology groups are isomorphic. In this way we obtain the following results.

Theorem 1. There are 2 non-equivariant face identifications of D^{(3)} in F^{(n)} yielding fixed point free fundamental dodecahedron tilings in S^{3} (Section 9). Both of them determine orientable spherical manifolds. Since their first homology groups are different, the fundamental groups are non-isomorphic, the manifolds are not isometric.

Remarks. S^{3}/G_{1}^{(3)} is the spherical dodecahedral space of H. Poincaré with the first homology group 0. The second dodecahedral space S^{3}/G_{2}^{(3)} was mentioned by P. J. Lorimer in [4] and B. Everitt in [3]. Our result also shows that non-orientable manifolds (and other orientable ones) do not exist.

Theorem 2. There are 12 non-equivariant face identifications of D^{(5)} yielding fixed point free fundamental dodecahedron tilings in H^{3} (Section 10). Each of them determines compact orientable hyperbolic manifolds. They can be classified into 7 classes according to their first homology groups, so they determine at least 7 different manifolds.

Remarks. The first compact hyperbolic dodecahedral space G_{6}^{(5)} was discovered by C. Weber and H. Seifert [12]. Then L. A. Best listed 7 more manifolds in [1]. Unfortunately, his cases 4, 5 and 6 are equivariant, in Section 10 the corresponding group is G_{2}^{(5)}. His other cases 1, 2, 3 and 7 appear in that section table at G_{5}^{(5)}, G_{4}^{(5)}, G_{8}^{(5)}, G_{12}^{(5)}, respectively. Among our other identifications there are at least 3 new manifolds according to their homology groups. These are G_{7}^{(5)}, G_{9}^{(5)}, moreover, G_{10}^{(5)} and G_{11}^{(5)} (G_{10}^{(5)} and G_{11}^{(5)} may be isomorphic). Similarly to the spherical cases our result also shows that non-orientable manifolds cannot appear.

In the course of the classification of non-compact hyperbolic space forms we can apply a new approach to the classes determined by the first homology group. In the tiling D^{(6)} the vertices are ideal points and their domains, intersecting by a horosphere, form an infinite regular triangle tiling around the ends. A fixed point free space group G^{(6)} glues together these ends defining a so-called cusp. Gluing together the corresponding triangles, we can form a fundamental domain (with side pairing generators) of the vertex stabilizer subgroup, which is a planar Euclidean crystallographic group. Since G^{(6)} is fixed point free, the stabilizer must also be a fixed point free discontinuous group of the horosphere, i.e. of the Euclidean plane, and each stabilizer is either p1 or pg.

We compute the cusps for each identification of F^{(6)} and determine their stabilizer group. Let us consider two identifications F_{i} and F_{j} from F^{(6)} and let x_{i} and x_{j} denote the number of cusps with stabilizer group p1, moreover y_{i} and y_{j} denote the number of cusps with pg determined by F_{i} and F_{j}, respectively. The numbers x_{i}, y_{i} and x_{j}, y_{j} are characteristic to the manifolds described by F_{i} and F_{j}, respectively, namely if the ordered pairs (x_{i}, y_{i}) and (x_{j}, y_{j}) are not equal then F_{i} and F_{j} generate non-isomorphic fundamental groups. Finally, we put the identifications F_{i} and F_{j} into the same class iff both of them are orientable or both of them are non-orientable, furthermore their first homology groups H_{1,i} and H_{1,j} are isomorphic and (x_{i}, y_{i}) = (x_{j}, y_{j}).

Theorem 3. There are 78 non-equivariant face identifications of D^{(6)} yielding fixed point free fundamental dodecahedron tilings in H^{3}. They determine 10 orientable (Section 11) and 68 non-orientable (Section 12) non-compact hyperbolic manifolds with finite volume. The 10 orientable manifolds (each of them has an enantiomorphic pair) can be classified into 6 classes according to their first homology groups and we get 7 classes taking the cusp structures into consideration as an additional aspect. The 68 non-orientable manifolds can be classified into 22 homology classes and we obtain 31 classes according to their cusp structures. In this way the identifications determine at least 7 different (non-isometric) orientable and 31 different non-orientable manifolds.

Remarks. In an algebraic way, looking for the decompositions of the symmetry group of the tiling Sym D^{(6)} as a product of the symmetry group of the dodecahedron Sym D^{(6)} and a space group G^{(6)}, B. Everitt found the 10 orientable cases about a year earlier [3]. (He distinguished the enantiomorphic pairs, thus he obtained 20 cases.) He used the group theory computer program LOWINDEX. In this manner we get the same result by applying different methods. Our results related with the orientable tetrahedral cubic and octahedral manifolds are also identical [6, 11, 9, resp.]. Intentionally, he did not examine the non-orientable cases.

8. An example for non-compact dodecahedral manifold

The table of Section 12 contains the group G_{74}^{(6)} in Class 36 in the following way:

G_{74}^{(6)} = \{a\left[{ABCDE \atop LRKAB}\right], u\left[{BLQMC \atop EDNT O}\right], b\left[{CMP ND \atop P F GQM}\right], v\left[{EOSKA \atop IJF GH}\right], c\left[{GQLRH \atop KRHIS}\right], d\left[{ISOT J \atop NT JF P}\right]

— 1 = aua^{2}cv̄ = uvdbcā = avcvb̄^{2} = udv̄d^{2}b̄ = c^{2}dūb̄ū}.

In the first row we can see the face identifying generators (Fig. 1). In our tables (Sections 9, 10, 11 and 12) a, b, c, d, e and f will denote the orientation preserving transformations (screw motions), moreover u, v, w, x, y and z will denote the orientation reversing transformations (glide reflections). Now, the face identifying generators are four screw motions a, b, c and d, moreover two glide reflections u and v:

\begin{aligned}a & : & a^{−1} & = & ABCDE & ⟼ & a & = & LRKAB, \\ b & : & b^{−1} & = & CMP ND & ⟼ & b & = & P F GQM, \\ c & : & c^{−1} & = & GQLRH & ⟼ & c & = & KRHIS, \\ d & : & d^{−1} & = & ISOT J & ⟼ & d & = & NT JF P, \\ u & : & u^{−1} & = & BLQMC & ⟼ & u & = & EDNT O, \\ v & : & v^{−1} & = & EOSKA & ⟼ & v & = & IJF GH. \end{aligned}

These transformations map the dodecahedron D^{(6)} onto its neighbours along the faces a, b, c, d, u and v, respectively. In Figure 1 the face identifications are denoted by corresponding “small flags” |^{▷} inside the faces. The letters A, B, C, …, T for vertices are valid for each identifications in the tables of Section 9, 10, 11 and 12.

In the second row we can see the defining relations, which are given by the Poincaré algorithm in equivalent inverse form to that of Section 4 (see also [5]). For example starting with the directed edge \overrightarrow{AE} and the face a^{−1} = \overrightarrow{AE}DCB along \overrightarrow{AE} we get the following cycle:

\begin{aligned}a & : & \overrightarrow{AE}DCB & ⟼ & \overrightarrow{LB}AKR & []↑[] & \overrightarrow{LB}CMQ \\ u & : & \overrightarrow{LB}CMQ & ⟼ & \overrightarrow{DE}OT N & []↑[] & \overrightarrow{DE}ABC \\ a & : & \overrightarrow{DE}ABC & ⟼ & \overrightarrow{AB}LRK & []↑[] & \overrightarrow{AB}CDE \\ a & : & \overrightarrow{AB}CDE & ⟼ & \overrightarrow{LR}KAB & []↑[] & \overrightarrow{LR}HGQ \\ c & : & \overrightarrow{LR}HGQ & ⟼ & \overrightarrow{HI}SKR & []↑[] & \overrightarrow{HI}JF G \\ v^{−1} & : & \overrightarrow{HI}JF G & ⟼ & \overrightarrow{AE}OSK & []↑[] & \overrightarrow{AE}DCB \end{aligned}

and the edge cycle \{\overrightarrow{AE}, \overrightarrow{LB}, \overrightarrow{DE}, \overrightarrow{AB}, \overrightarrow{LR}, \overrightarrow{HI}\}, moreover the relation aua^{2}cv^{−1} = 1. In our tables we use a bar for the inverse transformations, e.g. the notation ḡ := g^{−1} for the sake of brevity. In similar way we obtain the other edge cycles and relations. In Figure 1 the edges in different cycles are denoted by different types of arrows and we can see the corresponding relations. Now we can compute the first homology group:

H_{1,74}^{(6)} = Z_{6}×Z_{16}×Z, since v resp. b are expressible from the first and second relations, and the others provide the existence of the cyclic group components Z_{6} (of order 6) and Z_{16} by commutativity.

Figure 1

Then we look for the fundamental domains of the vertex (end) stabilizer subgroups (the cusps). For example we start with the end A. Intersecting the dodecahedron D̃ := D^{(6)} with a horosphere centred at the ideal point A we get a regular horospherical triangle. Its sides are the intersecting lines with the faces v^{−1}, a, and a^{−1}, moreover its vertices are the intersecting points with the edges denoted by “one”, “three” and “one” arrows (Figure 2). The “upwards” (“one”) arrows show that the directed edges point out from A and the “downwards” (“three”) arrow denotes that the corresponding edge points into A.

Now we choose a side of the triangle, for example a, and apply the corresponding transformation a. This transformation maps D̃^{1} = D̃ onto its neighbour D̃^{a} along the face a = ABLRK mapping a^{−1} = DEABC onto a. Then the horospherical triangle around the end D^{a} ≡ A in D̃^{a} is glued to the previous one, uniting the sides a of D̃^{1} and (a^{−1})^{a} of D̃^{a}. The endpoints of this side is derived from the edges of “one” and “three” arrows. Thus we obtain a horospherical rhomb from the previous triangles. Its sides are a^{−1}, v^{−1}, (b^{−1})^{a} and u^{a}. (In Figure 2 we use the brief notations D, b^{−1} and u for D^{a}, (b^{−1})^{a} and u^{a}, respectively, moreover we discard the notations of the transformed images at the other ends and their sides, too.)

We choose a side of the rhomb, for example (b^{−1})^{a} of D̃^{a}, and apply the corresponding transformation (determined by the face identification of D̃^{1}) a^{−1}b^{−1}a, which maps D̃^{a} onto its neighbour D̃^{b^{−1}a} along the face (b^{−1})^{a} = (DCMP N)^{a} of D̃^{a} mapping b^{a} = (MP F GQ)^{a} onto (b^{−1})^{a}. Then the horospherical triangle around M^{b^{−1}a} ≡ D^{a} ≡ A in D̃^{b^{−1}a} is glued to the rhomb forming a trapezium.

Figure 2

We can follow the previous method, by choosing a free side of the actual plane figure and gluing the corresponding horospherical triangle along it if its center end did not occur earlier. If we cannot find such a triangle, then the fundamental domain is ready with the identification of its sides. During this method we usually have some freedom in choosing the next side to glue a triangle along it. In our example we have obtained a “big” horospherical rhomb gluing together the triangles around the ends

A ∈ D̃^{1},_{L^{cv^{−1}} ∈ D̃^{cv^{−1}},} D^{a} ∈ D̃^{a},_{H^{v^{−1}} ∈ D̃^{v^{−1}},} M^{b^{−1}a} ∈ D̃^{b^{−1}a},

S^{c^{−1}v^{−1}} ∈ D̃^{c^{−1}v^{−1}},

T^{u^{−1}b^{−1}a} ∈ D̃^{u^{−1}b^{−1}a},
F^{v^{−1}c^{−1}v^{−1}} ∈ D̃^{v^{−1}c^{−1}v^{−1}}.

These ends, which are denoted by “grey” circles in our figures, form an equivalence class under the group G_{74}^{(6)}. (In this way, surrounding the edges of “three” arrows we also get the defining relation avcvb^{−2} = 1 again.)

Considering the triangle sides and the symbols of edge classes along the boundary of the domain we can recognize the side identifications and the generators of the stabilizer subgroup. The transformation g_{1} maps the side a^{−1} of D̃^{1} (at the end A) onto a^{cv^{−1}} of D̃^{cv^{−1}} (at the image end L) and maps the domain onto its neighbour along a^{cv^{−1}}. In this way we get the glide reflection g_{1} = acv^{−1}. Similarly, the transformation g_{2} maps the side (d^{−1})^{u^{−1}b^{−1}a} of D̃^{u^{−1}b^{−1}a} (at the image end T between the “black” arrows) onto d^{v^{−1}c^{−1}v^{−1}} of D̃^{v^{−1}c^{−1}v^{−1}} (at the image end F between the “black” arrows again) and maps the domain onto its neighbour along d^{v^{−1}c^{−1}v^{−1}}. So we obtain the glide reflection g_{2} = a^{−1}budv^{−1}c^{−1}v^{−1}, (i.e. we map back D̃^{u^{−1}b^{−1}a} onto D̃^{1} and map D̃^{1} onto the neighbouring dodecahedron of D̃^{v^{−1}c^{−1}v^{−1}} along d^{v^{−1}c^{−1}v^{−1}}). Finally, the identifications of u^{a} and (u^{−1})^{cv^{−1}} (at the image ends D and L, resp.), moreover d^{u^{−1}b^{−1}a} and (d^{−1})^{c^{−1}v^{−1}} (at the image ends T and S, resp.) involve the translations t_{1} = a^{−1}u^{−1}cv^{−1} and t_{2} = a^{−1}bud^{−1}c^{−1}v^{−1}, respectively. We can observe that t_{1} = t_{2} = g_{1}^{2} = g_{2}^{2}, and the sides u^{a} and (u^{−1})^{cv^{−1}} can be extended with d^{u^{−1}b^{−1}a} and (d^{−1})^{c^{−1}v^{−1}}, respectively, determining new longer sides. Topologically, this fundamental domain is a hexagon. The glide reflections g_{1}, g_{2} and the translation t := t_{1} = t_{2} generate the planar crystallographic group pg.

We know that the stabilizer groups should be either p1 or pg, and among them only pg contains orientation reversing transformations (horospherical glide reflections). Therefore, in general, we only have to examine whether orientation reversing side identification occurs. If so, then the stabilizer is pg else p1.

Figure 3

Starting with a vertex (end) that does not belong to any previous class, we can find the other classes. In our case there are two more classes (Figure 3). The vertices (ends) belonging to them are denoted by “black” and “white” circles, respectively. The side identifications of the fundamental domains determine the plane group pg again. So we get the cusp structure: “pg: 3 cusps”.

9. Table of the spherical dodecahedral space forms

Class 1 ∘ H_{1,1}^{(3)} = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{1}^{(3)} = \{a\left[{ABCDE \atop HGF JI}\right], b\left[{AKRLB \atop T JF P N}\right], c\left[{BLQMC \atop SIJT O}\right], d\left[{CMP ND \atop RHISK}\right], e\left[{DNT OE \atop QGHRL}\right], f\left[{EOSKA \atop P F GQM}\right] — 1 = beā = cfā = dbā = ecā = fdā = fcb̄ = bdc̄ = ced̄ = dfē = ebf̄}

Class 2 ∘ H_{1,2}^{(3)} = Z_{3}×Z_{5} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{2}^{(3)} = \{a\left[{ABCDE \atop BCMQL}\right], b\left[{AKRLB \atop CDNP M}\right], c\left[{DNT OE \atop HGF JI}\right], d\left[{EOSKA \atop P F GQM}\right], e\left[{GQLRH \atop RHISK}\right], f\left[{ISOT J \atop P NT JF}\right] — 1 = a^{2}b̄ = bad̄ = aec̄ = abd̄ = ceb = cfd̄ = fcd̄ = f^{2}c̄ = e^{2}d = efb̄} 10. Table of the compact hyperbolic dodecahedral space forms

Class 1 ∘ H_{1,1}^{(5)} = H_{1,2}^{(5)} = Z_{3}^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{1}^{(5)} = \{a\left[{ABCDE \atop QMP F G}\right], b\left[{AKRLB \atop HGF JI}\right], c\left[{BLQMC \atop NDEOT}\right], d\left[{CMP ND \atop KSOEA}\right], e\left[{GQLRH \atop IJT OS}\right], f\left[{HRKSI \atop NT JF P}\right] — 1 = bfdc̄ā = adēfc̄ = dbefā = cab̄fē = cdaeb̄ = dfb̄ec̄}

• G_{2}^{(5)} = \{a\left[{ABCDE \atop HRLQG}\right], b\left[{AKRLB \atop IHGF J}\right], c\left[{BLQMC \atop DEOT N}\right], d\left[{CMP ND \atop IJT OS}\right], e\left[{EOSKA \atop P F GQM}\right], f\left[{HRKSI \atop NT JF P}\right] — 1 = afc̄db̄ = cdeb̄ā = acef̄d̄ = caēfb̄ = edf̄bā = bfdc̄ē}

Class 2 ∘ H_{1,3}^{(5)} = … = H_{1,6}^{(5)} = Z_{5}^{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{3}^{(5)} = \{a\left[{ABCDE \atop MQLBC}\right], b\left[{AKRLB \atop LQGHR}\right], c\left[{CMP ND \atop NDEOT}\right], d\left[{EOSKA \atop F GQMP}\right], e\left[{F GHIJ \atop ISKRH}\right], f\left[{ISOT J \atop T JF P N}\right] — 1 = b^{2}edā = dca^{2}b̄ = abēfc̄ = ac^{2}fd̄ = def^{2}c = fe^{2}bd̄}

• G_{4}^{(5)} = \{a\left[{ABCDE \atop QGHRL}\right], b\left[{AKRLB \atop HGF JI}\right], c\left[{BLQMC \atop T NDEO}\right], d\left[{CMP ND \atop JT OSI}\right], e\left[{EOSKA \atop F GQMP}\right], f\left[{HRKSI \atop NT JF P}\right] — 1 = bfdeā = edc̄ab̄ = afc̄bd̄ = ecabf̄ = acdfē = ceb̄fd̄}

• G_{5}^{(5)} = \{a\left[{ABCDE \atop HRLQG}\right], b\left[{AKRLB \atop F JIHG}\right], c\left[{BLQMC \atop DEOT N}\right], d\left[{CMP ND \atop OSIJT}\right], e\left[{EOSKA \atop QMP F G}\right], f\left[{HRKSI \atop JF P NT}\right] — 1 = afb̄eb̄ = abfd̄c̄ = acecd̄ = caēab̄ = dedfc̄ = fdb̄fē}

• G_{6}^{(5)} = \{a\left[{ABCDE \atop JIHGF}\right], b\left[{AKRLB \atop P NT JF}\right], c\left[{BLQMC \atop T OSIJ}\right], d\left[{CMP ND \atop SKRHI}\right], e\left[{DNT OE \atop RLQGH}\right], f\left[{EOSKA \atop QMP F G}\right] — 1 = dfb̄ac̄ = ebc̄ad̄ = fcd̄aē = bdēaf̄ = cef̄ab̄ = becfd}

Class 3 ∘ H_{1,7}^{(5)} = Z_{29} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{7}^{(5)} = \{a\left[{ABCDE \atop NDEOT}\right], b\left[{AKRLB \atop MCDNP}\right], c\left[{BLQMC \atop JIHGF}\right], d\left[{EOSKA \atop GHRLQ}\right], e\left[{F P MQG \atop ISKRH}\right], f\left[{ISOT J \atop P NT JF}\right] — 1 = bedbā = a^{2}f^{2}c̄ = ebadc̄ = ded̄fā = bcec̄d̄ = cfefb̄}

Class 4 ∘ H_{1,8}^{(5)} = Z_{3}×Z_{5}^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{8}^{(5)} = \{a\left[{ABCDE \atop GF JIH}\right], b\left[{AKRLB \atop NP MCD}\right], c\left[{BLQMC \atop T NDEO}\right], d\left[{EOSKA \atop P F GQM}\right], e\left[{GQLRH \atop T OSIJ}\right], f\left[{HRKSI \atop NT JF P}\right] — 1 = edābc̄ = dec̄af̄ = baēfc̄ = dcafb̄ = aef̄bd̄ = bcdf̄ē}

Class 5 ∘ H_{1,9}^{(5)} = Z_{3}^{2}×Z_{5}^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{9}^{(5)} = \{a\left[{ABCDE \atop QGHRL}\right], b\left[{AKRLB \atop NP MCD}\right], c\left[{BLQMC \atop T NDEO}\right], d\left[{EOSKA \atop QMP F G}\right], e\left[{F GHIJ \atop ISKRH}\right], f\left[{ISOT J \atop T JF P N}\right] — 1 = acb̄ad̄ = aedf̄c̄ = baēfc̄ = abcdc = def^{2}b̄ = fe^{2}bd̄}

Class 6 ∘ H_{1,10}^{(5)} = H_{1,11}^{(5)} = Z_{3}×Z_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{10}^{(5)} = \{a\left[{ABCDE \atop IHGF J}\right], b\left[{AKRLB \atop DEOT N}\right], c\left[{BLQMC \atop EAKSO}\right], d\left[{CMP ND \atop F GQMP}\right], e\left[{GQLRH \atop HISKR}\right], f\left[{ISOT J \atop JF P NT}\right] — 1 = bd^{2}eā = ae^{2}bc̄ = cfd̄ad̄ = cbaf̄ē = caf^{2}b̄ = bfdcē}

• G_{11}^{(5)} = \{a\left[{ABCDE \atop GF JIH}\right], b\left[{AKRLB \atop P MCDN}\right], c\left[{BLQMC \atop P F GQM}\right], d\left[{DNT OE \atop HRLQG}\right], e\left[{EOSKA \atop KRHIS}\right], f\left[{ISOT J \atop JF P NT}\right] — 1 = dcābf̄ = efācb̄ = af^{2}db = e^{2}ādā = bdēfc̄ = ebc^{2}d̄}

Class 7 ∘ H_{1,12}^{(5)} = Z_{5}×Z_{7} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{12}^{(5)} = \{a\left[{ABCDE \atop MQLBC}\right], b\left[{AKRLB \atop HGF JI}\right], c\left[{CMP ND \atop DEOT N}\right], d\left[{EOSKA \atop QGHRL}\right], e\left[{F P MQG \atop ISKRH}\right], f\left[{ISOT J \atop T JF P N}\right] — 1 = aebeb̄ = aca^{2}d̄ = abfc̄^{2} = dbf̄db̄ = cded̄ē = ef^{2}cf} 11. Table of the non-compact orientable hyperbolic dodecahedral space forms

Class 1 ∘ H_{1,1}^{(6)} = Z_{2}×Z ∘ p1: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{1}^{(6)} = \{a\left[{ABCDE \atop RKABL}\right], b\left[{BLQMC \atop QGHRL}\right], c\left[{CMP ND \atop OT NDE}\right], d\left[{EOSKA \atop JIHGF}\right], e\left[{F P MQG \atop KRHIS}\right], f\left[{ISOT J \atop P NT JF}\right]

— 1 = cdfeā^{2} = adedb̄^{2} = abefc^{2} = cf^{2}d̄ab̄ = bēcf̄dē} Class 2 ∘ H_{1,2}^{(6)} = H_{1,3}^{(6)} = H_{1,4}^{(6)} = Z_{2}×Z^{2} ∘ p1: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{2}^{(6)} = \{a\left[{ABCDE \atop GHRLQ}\right], b\left[{AKRLB \atop LBCMQ}\right], c\left[{CMP ND \atop OEAKS}\right], d\left[{DNT OE \atop IHGF J}\right], e\left[{F P MQG \atop RHISK}\right], f\left[{ISOT J \atop T JF P N}\right] — 1 = dfdābā = debaēf̄ = abcdf̄c̄ = caec̄dē = b^{2}ef^{2}c}

• G_{3}^{(6)} = \{a\left[{ABCDE \atop NP MCD}\right], b\left[{AKRLB \atop BCMQL}\right], c\left[{DNT OE \atop GF JIH}\right], d\left[{EOSKA \atop MP F GQ}\right], e\left[{GQLRH \atop HISKR}\right], f\left[{ISOT J \atop T JF P N}\right] — 1 = b^{2}ef^{2}ā = dec̄dāb̄ = ce^{2}bā^{2} = edc̄ādb̄ = cfcf̄df̄}

• G_{4}^{(6)} = \{a\left[{ABCDE \atop RLQGH}\right], b\left[{AKRLB \atop QLBCM}\right], c\left[{CMP ND \atop F JIHG}\right], d\left[{DNT OE \atop ISKRH}\right], e\left[{EOSKA \atop P F GQM}\right], f\left[{ISOT J \atop P NT JF}\right]

— 1 = abab̄eb̄ = fdeācē = dfcd̄ac̄ = ecfēdā = db^{2}cf̄^{2}} Class 3 ∘ H_{1,5}^{(6)} = H_{1,6}^{(6)} = Z_{2}^{2}×Z ∘ p1: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{5}^{(6)} = \{a\left[{ABCDE \atop CMQLB}\right], b\left[{AKRLB \atop DNP MC}\right], c\left[{DNT OE \atop QGHRL}\right], d\left[{EOSKA \atop IHGF J}\right], e\left[{F P MQG \atop ISKRH}\right], f\left[{ISOT J \atop NT JF P}\right] — 1 = bac̄abā = a^{2}ebf̄d̄ = bcec̄fd̄ = cbeded̄ = f^{2}efcd̄}

• G_{6}^{(6)} = \{a\left[{ABCDE \atop P F GQM}\right], b\left[{AKRLB \atop T NDEO}\right], c\left[{BLQMC \atop IHGF J}\right], d\left[{CMP ND \atop SOEAK}\right], e\left[{GQLRH \atop OSIJT}\right], f\left[{HRKSI \atop NT JF P}\right]

— 1 = efābēc̄ = bacācē = dfc̄dēā = cfdadb̄ = bdbf̄ef̄} Class 4 ∘ H_{1,7}^{(6)} = Z_{2}^{2}×Z^{2} ∘ p1: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{7}^{(6)} = \{a\left[{ABCDE \atop HGF JI}\right], b\left[{AKRLB \atop SOEAK}\right], c\left[{BLQMC \atop QMP F G}\right], d\left[{CMP ND \atop LQGHR}\right], e\left[{DNT OE \atop KRHIS}\right], f\left[{ISOT J \atop NT JF P}\right]

— 1 = afdābē = dcdc̄ac̄ = dbaēfā = b^{2}f^{2}c̄^{2} = ebefed̄} Class 5 ∘ H_{1,8}^{(6)} = Z_{2}^{2}×Z^{2} ∘ p1: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{8}^{(6)} = \{a\left[{ABCDE \atop CMQLB}\right], b\left[{AKRLB \atop DNP MC}\right], c\left[{DNT OE \atop QGHRL}\right], d\left[{EOSKA \atop F JIHG}\right], e\left[{F P MQG \atop KRHIS}\right], f\left[{ISOT J \atop NT JF P}\right]

— 1 = bac̄abā = dedēā^{2} = bcefcd̄ = cbec̄fd̄ = f^{2}ebf̄d̄} Class 6 ∘ H_{1,9}^{(6)} = Z_{2}×Z_{3}×Z^{2} ∘ p1: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{9}^{(6)} = \{a\left[{ABCDE \atop DEOT N}\right], b\left[{AKRLB \atop LBCMQ}\right], c\left[{CMP ND \atop OEAKS}\right], d\left[{F GHIJ \atop RLQGH}\right], e\left[{F P MQG \atop RHISK}\right], f\left[{ISOT J \atop T JF P N}\right]

— 1 = a^{2}f̄d^{2}b̄ = dbcāb̄ē = cfdēf̄ā = edec̄āc̄ = b^{2}ef^{2}c} Class 7 ∘ H_{1,10}^{(6)} = Z_{2}^{2}×Z_{3}×Z^{2} ∘ p1: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{10}^{(6)} = \{a\left[{ABCDE \atop HRLQG}\right], b\left[{AKRLB \atop P F GQM}\right], c\left[{BLQMC \atop AKSOE}\right], d\left[{CMP ND \atop RHISK}\right], e\left[{DNT OE \atop GF JIH}\right], f\left[{ISOT J \atop NT JF P}\right] — 1 = bdēc̄dā = caēab̄ā = dbēdc̄ā = bcf^{2}b̄c̄ = efdfef̄}

12. Table of the non-compact non-orientable hyperbolic dodecahedral space forms

Class 8 ∘ H_{1,11}^{(6)} = … = H_{1,14}^{(6)} = Z ∘ p1: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{11}^{(6)} = \{u\left[{ABCDE \atop HGQLR}\right], v\left[{AKRLB \atop BLQMC}\right], w\left[{CMP ND \atop SIHRK}\right], x\left[{DNT OE \atop F GHIJ}\right], y\left[{EOSKA \atop MQGF P}\right], z\left[{ISOT J \atop JT NP F}\right] — 1 = vuȳzxū = wyx̄wvū = uvȳxzx̄ = ywx̄zwū = v^{2}wz^{2}ȳ} • G_{12}^{(6)} = \{u\left[{ABCDE \atop LRHGQ}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{CMP ND \atop HRKSI}\right], x\left[{DNT OE \atop GHIJF}\right], y\left[{EOSKA \atop F P MQG}\right], z\left[{ISOT J \atop T NP F J}\right]

— 1 = v^{2}uw̄v̄ū = wzxw̄ux̄ = xȳuv̄yū = xz^{2}x̄yz̄ = zwyv̄w̄ȳ}

• G_{13}^{(6)} = \{u\left[{ABCDE \atop GQLRH}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{CMP ND \atop KSIHR}\right], x\left[{DNT OE \atop GHIJF}\right], y\left[{EOSKA \atop F P MQG}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = v^{2}uv̄yū = uvwyv̄w̄ = xȳux̄wū = xz^{2}x̄yz̄ = ywzxw̄z̄}

• G_{14}^{(6)} = \{u\left[{ABCDE \atop GQLRH}\right], v\left[{AKRLB \atop MCBLQ}\right], w\left[{CMP ND \atop KSIHR}\right], x\left[{DNT OE \atop GHIJF}\right], y\left[{EOSKA \atop F P MQG}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = vwyv̄uȳ = uv^{2}ūv̄w̄ = xȳux̄wū = xz^{2}x̄yz̄ = ywzxw̄z̄}

Class 9 ∘ H_{1,15}^{(6)} = Z^{2} ∘ p1: 1 cusp, pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{15}^{(6)} = \{u\left[{ABCDE \atop HIJF G}\right], v\left[{AKRLB \atop QGF P M}\right], w\left[{BLQMC \atop OSKAE}\right], x\left[{CMP ND \atop RKSIH}\right], y\left[{DNT OE \atop GQLRH}\right], z\left[{ISOT J \atop JT NP F}\right] — 1 = vwvȳxū = wyx̄uz̄ū = xvūyūw̄ = wzywx̄v̄ = zxz^{2}v̄ȳ}

Class 10 ∘ H_{1,16}^{(6)} = H_{1,17}^{(6)} = Z_{2}×Z^{2} ∘ p1: 1 cusp, pg: 1 cusp . . . . . . . . . . . . . . . . . . . . .

• G_{16}^{(6)} = \{a\left[{ABCDE \atop EOT ND}\right], u\left[{AKRLB \atop CMP ND}\right], b\left[{BLQMC \atop LQGHR}\right], v\left[{EOSKA \atop GF P MQ}\right], w\left[{F GHIJ \atop SIHRK}\right], x\left[{ISOT J \atop T NP F J}\right] — 1 = avwxāū = bux̄vx̄ā = b^{2}v̄a^{2}ū = vbw^{2}b̄ū = x^{2}wvūw̄}

• G_{17}^{(6)} = \{u\left[{ABCDE \atop QGF P M}\right], v\left[{AKRLB \atop GHIJF}\right], w\left[{BLQMC \atop DNT OE}\right], x\left[{CMP ND \atop HGQLR}\right], y\left[{EOSKA \atop RKSIH}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = wuxūvū = uwzūxȳ = wyvȳvx̄ = wxvz^{2}v̄ = y^{2}zw̄x̄z̄}

Class 11 ∘ H_{1,18}^{(6)} = H_{1,19}^{(6)} = Z_{2}^{2}×Z ∘ pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{18}^{(6)} = \{a\left[{ABCDE \atop HGF JI}\right], b\left[{AKRLB \atop MP F GQ}\right], u\left[{BLQMC \atop DNT OE}\right], v\left[{CMP ND \atop GQLRH}\right], c\left[{EOSKA \atop RHISK}\right], d\left[{ISOT J \atop NT JF P}\right] — 1 = budāvā = uadvbā = cbd̄^{2}cā = c^{2}dūv̄b̄ = uvc̄ūvb̄}

• G_{19}^{(6)} = \{u\left[{ABCDE \atop F GHIJ}\right], a\left[{AKRLB \atop MP F GQ}\right], b\left[{BLQMC \atop EOT ND}\right], c\left[{CMP ND \atop KRHIS}\right], d\left[{EOSKA \atop HRLQG}\right], v\left[{ISOT J \atop JT NP F}\right] — 1 = abv̄daū = buvūdū = cdbvcū = bdc̄ādā = bcv^{2}āc̄}

Class 12 ∘ H_{1,20}^{(6)} = H_{1,21}^{(6)} = Z_{2}^{2}×Z ∘ pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{20}^{(6)} = \{a\left[{ABCDE \atop LQGHR}\right], b\left[{AKRLB \atop NP MCD}\right], c\left[{BLQMC \atop DEOT N}\right], d\left[{EOSKA \atop GQMP F}\right], u\left[{F GHIJ \atop SIHRK}\right], v\left[{ISOT J \atop T NP F J}\right] — 1 = acdācb̄ = bau^{2}āc̄ = duvc̄b̄ā = dcvd̄bv̄ = v^{2}udb̄ū}

• G_{21}^{(6)} = \{a\left[{ABCDE \atop F JIHG}\right], b\left[{AKRLB \atop F GQMP}\right], c\left[{BLQMC \atop OT NDE}\right], d\left[{CMP ND \atop RLQGH}\right], u\left[{EOSKA \atop RKSIH}\right], v\left[{ISOT J \atop P F JT N}\right] — 1 = u^{2}vb̄av̄ = cubd̄v̄ā = dūab̄uā = dbcdāc̄ = cv^{2}c̄d̄b̄}

Class 13 ∘ H_{1,22}^{(6)} = H_{1,23}^{(6)} = Z_{2}^{2}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{22}^{(6)} = \{u\left[{ABCDE \atop MQGF P}\right], v\left[{AKRLB \atop F GHIJ}\right], w\left[{BLQMC \atop OEDNT}\right], x\left[{CMP ND \atop GQLRH}\right], y\left[{EOSKA \atop IHRKS}\right], z\left[{ISOT J \atop P F JT N}\right] — 1 = uwxȳzv̄ = wz^{2}w̄xū = y^{2}vx̄uv̄ = uxwuz̄ȳ = vzxvȳw̄}

• G_{23}^{(6)} = \{u\left[{ABCDE \atop MQGF P}\right], v\left[{AKRLB \atop F GHIJ}\right], w\left[{BLQMC \atop MP NDC}\right], x\left[{DNT OE \atop HRKSI}\right], y\left[{EOSKA \atop LRHGQ}\right], z\left[{ISOT J \atop P F JT N}\right] — 1 = uwxȳzv̄ = w^{2}uv̄yū = xzūxv̄ȳ = ywz̄v̄wū = z^{2}xvȳx̄}

Class 14 ∘ H_{1,24}^{(6)} = H_{1,25}^{(6)} = Z_{2}^{2}×Z^{2} ∘ p1: 1 cusp, pg: 1 cusp . . . . . . . . . . . . . . . . . . . . .

• G_{24}^{(6)} = \{u\left[{ABCDE \atop IJF GH}\right], v\left[{AKRLB \atop MQGF P}\right], a\left[{BLQMC \atop NP MCD}\right], w\left[{DNT OE \atop RKSIH}\right], x\left[{EOSKA \atop HGQLR}\right], b\left[{ISOT J \atop T JF P N}\right] — 1 = ubwxav̄ = awvx̄bū = va^{2}uv̄x̄ = wx̄uw̄xū = wb^{2}āvb̄}

• G_{25}^{(6)} = \{a\left[{ABCDE \atop RKABL}\right], b\left[{BLQMC \atop JIHGF}\right], c\left[{CMP ND \atop EAKSO}\right], u\left[{DNT OE \atop QLRHG}\right], v\left[{F P MQG \atop IHRKS}\right], d\left[{ISOT J \atop NT JF P}\right] — 1 = cub̄vā^{2} = bd̄uv̄cā = abdcv̄ū = bvduāc̄ = cd^{2}vb̄ū} Class 15 ∘ H_{1,26}^{(6)} = Z_{2}^{2}×Z^{2} ∘ p1: 1 cusp, pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{26}^{(6)} = \{u\left[{ABCDE \atop JF GHI}\right], a\left[{AKRLB \atop SOEAK}\right], b\left[{BLQMC \atop QMP F G}\right], v\left[{CMP ND \atop RKSIH}\right], c\left[{DNT OE \atop HRLQG}\right], d\left[{ISOT J \atop NT JF P}\right] — 1 = cbvāud̄ = vacb̄ub̄ = vc̄vūcū = udvdca = a^{2}d^{2}b̄^{2}}

Class 16 ∘ H_{1,27}^{(6)} = … = H_{1,38}^{(6)} = Z_{2}^{2}×Z^{2} ∘ p1: 1 cusp, pg: 3 cusps . . . . . . . . . . . . . .

• G_{27}^{(6)} = \{u\left[{ABCDE \atop HGQLR}\right], v\left[{AKRLB \atop DCMP N}\right], w\left[{BLQMC \atop GHIJF}\right], x\left[{DNT OE \atop OSKAE}\right], y\left[{F P MQG \atop IHRKS}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = vxzv̄wū = wyzxȳū = xvuwȳz̄ = x^{2}uȳv̄ū = wz^{2}w̄v̄ȳ}

• G_{28}^{(6)} = \{u\left[{ABCDE \atop MQGF P}\right], v\left[{AKRLB \atop T OEDN}\right], w\left[{BLQMC \atop OSKAE}\right], x\left[{CMP ND \atop RHGQL}\right], y\left[{F GHIJ \atop SIHRK}\right], z\left[{ISOT J \atop P F JT N}\right] — 1 = uwvz^{2}v̄ = zxūwv̄ȳ = xvuz̄ȳū = wuxy^{2}x̄ = wzyw̄x̄v̄}

• G_{29}^{(6)} = \{u\left[{ABCDE \atop GHIJF}\right], v\left[{AKRLB \atop MQGF P}\right], w\left[{BLQMC \atop OEDNT}\right], x\left[{CMP ND \atop LRHGQ}\right], y\left[{EOSKA \atop IHRKS}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = vxȳzxū = wzv̄wyū = xwuz̄^{2}ū = yzw̄xvū = y^{2}vx̄w̄v̄}

• G_{30}^{(6)} = \{u\left[{ABCDE \atop LBAKR}\right], v\left[{BLQMC \atop NDCMP}\right], w\left[{DNT OE \atop OSKAE}\right], x\left[{F GHIJ \atop QGF P M}\right], y\left[{GQLRH \atop SIHRK}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = u^{2}vwzv̄ = yxz̄wūv̄ = w^{2}uy^{2}ū = z^{2}xv^{2}x̄ = x^{2}yzwȳ}

• G_{31}^{(6)} = \{u\left[{ABCDE \atop T OEDN}\right], v\left[{AKRLB \atop SKAEO}\right], w\left[{BLQMC \atop HGQLR}\right], x\left[{CMP ND \atop GF P MQ}\right], y\left[{F GHIJ \atop SIHRK}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = vzx^{2}z̄ū = wy^{2}w̄uv̄ = u^{2}xw^{2}x̄ = xyzūv̄w̄ = z^{2}yv̄^{2}ȳ}

• G_{32}^{(6)} = \{u\left[{ABCDE \atop T OEDN}\right], v\left[{AKRLB \atop AEOSK}\right], w\left[{BLQMC \atop NDCMP}\right], x\left[{F GHIJ \atop F P MQG}\right], y\left[{GQLRH \atop RKSIH}\right], z\left[{ISOT J \atop P F JT N}\right] — 1 = v^{2}uz̄^{2}ū = xyvūwz̄ = wu^{2}w̄vȳ = y^{2}xw^{2}x̄ = vzx^{2}z̄ȳ}

• G_{33}^{(6)} = \{u\left[{ABCDE \atop AKRLB}\right], v\left[{BLQMC \atop NDCMP}\right], w\left[{DNT OE \atop F GHIJ}\right], x\left[{EOSKA \atop QGF P M}\right], y\left[{GQLRH \atop RKSIH}\right], z\left[{ISOT J \atop P F JT N}\right] — 1 = xv^{2}x̄ū^{2} = xyūvz̄w̄ = vuyzx̄ȳ = uvwx̄zw̄ = z^{2}wy^{2}w̄}

• G_{34}^{(6)} = \{u\left[{ABCDE \atop HIJF G}\right], v\left[{AKRLB \atop AEOSK}\right], w\left[{BLQMC \atop EDNT O}\right], x\left[{CMP ND \atop P MQGF}\right], y\left[{GQLRH \atop RKSIH}\right], z\left[{ISOT J \atop P F JT N}\right] — 1 = v^{2}uy^{2}ū = uzxyvw̄ = vzūxz̄ȳ = wxūw̄vȳ = x^{2}wz̄^{2}w̄}

• G_{35}^{(6)} = \{u\left[{ABCDE \atop QGF P M}\right], v\left[{AKRLB \atop HIJF G}\right], w\left[{BLQMC \atop NDCMP}\right], x\left[{DNT OE \atop OSKAE}\right], y\left[{GQLRH \atop SIHRK}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = uyzxȳv̄ = wxzw̄uv̄ = xvȳwuz̄ = x^{2}uw^{2}ū = y^{2}vz^{2}v̄}

• G_{36}^{(6)} = \{u\left[{ABCDE \atop T OEDN}\right], v\left[{AKRLB \atop SKAEO}\right], w\left[{BLQMC \atop HGQLR}\right], x\left[{CMP ND \atop F GHIJ}\right], y\left[{F P MQG \atop IHRKS}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = vzxȳz̄ū = wȳxw̄vū = u^{2}xz^{2}x̄ = xyzūv̄w̄ = w^{2}yv̄^{2}ȳ}

• G_{37}^{(6)} = \{u\left[{ABCDE \atop MP NDC}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{DNT OE \atop JF GHI}\right], x\left[{EOSKA \atop F P MQG}\right], y\left[{GQLRH \atop RKSIH}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = v^{2}uz̄xū = wz^{2}w̄ū^{2} = yzwx̄uv̄ = xyvx̄ȳv̄ = wy^{2}w̄xz̄}

• G_{38}^{(6)} = \{u\left[{ABCDE \atop T OEDN}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{CMP ND \atop F GHIJ}\right], x\left[{EOSKA \atop F P MQG}\right], y\left[{GQLRH \atop RKSIH}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = v^{2}uxz̄ū = u^{2}wz^{2}w̄ = yzūxw̄v̄ = xyvx̄ȳv̄ = xwy^{2}w̄z̄}

Class 17 ∘ H_{1,39}^{(6)} = Z_{2}^{4}×Z ∘ pg: 4 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{39}^{(6)} = \{a\left[{ABCDE \atop F JIHG}\right], u\left[{AKRLB \atop LQMCB}\right], b\left[{CMP ND \atop F GQMP}\right], c\left[{DNT OE \atop OEAKS}\right], d\left[{GQLRH \atop RHISK}\right], v\left[{ISOT J \atop T NP F J}\right] — 1 = u^{2}av^{2}ā = cudābv̄ = cvbūd̄ā = dvcab̄ū = b^{2}d^{2}c̄^{2}}

Class 18 ∘ H_{1,40}^{(6)} = Z_{4}×Z ∘ pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{40}^{(6)} = \{u\left[{ABCDE \atop P MQGF}\right], a\left[{AKRLB \atop T NDEO}\right], v\left[{BLQMC \atop GHIJF}\right], b\left[{CMP ND \atop SOEAK}\right], c\left[{GQLRH \atop OSIJT}\right], d\left[{HRKSI \atop NT JF P}\right] — 1 = ubāvcā = uvc̄auv̄ = bdv̄bc̄ū = vdbud̄c̄ = abad̄cd̄} Class 19 ∘ H_{1,41}^{(6)} = H_{1,42}^{(6)} = Z_{4}×Z ∘ pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{41}^{(6)} = \{u\left[{ABCDE \atop GHIJF}\right], a\left[{AKRLB \atop NDEOT}\right], v\left[{BLQMC \atop P MQGF}\right], b\left[{CMP ND \atop RHISK}\right], w\left[{EOSKA \atop RHGQL}\right], x\left[{ISOT J \atop T NP F J}\right] — 1 = xbxāuw̄ = vbūvx̄ā = baux̄^{2}ū = wawb̄vū = wv^{2}w̄b̄ā}

• G_{42}^{(6)} = \{a\left[{ABCDE \atop MP F GQ}\right], b\left[{AKRLB \atop HGF JI}\right], c\left[{BLQMC \atop T NDEO}\right], u\left[{CMP ND \atop RKSIH}\right], d\left[{EOSKA \atop HRLQG}\right], v\left[{ISOT J \atop JT NP F}\right]

— 1 = audcub̄ = cvuv^{2}ā = ubāud̄c̄ = acad̄bd̄ = dbv̄b̄cv̄} Class 20 ∘ H_{1,43}^{(6)} = H_{1,44}^{(6)} = Z_{4}×Z^{2} ∘ p1: 1 cusp, pg: 1 cusp . . . . . . . . . . . . . . . . . . . . .

• G_{43}^{(6)} = \{a\left[{ABCDE \atop LRKAB}\right], u\left[{BLQMC \atop OSKAE}\right], b\left[{CMP ND \atop P F GQM}\right], v\left[{DNT OE \atop F GHIJ}\right], c\left[{GQLRH \atop KRHIS}\right], d\left[{ISOT J \atop NT JF P}\right] — 1 = a^{2}cv̄dv̄ = uvdbcā = b^{2}v̄buā = uaud^{2}b̄ = c^{2}dvcū}

• G_{44}^{(6)} = \{a\left[{ABCDE \atop QMP F G}\right], b\left[{AKRLB \atop DEOT N}\right], c\left[{BLQMC \atop HGF JI}\right], u\left[{CMP ND \atop KAEOS}\right], v\left[{GQLRH \atop T JISO}\right], d\left[{HRKSI \atop JF P NT}\right]

— 1 = buv̄dc̄ā = auavd̄c̄ = udub̄dā = ubac̄vc̄ = cvb̄vdb̄} Class 21 ∘ H_{1,45}^{(6)} = Z_{2}×Z_{4}×Z ∘ pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{45}^{(6)} = \{a\left[{ABCDE \atop CMQLB}\right], b\left[{AKRLB \atop DNP MC}\right], c\left[{DNT OE \atop QGHRL}\right], u\left[{EOSKA \atop IJF GH}\right], v\left[{F P MQG \atop HRKSI}\right], d\left[{ISOT J \atop NT JF P}\right]

— 1 = bac̄abā = a^{2}vuvū = bcvdcū = cbvbd̄ū = d^{2}vc̄dū} Class 22 ∘ H_{1,46}^{(6)} = H_{1,47}^{(6)} = H_{1,48}^{(6)} = Z_{2}×Z_{4}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . .

• G_{46}^{(6)} = \{a\left[{ABCDE \atop MP F GQ}\right], u\left[{AKRLB \atop MCBLQ}\right], b\left[{CMP ND \atop EAKSO}\right], c\left[{DNT OE \atop IHGF J}\right], v\left[{GQLRH \atop RKSIH}\right], d\left[{ISOT J \atop P NT JF}\right] — 1 = abubaū = avuad̄c̄ = bcd̄^{2}cā = vdbv̄ū^{2} = bdcv^{2}c̄}

• G_{47}^{(6)} = \{u\left[{ABCDE \atop T OEDN}\right], v\left[{AKRLB \atop AEOSK}\right], a\left[{BLQMC \atop IHGF J}\right], b\left[{CMP ND \atop P F GQM}\right], w\left[{GQLRH \atop RKSIH}\right], x\left[{ISOT J \atop P F JT N}\right] — 1 = v^{2}ux̄^{2}ū = axbwvū = u^{2}bab̄^{2} = aw^{2}āvw̄ = vxābx̄w̄}

• G_{48}^{(6)} = \{u\left[{ABCDE \atop T OEDN}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{CMP ND \atop GF P MQ}\right], a\left[{EOSKA \atop F JIHG}\right], x\left[{GQLRH \atop RKSIH}\right], b\left[{ISOT J \atop NT JF P}\right]

— 1 = v^{2}uab̄ū = wxvw̄ū^{2} = xbūaw̄v̄ = ax^{2}āx̄v̄ = abw^{2}b̄^{2}} Class 23 ∘ H_{1,49}^{(6)} = H_{1,50}^{(6)} = Z_{2}^{2}×Z_{4}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{49}^{(6)} = \{a\left[{ABCDE \atop F JIHG}\right], u\left[{AKRLB \atop LQMCB}\right], b\left[{CMP ND \atop LQGHR}\right], c\left[{DNT OE \atop MP F GQ}\right], d\left[{EOSKA \atop SKRHI}\right], v\left[{ISOT J \atop T NP F J}\right] — 1 = u^{2}av^{2}ā = bubūdā = vbācūd̄ = dvcv̄cā = cbc̄d^{2}b̄}

• G_{50}^{(6)} = \{u\left[{ABCDE \atop JF GHI}\right], a\left[{AKRLB \atop P F GQM}\right], b\left[{BLQMC \atop AKSOE}\right], c\left[{CMP ND \atop ISKRH}\right], v\left[{DNT OE \atop HGQLR}\right], d\left[{ISOT J \atop NT JF P}\right]

— 1 = acb̄v̄dū = budcaū = cūvc̄vū = abd^{2}āb̄ = cdvāv̄b̄} Class 24 ∘ H_{1,51}^{(6)} = Z_{2}×Z_{8}×Z ∘ pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{51}^{(6)} = \{a\left[{ABCDE \atop QMP F G}\right], u\left[{AKRLB \atop HIJF G}\right], b\left[{BLQMC \atop EOT ND}\right], v\left[{CMP ND \atop AEOSK}\right], c\left[{GQLRH \atop RHISK}\right], d\left[{ISOT J \atop T JF P N}\right]

— 1 = ucudb̄ā = baūbv̄ā = vuc̄bdā = vac^{2}v̄b̄ = cd^{2}vdū} Class 25 ∘ H_{1,52}^{(6)} = H_{1,53}^{(6)} = H_{1,54}^{(6)} = Z_{2}×Z_{8}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . .

• G_{52}^{(6)} = \{u\left[{ABCDE \atop RLBAK}\right], v\left[{BLQMC \atop OEDNT}\right], w\left[{CMP ND \atop EOSKA}\right], x\left[{F GHIJ \atop F P MQG}\right], y\left[{GQLRH \atop SIHRK}\right], z\left[{ISOT J \atop F JT NP}\right] — 1 = u^{2}y^{2}ūw̄ = vzv̄wv̄ū = yxvwūv̄ = xyzx^{2}z̄ = xwz^{2}wȳ}

• G_{53}^{(6)} = \{u\left[{ABCDE \atop BLQMC}\right], v\left[{AKRLB \atop T OEDN}\right], w\left[{CMP ND \atop EOSKA}\right], x\left[{F GHIJ \atop F P MQG}\right], y\left[{GQLRH \atop SIHRK}\right], z\left[{ISOT J \atop F JT NP}\right] — 1 = uvwvzv̄ = wu^{2}yxū = y^{2}vw̄ūv̄ = xyzx^{2}z̄ = xwz^{2}wȳ}

• G_{54}^{(6)} = \{u\left[{ABCDE \atop IJF GH}\right], a\left[{AKRLB \atop OT NDE}\right], v\left[{BLQMC \atop P MQGF}\right], w\left[{CMP ND \atop QLRHG}\right], x\left[{EOSKA \atop RKSIH}\right], b\left[{ISOT J \atop JF P NT}\right] — 1 = axab̄^{2}ū = x^{2}būvb̄ = wv^{2}w̄uv̄ = vwawūā = abwx̄ux̄} Class 26 ∘ H_{1,55}^{(6)} = Z_{2}^{2}×Z_{8}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{55}^{(6)} = \{u\left[{ABCDE \atop CBLQM}\right], a\left[{AKRLB \atop NDEOT}\right], v\left[{CMP ND \atop HGQLR}\right], w\left[{EOSKA \atop QGF P M}\right], b\left[{F GHIJ \atop SKRHI}\right], x\left[{ISOT J \atop P F JT N}\right] — 1 = u^{2}ax^{2}ā = xvūvb̄^{2} = uvbauw̄ = avawv̄w̄ = xbxw̄b̄w̄}

Class 27 ∘ H_{1,56}^{(6)} = Z_{2}×Z_{16}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{56}^{(6)} = \{a\left[{ABCDE \atop F GQMP}\right], u\left[{AKRLB \atop NT OED}\right], v\left[{BLQMC \atop GHIJF}\right], b\left[{CMP ND \atop IJT OS}\right], c\left[{EOSKA \atop LQGHR}\right], d\left[{HRKSI \atop P NT JF}\right] — 1 = ubcāvā = bdv̄bv̄ā = uabd̄cv̄ = cucvdā = cdbūdū}

Class 28 ∘ H_{1,57}^{(6)} = H_{1,58}^{(6)} = Z_{4}×Z_{4}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{57}^{(6)} = \{u\left[{ABCDE \atop CBLQM}\right], v\left[{AKRLB \atop DCMP N}\right], w\left[{DNT OE \atop OSKAE}\right], x\left[{F GHIJ \atop HGQLR}\right], y\left[{F P MQG \atop IHRKS}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = vwzv̄ū^{2} = zyxūv̄w̄ = w^{2}uv̄ȳū = yzwȳx̄^{2} = z^{2}xȳv̄x̄}

• G_{58}^{(6)} = \{u\left[{ABCDE \atop LBAKR}\right], v\left[{BLQMC \atop HGQLR}\right], w\left[{CMP ND \atop F GHIJ}\right], x\left[{DNT OE \atop KAEOS}\right], y\left[{F P MQG \atop IHRKS}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = u^{2}vw̄yv̄ = wz^{2}w̄ux̄ = v^{2}yx̄uȳ = wyzxuv̄ = x^{2}zwȳz̄}

Class 29 ∘ H_{1,59}^{(6)} = Z_{3}×Z^{2} ∘ p1: 1 cusp, pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{59}^{(6)} = \{u\left[{ABCDE \atop JF GHI}\right], v\left[{AKRLB \atop BLQMC}\right], w\left[{CMP ND \atop EDNT O}\right], x\left[{EOSKA \atop F P MQG}\right], y\left[{GQLRH \atop IHRKS}\right], z\left[{ISOT J \atop JT NP F}\right] — 1 = vux̄uzū = uyz^{2}x̄w̄ = v^{2}wuȳx̄ = xw^{2}zwz̄ = yvy^{2}xv̄}

Class 30 ∘ H_{1,60}^{(6)} = Z_{2}^{2}×Z_{3}×Z ∘ pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{60}^{(6)} = \{u\left[{ABCDE \atop F GHIJ}\right], v\left[{AKRLB \atop CMP ND}\right], w\left[{BLQMC \atop T OEDN}\right], x\left[{EOSKA \atop F P MQG}\right], y\left[{GQLRH \atop OT JIS}\right], z\left[{HRKSI \atop JT NP F}\right] — 1 = vuzūxū = uyxv̄zw̄ = xwuȳvz̄ = xyw̄vw̄v̄ = wxz̄ȳzȳ}

Class 31 ∘ H_{1,61}^{(6)} = Z_{2}^{2}×Z_{3}×Z ∘ pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{61}^{(6)} = \{u\left[{ABCDE \atop IJF GH}\right], v\left[{AKRLB \atop CBLQM}\right], a\left[{CMP ND \atop HRLQG}\right], b\left[{DNT OE \atop ISKRH}\right], w\left[{EOSKA \atop F P MQG}\right], x\left[{ISOT J \atop F JT NP}\right] — 1 = vab̄wx̄ū = bxūv̄wā = aūbūwū = xbwv̄^{2}b̄ = wavāx̄^{2}}

Class 32 ∘ H_{1,62}^{(6)} = … = H_{1,65}^{(6)} = Z_{2}^{2}×Z_{3}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . .

• G_{62}^{(6)} = \{u\left[{ABCDE \atop F GHIJ}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{CMP ND \atop OEDNT}\right], x\left[{EOSKA \atop MQGF P}\right], y\left[{GQLRH \atop SIHRK}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = v^{2}uyxū = wzx̄vyū = xwuz^{2}ū = vwxv̄ȳ^{2} = xyzw̄^{2}z̄}

• G_{63}^{(6)} = \{u\left[{ABCDE \atop F GHIJ}\right], v\left[{AKRLB \atop LQMCB}\right], w\left[{CMP ND \atop P MQGF}\right], x\left[{DNT OE \atop HGQLR}\right], y\left[{EOSKA \atop RKSIH}\right], z\left[{ISOT J \atop T NP F J}\right] — 1 = v^{2}ux̄wū = wz̄xv̄yū = uz^{2}ūyx̄ = xvw^{2}v̄ȳ = y^{2}zxw̄z̄}

• G_{64}^{(6)} = \{u\left[{ABCDE \atop IJF GH}\right], v\left[{AKRLB \atop NDCMP}\right], w\left[{BLQMC \atop EDNT O}\right], x\left[{EOSKA \atop GF P MQ}\right], y\left[{GQLRH \atop JISOT}\right], z\left[{HRKSI \atop P F JT N}\right] — 1 = xyūvz̄ū = wxūv̄zū = wuyz̄xv̄ = xwz̄ȳwv̄ = yxz̄yw̄v̄}

• G_{65}^{(6)} = \{u\left[{ABCDE \atop IJF GH}\right], v\left[{AKRLB \atop NDCMP}\right], w\left[{BLQMC \atop T JISO}\right], x\left[{DNT OE \atop MQGF P}\right], y\left[{EOSKA \atop LRHGQ}\right], z\left[{HRKSI \atop P F JT N}\right] — 1 = ywūvz̄ū = wxūv̄zū = xv̄wz̄yū = vxwzxȳ = yvwyzx̄}

Class 33 ∘ H_{1,66}^{(6)} = … = H_{1,71}^{(6)} = Z_{3}×Z_{4}×Z ∘ pg: 2 cusps . . . . . . . . . . . . . . . . . . . . . . .

• G_{66}^{(6)} = \{u\left[{ABCDE \atop HIJF G}\right], a\left[{AKRLB \atop LBCMQ}\right], v\left[{CMP ND \atop RHGQL}\right], w\left[{DNT OE \atop SIHRK}\right], x\left[{EOSKA \atop QGF P M}\right], b\left[{ISOT J \atop NT JF P}\right] — 1 = wbwūav̄ = waubvx̄ = vavw̄bū = wxb̄^{2}xū = xvūxā^{2}}

• G_{67}^{(6)} = \{a\left[{ABCDE \atop F JIHG}\right], u\left[{AKRLB \atop MCBLQ}\right], b\left[{CMP ND \atop RLQGH}\right], v\left[{DNT OE \atop SIHRK}\right], w\left[{EOSKA \atop QGF P M}\right], x\left[{ISOT J \atop P F JT N}\right] — 1 = uw̄aw̄xā = vuaxbw̄ = x^{2}vābv̄ = vwx̄v̄bā = ubu^{2}b̄w̄} • G_{68}^{(6)} = \{u\left[{ABCDE \atop HIJF G}\right], v\left[{AKRLB \atop MCBLQ}\right], a\left[{CMP ND \atop RLQGH}\right], b\left[{DNT OE \atop HISKR}\right], w\left[{EOSKA \atop QGF P M}\right], x\left[{ISOT J \atop P F JT N}\right]

— 1 = vw̄uābū = bvuxaw̄ = ab̄uw̄xū = vav^{2}āw̄ = x^{2}bxw̄b̄}

• G_{69}^{(6)} = \{u\left[{ABCDE \atop RHGQL}\right], a\left[{AKRLB \atop QLBCM}\right], v\left[{CMP ND \atop KSIHR}\right], w\left[{DNT OE \atop JF GHI}\right], x\left[{EOSKA \atop P MQGF}\right], b\left[{ISOT J \atop T JF P N}\right] — 1 = ax̄bw̄vū = uauw̄bx̄ = va^{2}vxū = wbwx̄aū = xvb^{2}vw̄}

• G_{70}^{(6)} = \{a\left[{ABCDE \atop LQGHR}\right], u\left[{AKRLB \atop LQMCB}\right], v\left[{CMP ND \atop SIHRK}\right], b\left[{DNT OE \atop HGF JI}\right], w\left[{EOSKA \atop F P MQG}\right], x\left[{ISOT J \atop T NP F J}\right] — 1 = u^{2}aw̄uā = vwūv̄bā = xvābv̄w̄ = auvxbw̄ = bx^{2}b̄xw̄}

• G_{71}^{(6)} = \{u\left[{ABCDE \atop RHGQL}\right], v\left[{AKRLB \atop LQMCB}\right], a\left[{CMP ND \atop ISKRH}\right], b\left[{DNT OE \atop HGF JI}\right], w\left[{EOSKA \atop F P MQG}\right], x\left[{ISOT J \atop T NP F J}\right]

— 1 = v^{2}ub̄aū = ab̄uv̄wū = uvaxbw̄ = bx^{2}b̄xw̄ = wawv̄āx̄} Class 34 ∘ H_{1,72}^{(6)} = Z_{2}×Z_{3}×Z_{4}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{72}^{(6)} = \{a\left[{ABCDE \atop QMP F G}\right], b\left[{AKRLB \atop DEOT N}\right], c\left[{BLQMC \atop HGF JI}\right], u\left[{CMP ND \atop KSIHR}\right], v\left[{EOSKA \atop LRHGQ}\right], d\left[{ISOT J \atop T JF P N}\right]

— 1 = buv̄dc̄ā = aud^{2}uc̄ = ubvbdā = vcāb̄vā = cdb̄cv̄ū} Class 35 ∘ H_{1,73}^{(6)} = Z_{3}×Z_{8}×Z ∘ pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{73}^{(6)} = \{u\left[{ABCDE \atop GHIJF}\right], a\left[{AKRLB \atop MQLBC}\right], v\left[{CMP ND \atop QLRHG}\right], w\left[{DNT OE \atop RKSIH}\right], x\left[{EOSKA \atop P MQGF}\right], b\left[{ISOT J \atop JF P NT}\right]

— 1 = avāw̄vū = xva^{2}uw̄ = ub^{2}wxv̄ = wbūwv̄b̄ = xb̄xāxū} Class 36 ∘ H_{1,74}^{(6)} = Z_{2}×Z_{3}×Z_{16}×Z ∘ pg: 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{74}^{(6)} = \{a\left[{ABCDE \atop LRKAB}\right], u\left[{BLQMC \atop EDNT O}\right], b\left[{CMP ND \atop P F GQM}\right], v\left[{EOSKA \atop IJF GH}\right], c\left[{GQLRH \atop KRHIS}\right], d\left[{ISOT J \atop NT JF P}\right]

— 1 = aua^{2}cv̄ = uvdbcā = avcvb̄^{2} = udv̄d^{2}b̄ = c^{2}dūb̄ū} Class 37 ∘ H_{1,75}^{(6)} = H_{1,76}^{(6)} = H_{1,77}^{(6)} = Z_{2}^{2}×Z_{5}×Z ∘ pg: 1 cusp . . . . . . . . . . . . . . . . . . . . .

• G_{75}^{(6)} = \{a\left[{ABCDE \atop CMQLB}\right], b\left[{AKRLB \atop DNP MC}\right], c\left[{DNT OE \atop QGHRL}\right], u\left[{EOSKA \atop F GHIJ}\right], v\left[{F P MQG \atop SIHRK}\right], d\left[{ISOT J \atop NT JF P}\right] — 1 = bac̄abā = a^{2}vc̄dū = bcvbd̄ū = uvuv̄b̄c̄ = d^{2}vdcū}

• G_{76}^{(6)} = \{a\left[{ABCDE \atop CMQLB}\right], b\left[{AKRLB \atop DNP MC}\right], c\left[{DNT OE \atop QGHRL}\right], u\left[{EOSKA \atop GF P MQ}\right], d\left[{F GHIJ \atop HISKR}\right], v\left[{ISOT J \atop JT NP F}\right] — 1 = bac̄abā = bcūa^{2}ū = ud^{2}ub̄c̄ = vcdv^{2}ū = vdc̄vb̄d̄}

• G_{77}^{(6)} = \{a\left[{ABCDE \atop CMQLB}\right], b\left[{AKRLB \atop DNP MC}\right], c\left[{DNT OE \atop QGHRL}\right], u\left[{EOSKA \atop MQGF P}\right], v\left[{F GHIJ \atop IHRKS}\right], d\left[{ISOT J \atop NT JF P}\right]

— 1 = bac̄abā = cbūa^{2}ū = bcūd^{2}ū = vuv^{2}bd̄ = dvdcvc̄} Class 38 ∘ H_{1,78}^{(6)} = Z_{3}×Z_{5}×Z ∘ pg: 1 cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• G_{78}^{(6)} = \{u\left[{ABCDE \atop HGQLR}\right], v\left[{AKRLB \atop DCMP N}\right], w\left[{BLQMC \atop OEDNT}\right], x\left[{EOSKA \atop QGF P M}\right], y\left[{F GHIJ \atop IHRKS}\right], z\left[{ISOT J \atop JT NP F}\right] — 1 = uyūxwv̄ = wzv̄wxū = vuwuvx̄ = xy^{2}vwz̄ = zyz^{2}x̄ȳ}

Acknowledgement. I am indebted to my colleague Prof. Emil Molnár for his valuable suggestions.

References

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[9] Molnár, E.; Prok, I.; Šarac, M.: The classification of the non-compact hyperbolic octahedron space forms of finite volume. Acta Math. Hung. (manuscript 1996).

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Received November 7, 1997