On Subgroups of Tetrahedron Groups
Ma. Louise N. De Las Peñas1, Rene P. Felix2 and Glenn R. Laigo3 1Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines. [email protected] 2Institute of Mathematics, University of the Philippines – Diliman, Diliman, Quezon City, Philippines. [email protected] 3Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines. [email protected] Abstract. We present a framework to determine subgroups of tetrahedron groups and tetrahedron Kleinian groups, based on tools in color symmetry theory. Mathematics Subject Classification (2000). 20F55, 20F67 Key words. Coxeter tetrahedra, tetrahedron groups, tetrahedron Kleinian groups, color symmetry
1 Introduction
In [6, 7] the determination of the subgroups of two-dimensional symmetry groups was facilitated using a geometric approach and applying concepts in color symmetry theory. In particular, these works derived low-indexed subgroups of a triangle group *abc, a group generated by reflections along the sides of a given triangle ∆ with interior angles π/a, π/b and π/c where a, b, c are integers ≥ 2. The elements of *abc are symmetries of the tiling formed by copies of ∆. The subgroups of *abc were calculated from colorings of this given tiling, assuming a corresponding group of color permutations. In this work, we pursue the three dimensional case and extend the problem to determine subgroups of tetrahedron groups, groups generated by reflections along the faces of a Coxeter tetrahedron, which are groups of symmetries of tilings in space by tetrahedra. The study will also address determining subgroups of other types of three dimensional symmetry groups such as the tetrahedron Kleinian groups.
2 Coxeter tetrahedra, tetrahedron groups and tetrahedron Kleinian groups
Let X3 = S3, E3, or H3 denote either the spherical, Euclidean or hyperbolic 3-space. Consider a tetrahedron t in X3 all of whose dihedral angles are submultiples of . t is called a Coxeter tetrahedron and if its dihedral angles are given by /p, /q, /r, /s, /t, /u we denote t by [p, q, r, s, t, u]. Five of the known Coxeter tetrahedra are spherical: [3, 3, 3, 2, 2, 2], [4, 3, 3, 2, 2, 2], [3, 4, 3, 2, 2, 2], [5, 3, 3, 2, 2, 2], and [3, 3, 2, 3, 2, 2]; and three are Euclidean: [4, 3, 4, 2, 2, 2], [3, 4, 2, 3, 2, 2], and [3, 3, 3, 2, 2, 3][2]. In the hyperbolic case, there are 9 compact and 23 non-compact Coxeter tetrahedra of finite volume [5, 13]. The compact types have no ideal vertices or vertices at infinity; while the 23 non- compact ones have at least one ideal vertex. Given a Coxeter tetrahedron t in X3, the reflections in its faces generate a group called a tetrahedron group or a Coxeter tetrahedron group. The subgroup of index 2 in the tetrahedron group consisting of orientation preserving isometries is referred to as a tetrahedron Kleinian group. We address in this paper the problem of determining the subgroups of tetrahedron groups and tetrahedron
1 Kleinian groups corresponding to hyperbolic Coxeter tetrahedra of finite volume. Studies have been carried out on the subgroup structure of tetrahedron groups and tetrahedron Kleinian groups associated with spherical and Euclidean Coxeter tetrahedra, but a lot is yet to be discovered for the hyperbolic case. In our work, the determination of the subgroups of tetrahedron groups and tetrahedron Kleinian groups is facilitated by the link of group theory and color symmetry theory, as will be discussed in the next section.
3 Colorings of three-dimensional tilings by tetrahedra
We begin by first explaining the setting we will be working with in studying tetrahedron groups and their subgroups, and the notation we will use in the results that follow. Consider a Coxeter tetrahedron t := [p, q, r, s, t, u] of finite volume with vertices P0, P1, P2, P3 and dihedral angles /p, /q, /r, /s, /t, /u, as shown in Fig. 1, lying on either X3 = S3, E3 or H3. Repeatedly reflecting t in its faces results in a tiling T of X3 by copies of t. Let P, Q, R, S denote reflections along the respective faces P1P2P3, P0P2P3, P0P1P3, P0P1P2 of t. The group H generated by P, Q, R, S is called a tetrahedron group. The subgroup of the tetrahedron group consisting of orientation preserving isometries, and in particular generated by PQ, QR, and RS is the tetrahedron Kleinian group K.
Fig.1. A tetrahedron t with vertices P0, P1, P2, P3. A label k at an edge of t means that the dihedral angle of t at this edge is /k.
As in [26], we have the following result on the tetrahedron group H:
Lemma 1. The tiling T is the H-orbit of t, that is, T = {ht : h ∈ H}. Moreover, t forms a fundamental region for H and StabH(t) = {h ∈ H : ht = t} is the trivial group {e}.
By this lemma we mean that if h ∈ H and ht is the image of t under h then the union of these images as h varies over the elements of H is X3 and, moreover, if h1, h2 ∈ H, h1 ≠ h2, the respective interiors of h1t and h2t are disjoint. Each tetrahedron in T is the image of t under a uniquely determined h ∈ H. Suppose M is a subgroup of H and S = Mt = {mt : m ∈ M}, the M-orbit of t. Then StabM(t) = {e} and M acts transitively on S. Consequently, there is a one-to-one correspondence between M and S given by m mt, m ∈ M. The S S action of M on is regular, where m’∈ Μ acts on mt ∈ by sending it to its image under m'.
2 In this paper, the study of the subgroups of the tetrahedron group H and tetrahedron Kleinian group K is addressed via tools in color symmetry theory, and in particular, using colorings of T and Kt, respectively. We give the definition of a coloring of the M-orbit of t for a subgroup M of H as follows:
Consider S = Mt = {mt : m ∈ M}, M a subgroup of H. If C = {c1, c2, …, cn} is a set of n colors, an onto function f : S C is called an n-coloring of S. To each mt ∈ S is assigned a color in C. The coloring determines a partition T = -1 -1 {f (ci) : ci ∈ C} where f (ci) is the set of elements of S assigned color ci. Equivalently, we may think of the coloring as a partition of S. We look at n-colorings of S for which the group M has a transitive action on the set C of n colors. We will refer to such a coloring as an M-transitive n- coloring of S. To arrive at such a coloring, consider the following lemma.
Lemma 2. Let T = {X1, X2, …, Xn} be a partition of S into n sets. Suppose M acts transitively on T, then T = {mLt : m ∈ M} for some subgroup L of M of index n.
Proof
Assume t ∈ X1 and let L = StabM(X1). Using the fact that M acts transitively on S and M acts transitively on T, it follows that X1 = Lt and each set in the partition T is equal to mX1 for some m ∈ M. Thus, T = {mLt : m ∈ M}.
Moreover, by the orbit-stabilizer theorem, n = |T| = [M : StabM(X1)] = [M : L].
Let L be a subgroup of M of index n and {m1, m2, …, mn} be a complete set of left coset representatives of L in M. Define the coloring f : S = Mt C = {c1, -1 -1 -1 c2, …, cn} by f(x) = ci if x ∈ miLt. Then f (ci) = miLt and T = {f (c1), f (c2), -1 …, f (cn)} = {m1Lt, m2Lt, …, mnLt}. The group M acts transitively on T with m ∈ M sending miLt to its image mmiLt. Consequently, we get a transitive action -1 -1 of M on C by defining for m ∈ M, mci = cj if and only if mf (ci) = f (cj). The n- coloring of S determined by f is M- transitive.
Our main result which is a consequence of the preceding discussions and forms the basis of our calculations is given as follows.
Theorem. Let M be a subgroup of H =
3
Remark: Note that given a subgroup L of M of index n and a set of n colors {c1, c2, …, cn}, then there correspond (n – 1)! M-transitive n-colorings of S with L fixing c1. In an M-transitive n-coloring of S with L fixing c1, the set of tetrahedra Lt is assigned c1 and the remaining n – 1 colors are distributed among the miLt, mi ∉ L.
4 Index 2, 3 and 4 subgroups of tetrahedron groups
In applying the given theorem to derive the index 2, 3 and 4 subgroups of the tetrahedron group H, we construct respectively, two, three and four colorings of T where all elements of H effect color permutations and H acts transitively on the set of colors. For such an n-coloring of T, n ∈ {2, 3, 4}, a homomorphism
π: H Sn is defined. The group H is generated by P, Q, R and S, thus π is completely determined when π(P), π(Q), π(R) and π(S) are specified. To construct a coloring, we consider a fundamental region for H and assign to it color c1. The arguments to obtain 2-, 3- or 4-colorings of T where each of the generators P, Q, R, S permutes the colors, that is, the associated color permutation to a generator is of order 1 or 2, are as follows:
(i) In constructing 2-colorings of T, each of P, Q, R, S may be assigned either the identity permutation (1) or the permutation (12). The possible 2-colorings of T are listed in Table 1. (ii) In arriving at 3-colorings of T, this time each of P, Q, R, S may be assigned either the identity permutation or either of the permutations (12), (13) or (23). Now, since we assume H acts transitively on the set of 3 colors, we consider the assignments that bring forth a π(H) which is a transitive subgroup of S3. Note that there are two transitive subgroups of S3 namely S3 and the cyclic group Z3. It is not possible for π(H) to be Z3 since this group cannot be generated by elements of order two. Thus we only consider the case when π(H) is S3. In order to generate S3, two of the three permutations (12), (13) or (23) must be associated with two from among the generators P, Q, R and S. The remaining third and fourth generators are associated either with the identity, with one of the two already assigned earlier or with the third permutation of order 2. The possible 3-colorings that will yield distinct subgroups of index 3 distinct up to conjugacy in H are given in Table 2. (iii) In constructing 4-colorings of T that will yield index 4 subgroups of H, each of P, Q, R, S is assigned either the identity permutation; a 2-cycle or a product of two disjoint 2-cycles: (12), (13), (14), (23), (24), (34), (12)(34),(13)(24) or (14)(23). We only look at the situations that arise when π(H) is a transitive subgroup of S4. In particular, we consider the cases when π(H) is either S4, D4 or V = {e, (12)(34), (13)(24), (14)(23)}, a Klein 4 group. The possible H-transitive four-colorings of T that can be constructed when π(H) ≅ V, D4 and S4 are given respectively in Tables 3, 4 and 5. These colorings will give rise to distinct subgroups of index 4 distinct up to conjugacy in H.
To illustrate the process in determining subgroups of tetrahedron groups, let us consider the tetrahedron group H generated by the reflections P, Q, R, S
4 along the faces of the Coxeter tetrahedron t10 := [3, 3, 6, 2, 2, 2] (refer to Table 7 for the complete list of hyperbolic Coxeter tetrahedra). t10 is a tetrahedron with one vertex at infinity and four right angled triangles for its faces. The symmetry group G of the tiling T by copies of t10 is H = < P, Q, R, S > with defining relations P2 = Q2 = R2 = S2 = (PQ)3 = (QR)3 = (RS)6 = (PR)2 = (PS)2 = (QS)2 = e. There are only three index 2 subgroups of H, namely,
5 Index 2, 3 and 4 subgroups of tetrahedron Kleinian groups
In determining the subgroups of tetrahedron Kleinian groups, we carry over the same setting given in the previous section, that is, we consider a Coxeter tetrahedron t := [p, q, r, s, t] of finite volume and the three-dimensional tiling T which results by reflecting t in its faces. The group generated by the reflections P, Q, R, S in the faces of t, is the tetrahedron group H. The subgroup of the tetrahedron group generated by PQ, QR, and RS is the tetrahedron Kleinian group K. In applying our theorem to derive the subgroups of K, we consider the K- orbit of t, namely S = Kt = {kt : k ∈ K} and construct K-transitive colorings of S. In determining respectively, the index 2, 3 and 4 subgroups of K, we construct
5 two, three and four colorings of S. For an n-coloring of S, n ∈ {2, 3, 4}, a homomorphism π: K Sn is defined. The group K is generated by PQ, QR and RS, thus π is completely determined when π(PQ), π(QR), π(RS) are specified. To obtain 2-, 3- or 4-colorings of S where each of the generators PQ, QR, RS permutes the colors, the orders of the color permutations associated to PQ, QR, RS should be divisors respectively of the orders of PQ, QR, RS. Construction of tables similar to Tables 1-5, this time pertaining to assignments of permutations to PQ, QR and RS, will yield transitive two, three and four colorings of S. To illustrate the above ideas, let us consider the tetrahedron Kleinian subgroup K =
6 Low index subgroups of tetrahedron groups and tetrahedron Kleinian groups
In this part of the paper, we present the number of low index subgroups of the tetrahedron groups and tetrahedron Kleinian groups arising from the Coxeter tetrahedra of finite volume. The results for the number of index 2, 3, 4 subgroups of these groups, up to conjugacy, are shown in Table 7. In column 1, we list the compact tetrahedra (t1 – t9) and the non-compact ones (t10 – t32). The number of vertices at infinity is also given, for each non-compact tetrahedron.
6 7 Future Outlook
In this study, a method is provided which facilitates the enumeration of the subgroups of a group of symmetries of a three dimensional tiling by a Coxeter tetrahedron. The approach we have presented here allows one to study three dimensional Coxeter groups and their subgroups from a geometric perspective using concepts in color symmetry theory. In our work, we focused on deriving the index 2, 3 and 4 subgroups of hyperbolic tetrahedron groups and tetrahedron Kleinian groups realized by Coxeter tetrahedra of finite volume. In coming up with the list of subgroups, most of the information pertaining to the subgroups relied on the construction of a table of permutation assignments. As a continuation of this study, it would be interesting to look at the subgroup structure of the other subgroups of a tetrahedron group using the approach presented here. The Picard group for example, which is an index 4 subgroup of the tetrahedron group associated with t11 [17] has a very interesting subgroup structure. The Picard group has especially been of great interest in many fields of mathematics, for example, number theory and automorphic function theory. Particular examples of subgroups of this group have been studied in existing literature [9, 12, 21, 24, 25]. Continuing the study of the subgroup structure of the tetrahedron group will also include determining its two dimensional subgroups. For example, the Picard group contains the modular group PSL(2,Z) as a subgroup [8]. It would also be interesting to determine the classes of triangle fuchsian groups contained in the tetrahedron Kleinian groups. The study of tetrahedra with infinite volume is still at its infancy, and not much has been said about its corresponding tetrahedron groups [1, 14]. Studying these groups and their subgroups would be worth looking at. There are initial studies on truncated tetrahedra, which are polyhedra with some or all of its vertices cut off arising from tetrahedra of infinite volume. When there are m truncations, a corresponding group arises generated by m + 4 reflections [10, 18]. It would also be interesting to look at the subgroup structure of these types of reflection groups. Finally, another avenue in the study of hyperbolic Coxeter groups and their subgroup structure would be to extend the approach given in this work to look at the higher dimensional cases. For example, there are 5 compact and 9 non- compact hyperbolic simplices in 4 dimensions; and 12 non-compact hyperbolic simplices in 5 dimensions. There are also existing non-compact hyperbolic simplices of finite volume in dimensions 6 to 9 [13]. The next step would be to determine the subgroups of the Coxeter groups corresponding to these simplices.
Acknowledgements. Louise De Las Peñas acknowledges the support from the FEBTC- David G. Choa Professorial Chair Faculty Grant, Ateneo de Manila University. Glenn R. Laigo is grateful for a grant from the Commission on Higher Education – Higher Education Development Program. It is a pleasure to acknowledge Marjorie Senechal for suggestions on the problem.
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generators for PQ QR RS PR PS QS no P Q R S the subgroup /p /q /r /s /t /u fixing color 1 Q, R, S, PQP, PRP, 1 (12) (1) (1) (1) (12) (1) (1) (12) (12) (1) PSP P, R, S, QPQ, QRQ, 2 (1) (12) (1) (1) (12) (12) (1) (1) (1) (12) QSQ P, Q, S, RPR, RQR, 3 (1) (1) (12) (1) (1) (12) (12) (12) (1) (1) RSR P, Q, R, SPS, SQS, 4 (1) (1) (1) (12) (1) (1) (12) (1) (12) (12) SRS 5 (1) (1) (12) (12) (1) (12) (1) (12) (12) (12) P, Q, SR, RPR, RQR
8 6 (1) (12) (1) (12) (12) (12) (12) (1) (12) (1) P, R, SQ, QPQ, QRQ 7 (1) (12) (12) (1) (12) (1) (12) (12) (1) (12) P, RQ, S, QPQ, QSQ 8 (12) (12) (1) (1) (1) (12) (1) (12) (12) (12) QP, R, S, PRP, PSP 9 (12) (1) (12) (1) (12) (12) (12) (1) (12) (1) Q, RP, S, PQP, PSP 10 (12) (1) (1) (12) (12) (1) (12) (12) (1) (12) Q, R, SP, PQP, PRP 11 (1) (12) (12) (12) (12) (1) (1) (12) (12) (1) P, RQ, SQ, QPQ 12 (12) (1) (12) (12) (12) (12) (1) (1) (1) (12) Q, RP, SP, PQP 13 (12) (12) (1) (12) (1) (12) (12) (12) (1) (1) QP, R, SP, PRP 14 (12) (12) (12) (1) (1) (1) (12) (1) (12) (12) QP, RP, S, PSP 15 (12) (12) (12) (12) (1) (1) (1) (1) (1) (1) PQ, QR, RS Table 1. H-transitive colorings that will give rise to index 2 subgroups of H.
generators for PQ QR RS PR PS QS no P Q R S the subgroup /p /q /r /s /t /u fixing color 1 R, S, QPQ, QRQ, QSQ, 1 (12) (13) (1) (1) (132) (13) (1) (12) (12) (13) PQP, PRP, PSP Q, S, RPR, RQR, RSR, 2 (12) (1) (13) (1) (12) (13) (13) (132) (12) (1) PQP, PRP, PSP Q, R, SPS, SQS, SRS, 3 (12) (1) (1) (13) (12) (1) (13) (12) (132) (13) PQP, PRP, PSP P, S, RPR, RQR, RSR, 4 (1) (12) (13) (1) (12) (132) (13) (13) (1) (12) QPQ, QRQ, QSQ P, R, SPS, SQS, SRS, 5 (1) (12) (1) (13) (12) (12) (13) (1) (13) (132) QPQ, QRQ, QSQ P, Q, SPS, SQS, SRS, 6 (1) (1) (12) (13) (1) (12) (132) (12) (13) (13) RPR, RQR, RSR R, QP, SPS, SQS, SRS, 7 (12) (12) (1) (13) (1) (12) (13) (12) (132) (132) PRP, PSP Q, RP, SPS, SQS, SRS, 8 (12) (1) (12) (13) (12) (12) (132) (1) (132) (13) PQP, PSP Q, SP, RPR, RQR, RSR, 9 (12) (1) (13) (12) (12) (13) (123) (132) (1) (12) PQP, PRP P, RQ, SPS, SQS, SRS, 10 (1) (12) (12) (13) (12) (1) (132) (12) (13) (132) QPQ, QSQ P, SQ, RPR, RQR, RSR, 11 (1) (12) (13) (12) (12) (132) (123) (13) (12) (1) QPQ, QRQ P, RS, RPR, RQR, QPQ, 12 (1) (13) (12) (12) (13) (12) (1) (12) (12) (123) QRQ, QSQ S, QP, RPR, RQR, RSR, 13 (12) (12) (13) (1) (1) (132) (13) (132) (12) (12) PRP, PSP S, RP, QPQ, QRQ, QSQ, 14 (12) (13) (12) (1) (132) (123) (12) (1) (12) (13) PQP, PSP R, SP, QPQ, QRQ, QSQ, 15 (12) (13) (1) (12) (132) (13) (12) (12) (1) (123) PQP, PRP S, QR, QPQ, QSQ, PQP, 16 (13) (12) (12) (1) (123) (1) (12) (123) (13) (12) PRP, PSP R, QS, QPQ, QRQ, PQP, 17 (13) (12) (1) (12) (123) (12) (12) (13) (123) (1) PRP, PSP Q, RS, RPR, RQR, PQP, 18 (13) (1) (12) (12) (13) (12) (1) (123) (123) (12) PRP, PSP R, SPS, SQS, SRP, SRQ, 19 (12) (12) (23) (13) (1) (123) (123) (123) (132) (132) SRSRS Q, SPS, SRS, SQP, SQR, 20 (12) (23) (12) (13) (123) (132) (132) (1) (132) (123) SQSQS Q, RPR, RSR, RQP, RQS, 21 (12) (23) (13) (12) (123) (123) (123) (132) (1) (132) RQRQR P, RQ, QPS, QSQ, SQS, 22 (23) (12) (12) (13) (132) (1) (132) (132) (123) (132) SRS P, SQ, QPR, QRQ, RQR, 23 (23) (12) (13) (12) (132) (132) (123) (123) (132) (1) RSR P, SR, QPR, QRQ, QSQ, 24 (23) (13) (12) (12) (123) (123) (1) (132) (132) (123) RQR P, S, RPR, RQR, QSR, 25 (1) (12) (13) (23) (12) (132) (132) (13) (23) (123) RSPSR, RSRSR
9 Q, S, RPR, RQR, PSR, 26 (12) (1) (13) (23) (12) (13) (132) (132) (123) (23) RSQSR, RSRSR R, S, QPQ, QRQ, PSQ, 27 (12) (13) (1) (23) (132) (13) (23) (12) (123) (132) QSQSQ, QSRSQ R, S, QPQ, QSQ, PRQ, 28 (12) (13) (23) (1) (132) (132) (23) (123) (12) (13) QRQRQ, QRSRQ QR, QS, QPQ, PQP, 29 (13) (12) (12) (12) (123) (1) (1) (123) (123) (1) PRP, PSP RP, SP, QPQ, QRQ, 30 (12) (13) (12) (12) (132) (123) (1) (1) (1) (123) QSQ, PQP QP, SP, RPR, RQR, RSR, 31 (12) (12) (13) (12) (1) (132) (123) (132) (1) (1) PRP QP, RP, SPS, SQS, SRS, 32 (12) (12) (12) (13) (1) (1) (132) (1) (132) (132) PSP RS, QP, RPR, RQR, PRP, 33 (12) (12) (13) (13) (1) (132) (1) (132) (132) (132) PSP QS, RP, QPQ, QRQ, 34 (12) (13) (12) (13) (132) (123) (132) (1) (132) (1) PQP, PSP QR, SP, QPQ, QSQ, 35 (12) (13) (13) (12) (132) (1) (123) (132) (1) (123) PQP, PRP Table 2. Colorings that will give rise to index 3 subgroups of H.
generators for generators for no P Q R S the subgroup no P Q R S the subgroup fixing color 1 fixing color 1 R, S, PRP, PSP, QPQP, QRQ, QSQ, QP, PRS, PSR, RPS, RQS 1 (12)(34) (13)(24) (1) (1) PQRQP, PQSQP 19 (12)(34) (12)(34) (14)(23) (13)(24) Q, S, PQP, PSP, RPRP, RQR, RSR, RP, PQS, PSQ, QPS, QRS 2 (12)(34) (1) (13)(24) (1) PRQRP, PRSRP 20 (12)(34) (14)(23) (12)(34) (13)(24) Q, R, PQP, PRP, SPSP, SQS, SRS, SP, PQR, PRQ, QPR, QSR 3 (12)(34) (1) (1) (13)(24) PSQSP, PSRSP 21 (12)(34) (14)(23) (13)(24) (12)(34) P, S, QPQ, QSQ, RPR, RQRQ, RSR, RQ, PQS, PRS, PSQ, QPS 4 (1) (12)(34) (13)(24) (1) QRPRQ, QRSRQ 22 (14)(23) (12)(34) (12)(34) (13)(24) P, R, QPQ, QRQ, SPS, SQSQ, SRS, SQ, PQR, PRQ, PSR, QPR 5 (1) (12)(34) (1) (13)(24) QSPSQ, QSRSQ 23 (14)(23) (12)(34) (13)(24) (12)(34) P, Q, RPR, RQR, SPS, SQS, SRSR, SR, PQR, PRQ, PSQ, QPR 6 (1) (1) (12)(34) (13)(24) RSPSR, RSQSR 24 (14)(23) (13)(24) (12)(34) (12)(34) QP, R, PRP, SPSP, SQSP, SRS, P, QPQ, QRS, QSR, RPR, RQS, SPS 7 (12)(34) (12)(34) (1) (13)(24) PSRSP 25 (1) (12)(34) (13)(24) (14)(23) Q, RP, PQP, SPSP, SQS, SRSP, Q, PQP, PRS, PSR, RPS, RQR, SQS 8 (12)(34) (1) (12)(34) (13)(24) PSQSP 26 (12)(34) (1) (13)(24) (14)(23) Q, SP, PQP, RPRP, RQR, RSRP, R, PQS, PRP, PSQ, QPS, QRQ, SRS 9 (12)(34) (1) (13)(24) (12)(34) PRQRP 27 (12)(34) (13)(24) (1) (14)(23) P, RQ, QPQ, SPS, SQSQ, SRSQ, S, PQR, PRQ, PSP, QPR, QSQ, 10 (1) (12)(34) (12)(34) (13)(24) QSPSQ 28 (12)(34) (13)(24) (14)(23) (1) RSR P, SQ, QPQ, RPR, RQRQ, RSRQ, RQ, SQ, PRQP, PSQP, QPQP 11 (1) (12)(34) (13)(24) (12)(34) QRPRQ 29 (13)(24) (12)(34) (12)(34) (12)(34) P, SR, QPQ, QSRQ, RPR, RQRQ, RP, SP, QPQP, QRQP, QSQP 12 (1) (13)(24) (12)(34) (12)(34) QRPRQ 30 (12)(34) (13)(24) (12)(34) (12)(34) QP, S, PSP, RPRP, RQRP, RSR, QP, SP, RPRP, RQRP, RSRP 13 (12)(34) (12)(34) (13)(24) (1) PRSRP 31 (12)(34) (12)(34) (13)(24) (12)(34) RP, S, PSP, QPQP, QRQP, QSQ, QP, RP, SPSP, SQSP, SRSP 14 (12)(34) (13)(24) (12)(34) (1) PQSQP 32 (12)(34) (12)(34) (12)(34) (13)(24) R, SP, PRP, QPQP, QRQ, QSQP, QP, SR, PSRP, RPRP, RQRP 15 (12)(34) (13)(24) (1) (12)(34) PQRQP 33 (12)(34) (12)(34) (13)(24) (13)(24) RQ, S, PRQP, PSP, QPQP, QSQ, RP, SQ, PSQP, QPQP, QRQP 16 (13)(24) (12)(34) (12)(34) (1) PQSQP 34 (12)(34) (13)(24) (12)(34) (13)(24) R, SQ, PRP, PSQP, QPQP, QRQ, RQ, SP, PRQP, QPQP, QSQP 17 (13)(24) (12)(34) (1) (12)(34) PQRQP 35 (12)(34) (13)(24) (13)(24) (12)(34) Q, SR, PQP, PSRP, RPRP, RQR,
18 (13)(24) (1) (12)(34) (12)(34) PRQRP
Table 3. Colorings that will give rise to index 4 subgroups of H, where (H) ≅ V.
generators for the generators for the no P Q R S subgroup fixing no P Q R S subgroup fixing color 1 color 1 R, S, QPQ, QRQ, QSQ, PRP, PSP, SPS, SQP, SQR, SRQ, SQSRS, SRPRS 1 (12) (13)(24) (1) (1) PQPQP, PQRQP, PQSQP 101 (12) (13)(24) (12)(34) (14)(23) Q, S, RPR, RQR, RSR, PQP, PSP, RPR, RQP, RQS, QSR, RSPSR, RQRSR 2 (12) (1) (13)(24) (1) PRPRP, PRQRP, PRSRP 102 (12) (13)(24) (14)(23) (12)(34) Q, PRP, PQP, SPS, SQS, SRS, PSPSP, QR, QS, QPQ, PRQP, PSQP, PQPQP 3 (12) (1) (1) (13)(24) PSQSP, PSRSP 103 (12) (13)(24) (13)(24) (13)(24) P, S, RPR, RQR, RSR, QPQ, QSQ, QR, QPQ, PSQ, SPS, QSQS, QSRS 4 (1) (12) (13)(24) (1) QRPRQ, QRQRQ, QRSRQ 104 (12) (13)(24) (13)(24) (14)(23)
10 P, R, SPS, SQS, SRS, QPQ, QRQ, QS, QPQ, PRQ, RPR, QRQR, QRSR 5 (1) (12) (1) (13)(24) QSPSQ, QSQSQ, QSRSQ 105 (12) (13)(24) (14)(23) (13)(24) P, Q, RPR, RQR, SPS, SQS, SRS, SR, QPQ, PRQ, RPR, QSRQ, QRQR 6 (1) (1) (12) (13)(24) RSPSR, RSQSR, RSRSR 106 (12) (14)(23) (13)(24) (13)(24) R, QP, SPS, SQS, SRS, PRP, PSPSP, Q, R, PQP, PRP, PSP, SQS, SRS, 7 (12) (12) (1) (13)(24) PSQSP, PSRSP 107 (12)(34) (1) (1) (13) SPQPS, SPRPS, SPSPS Q, RP, SPS, SQS, SRS, PQP, PSPSP, Q, S, PQP, PRP, PSP, RQR, RSR, 8 (12) (1) (12) (13)(24) PSQSP, PSRSP 108 (12)(34) (1) (13) (1) RPQPR, RPRPR, RPSPR Q, SP, RPR, RQR, RSR, PQP, PRPRP, R, S, PQP, PRP, PSP, QRQ, QSQ, 9 (12) (1) (13)(24) (12) PRQRP, PRSRP 109 (12)(34) (13) (1) (1) QPQPQ, QPRPQ, QPSPQ P, QR, SPS, SQS, SRS, QPQ, QSPSQ, RQR, RSR, RPQPR, RPSPR, PRPR, 10 (1) (12) (12) (13)(24) QSQSQ, QSRSQ 110 (12)(34) (12) (13)(24) (1) QRPR, SRPR P, SQ, RPR, RQR, RSR, QPQ, QRPRQ, R, QP, SQS, QRQ, PRP, PSPS, QPQP, 11 (1) (12) (13)(24) (12) QRQRQ, QRSRQ 111 (12)(34) (12) (1) (13)(24) QPRPQ, QPSPQ P, SR, QPQ, QRQ, QSQ, RPR, RQPQR, Q, SQS, SRS, PSPS, RSPS, SPQPS, 12 (1) (13)(24) (12) (12) RQRQR, RQSQR 112 (12)(34) (1) (12) (13)(24) SPRPS, SPSQSPS S, QP, RPR, RQR, RSR, PSP, PRPRP, R, S, PQP, PSP, QRQ, QSQ, PRPQ, 13 (12) (12) (13)(24) (1) PRQRP, PRSRP 113 (12)(34) (13) (24) (1) PRQRP, PRSRP S, RP, QPQ, QRQ, QSQ, PSP, PQPQP, R, S, PQP, PRP, QRQ, QSQ, PSPQ, 14 (12) (13)(24) (12) (1) PQRQP, PQSQP 114 (12)(34) (13) (1) (24) PSQSP, PSRSP R, SP, QPQ, QRQ, QSQ, PRP, PQPQP, Q, S, PQP, PRP, RQR, RSR, PSPR, 15 (12) (13)(24) (1) (12) PQRQP, PQSQP 115 (12)(34) (1) (13) (24) PSQSP, PSRSP S, RQ, PQP, PRP, PSP, QSQ, QPQPQ, S, PR, PQP, PSP, QSQ, QRPQ, 16 (13)(24) (12) (12) (1) QPRPQ, QPSPQ 116 (12)(34) (13) (12)(34) (1) QPQPQ, QPSPQ R, SQ, PQP, PRP, QRQ, PSPQ, R, PS, PQP, PRP, QRQ, QSPQ, 17 (13)(24) (12) (1) (12) PSQSP, PSRSP 117 (12)(34) (13) (1) (12)(34) QPQPQ, QPRPQ Q, SR, PQP, PRP, RQR, PSP, PSQSP, Q, PS, PQP, PRP, RQR, RSPR, 18 (13)(24) (1) (12) (12) PSRSP 118 (12)(34) (1) (13) (12)(34) RPQPR, RPRPR SPS, SQS, PSRS, QSRS, RSRS, SRPRS, S, RQ, PQP, PSP, QSQ, PRPQ, 19 (12) (12) (12)(34) (13)(24) SRQRS 119 (12)(34) (13) (13)(24) (1) PRQRP, PRSRP PRP, SPSP, PSQSP, PSRSP, SPS, SRS, PSQS, QSQS, RSQS, SQPQS, PSPQPSP, PSPRPSP, PQSPSP, 20 (12) (12)(34) (12) (13)(24) SQRQS 120 (12)(34) (13) (1) (13)(24) PSPSRSPSP RPR, RSR, PRQR, QRQR, SRQR, Q, SR, PQP, PRP, PSP, RQR, RPQPR, 21 (12) (12)(34) (13)(24) (12) RQPQR, RQSQR 121 (12)(34) (1) (13) (13)(24) RPRPR, RPSPR SQS, SRS, PSPS, QSPS, RSPS, SPQPS, S, PQP, PSP, QRP, RQR, RSR, PRPR, 22 (12)(34) (12) (12) (13)(24) SPRPS 122 (12)(34) (13) (14)(23) (1) PRSRP RS, RQR, RPRP, RPQPR, RPSPR, R, PQP, PRP, QSP, SQS, SRS, PSPS, 23 (12)(34) (12) (13)(24) (12) RPRQRPR, RPRSRPR 123 (12)(34) (13) (1) (14)(23) PSRSP QRQ, QSQ, PQPQ, RQPQ, SQPQ, Q, PQP, PRP, RSP, SQS, SRS, PSPS, 24 (12)(34) (13)(24) (12) (12) QPRPQ, QPSPQ 124 (12)(34) (1) (13) (14)(23) PSQSP P, R, SPS, SQS, QSRS, SRPRS, SRQRS, S, PQ, PRP, PSP, RSR, RQPR, RPRPR, 25 (1) (12) (34) (13)(24) SRSPSRS, SRSRSRS 125 (12)(34) (12)(34) (13) (1) RPSPR Q, R, SPS, SQS, PSRS, SRPRS, SRQRS, R, PQ, PRP, PSP, SRS, SQPS, SPRPS, 26 (12) (1) (34) (13)(24) SRSQSRS, SRSRSRS 126 (12)(34) (12)(34) (1) (13) SPSPS Q, R, SPS, SRS, PSQS, SQPQS, Q, PR, PQP, PSP, SQS, SRPS, SPQPS, 27 (12) (34) (1) (13)(24) SQRQS, SQSQSQS, SQSRSQS 127 (12)(34) (1) (12)(34) (13) SPSPS Q, S, RPR, RSR, PRQR, RQPQR, S, QRQ, QSQ, QPQP, QPQR, QPRPQ, 28 (12) (34) (13)(24) (1) RQSQR, RQRQRQR, RQRSRQR 128 (12)(34) (13)(24) (12) (1) QPSPQ, QPQSQPQ P, S, RPR, RQR, QPQ, QSQ, RSRQ, R, QRQ, QSQ, PQPQ, SQPQ, QPRPQ, 29 (1) (12) (13)(24) (34) RSPSR, RSQSR 129 (12)(34) (13)(24) (1) (12) QPSPQ, QPQRQPQ P, S, QPQ, QRQ, RPR, RSR, QSQR, Q, RQR, RSR, PRPR, SRPR, RPQPR, 30 (1) (13)(24) (12) (34) QSPSQ, QSRSQ 130 (12)(34) (1) (13)(24) (12) RPSPR, RPRQRPR P, S, QPQ, QRQ, QSQ, RPR, RSR, S, RQ, PRP, PSP, QSQ, PQPQ, 31 (1) (13)(24) (12) (1) RQPQR, RQRQR, RQSQR 131 (12)(34) (13)(24) (13) (1) PQRQP, PQSQP P, R, QPQ, QRQ, QSQ, SPS, SRS, R, SQ, PRP, PSP, QRQ, PQPQ, 32 (1) (13)(24) (1) (12) SQPQS, SQRQS, SQSQS 132 (12)(34) (13)(24) (1) (13) PQRQP, PQSQP PSP, RPRP, PRQRP, PRSRP, P, Q, RPR, RQR, RSR, SPS, SQS, PQRPRP, PRPQPRP, PRPSPRP, 33 (1) (1) (13)(24) (12) SRPRS, SRQRS, SRSRS 133 (12)(34) (1) (13)(24) (13) PRPRSRPRP P, RPR, RSR, QRQR, SRQR, RQPQR, S, QRQ, QSQ, QPR, PQPQ, QPSPQ, 34 (1) (12)(34) (13)(24) (12) RQSQR, RQRPRQR 134 (12)(34) (13)(24) (14) (1) QPQRQPQ, QPQSQPQ P, SPS, SRS, QSQS, RSQS, SQPQS, R, PRP, PSP, SQP, QRQ, QSQ, PQPQ, 35 (1) (12)(34) (12) (13)(24) SQRQS, SQSPSQS 135 (12)(34) (13)(24) (1) (14) PQRQP P, SPS, SQS, QSRS, RSRS, SRPRS, Q, PQP, PSP, SRP, RQR, RSR, PRPR, 36 (1) (12) (12)(34) (13)(24) SRQRS, SRSPSRS 136 (12)(34) (1) (13)(24) (14) PRQRP P, SR, QPQ, QSQ, RPR, QRQR, R, SQS, SRPS, PSPS, QSPS, SPQPS, 37 (1) (13)(24) (12)(34) (12) QRPRQ, QRSRQ 137 (12)(34) (12) (34) (13)(24) SPSRSPS P, SR, QPQ, QRQ, RPR, QSQR, S, RQR, RSPR, PRPR, QRPR, RPQPR, 38 (1) (13)(24) (12) (12)(34) QSPSQ, QSRSQ 138 (12)(34) (12) (13)(24) (34) RPRSRPR P, SQ, RPR, RQR, QPQ, RSRQ, RSPSR, S, QRQ, QPSQ, QPQP, QPQR, 39 (1) (12) (13)(24) (12)(34) RSQSR 139 (12)(34) (13)(24) (12) (34) QPRPQ, QPQSQPQ P, QR, QPQ, QSQ, SPS, SRQS, SQPQS, RSR, RQPR, PRPR, QRPR, SRPR, 40 (1) (13)(24) (13)(24) (12) SQSQS 140 (12)(34) (12)(34) (13)(24) (12) RPSPR P, QS, QPQ, QRQ, RPR, RSQR, SRS, SQPS, PSPS, QSPS, RSPS, SPRPS 41 (1) (13)(24) (12) (13)(24) RQPQR, RQRQR 141 (12)(34) (12)(34) (12) (13)(24) P, RS, RPR, RQR, QPQ, QSRQ, SQS, SRPS, PSPS, QSPS, RSPS, 42 (1) (12) (13)(24) (13)(24) QRPRQ, QRQRQ 142 (12)(34) (12) (12)(34) (13)(24) SPQPS P, QPQ, QSQ, SRQ, RPR, RSR, QRQR, QSQ, QRPQ, PQPQ, RQPQ, SQPQ, 43 (1) (13)(24) (14)(23) (12) QRPRQ 143 (12)(34) (13)(24) (12)(34) (12) QPSPQ QPSQ, QSRSQ, QSPRPSQ, P, QPQ, QRQ, RSQ, SPS, SRS, QSQS, QSQPQPSQ, QSQRQPSQ, 44 (1) (13)(24) (12) (14)(23) QSPSQ 144 (12)(34) (13)(24) (12) (12)(34) QSQSQPSQ P, SPS, SQS, SRQ, RSRS, SRPRS, RQR, RSPR, RPQPR, PRPR, QRPR, 45 (1) (12) (13)(24) (14)(23) SRSPSRS, SRSQSRS 145 (12)(34) (12) (13)(24) (12)(34) SRPR S, RPR, RSR, PRQR, QRQR, RQPQR, QR, QSQ, PQPQ, SQPQ, QPSPQ, 46 (12) (12)(34) (13)(24) (1) RQSQR, RQRSRQR 146 (12)(34) (13)(24) (13)(24) (12) QPRQPQ R, SPS, SRS, PSQS, QSQS, SQPQS, QS, QRQ, PQPQ, RQPQ, QPRPQ, 47 (12) (12)(34) (1) (13)(24) SQRQS, SQSRSQS 147 (12)(34) (13)(24) (12) (13)(24) QPSQPQ Q, SPS, SQS, PSRS, RSRS, SRPRS, RS, RQR, PRPR, QRPR, RPQPR, 48 (12) (1) (12)(34) (13)(24) SRQRS, SRSQSRS 148 (12)(34) (12) (13)(24) (13)(24) RPSRPR R, S, QPQ, QSQ, PRP, PSP, QRQP, RSR, RPQ, RQP, RQS, RPRQR, RPSPR 49 (12) (13)(24) (34) (1) QRPRQ, QRSRQ 149 (12)(34) (13)(24) (14)(23) (12) R, S, QPQ, QRQ, PRP, PSP, QSQP, SRS, SPQ, SQP, SQR, SPRPS, SPSQS 50 (12) (13)(24) (1) (34) QSPSQ, QSRSQ 150 (12)(34) (13)(24) (12) (14)(23) 11 Q, S, RPR, RQR, PQP, PSP, RSRP, SQS, SPR, SRP, SRQ, SPQPS, SPSRS 51 (12) (1) (13)(24) (34) RSPSR, RSQSR 151 (12)(34) (12) (13)(24) (14)(23) S, RP, QPQ, QSQ, PSP, QRQP, RQ, SQ, PQP, PRP, PSPQ, PSQSP, 52 (12) (13)(24) (12)(34) (1) QRPRQ, QRSRQ 152 (12)(34) (13) (13) (13) PSRSP R, SP, QPQ, QRQ, PRP, QSQP, S, RQ, PQP, PRP, QSQ, PSPQ, 53 (12) (13)(24) (1) (12)(34) QSPSQ, QSRSQ 153 (12)(34) (13) (13) (24) PSQSP, PSRSP Q, SP, RPR, RQR, PQP, RSRP, RSPSR, R, SQ, PQP, PSP, QRQ, PRPQ, 54 (12) (1) (13)(24) (12)(34) RSQSR 154 (12)(34) (13) (24) (13) PRQRP, PRSRP S, QR, QPQ, QSQ, PSP, PRQP, Q, SR, PRP, PSP, RPQP, RQR, 55 (12) (13)(24) (13)(24) (1) PQPQP, PQSQP 155 (12)(34) (24) (13) (13) PQRQP, PQSQP R, QS, QPQ, QRQ, PRP, PSQP, PS, RQ, PQP, PRP, QSPQ, QPQPQ, 56 (12) (13)(24) (1) (13)(24) PQPQP, PQRQP 156 (12)(34) (13) (13) (12)(34) QPRPQ Q, RS, RPR, RQR, PQP, PSRP, PRPRP, PR, SQ, PQP, PSP, QRPQ, QPQPQ, 57 (12) (1) (13)(24) (13)(24) PRQRP 157 (12)(34) (13) (12)(34) (13) QPSPQ S, QPQ, QSQ, PRQ, RPR, RSR, QRQR, SR, PQ, PRP, PSP, RQPR, RPRPR, 58 (12) (13)(24) (14)(23) (1) QRSRQ 158 (12)(34) (12)(34) (13) (13) RPSPR R, QPQ, QRQ, PSQ, SPS, SRS, QSQS, RQ, SQ, PQP, PRP, PSP, QPQPQ, 59 (12) (13)(24) (1) (14)(23) QSRSQ 159 (12)(34) (13) (13) (13)(24) QPRPQ, QPSPQ Q, RPR, RQR, PSR, SPS, SQS, RSRS, RQ, SQ, PQP, PSP, PRPQ, PRQRP, 60 (12) (1) (13)(24) (14)(23) RSQSR 160 (12)(34) (13) (13)(24) (13) PRSRP QP, RP, SPS, SQS, SRS, PSPSP, RQ, SQ, PRP, PSP, PQPQ, PQRQP, 61 (12) (12) (12) (13)(24) PSQSP, PSRSP 161 (12)(34) (13)(24) (13) (13) PQSQP QP, SP, RPR, RQR, RSR, PRPRP, PQP, PRP, QSP, RSP, SQS, SRS, PSPS 62 (12) (12) (13)(24) (12) PRQRP, PRSRP 162 (12)(34) (13) (13) (14)(23) RP, SP, QPQ, QRQ, QSQ, PQPQP, PQP, PSP, QRP, SRP, RQR, RSR, 63 (12) (13)(24) (12) (12) PQRQP, PQSQP 163 (12)(34) (13) (14)(23) (13) PRPR R, SPS, SQS, PSRS, QSRS, SRPRS, PRP, PSP, RQP, SQP, QRQ, QSQ, 64 (12) (12) (34) (13)(24) SRQRS, SRSRSRS 164 (12)(34) (14)(23) (13) (13) PQPQ S, QP, RPR, RQR, PSP, RSRP, RSPSR, R, PS, PQP, QRQ, PRPQ, PRSQ, 65 (12) (12) (13)(24) (34) RSQSR 165 (12)(34) (13) (24) (12)(34) PRQRP S, RP, QPQ, QRQ, PSP, QSQP, S, PR, PQP, QSQ, PSPQ, PSRQ, 66 (12) (13)(24) (12) (34) QSPSQ, QSRSQ 166 (12)(34) (13) (12)(34) (24) PSQSP RP, SP, QPQ, QSQ, QRQP, QRPRQ, S, PQ, PRP, RSR, PSPR, PSQR, PSRSP 67 (12) (13)(24) (12)(34) (12) QRSRQ 167 (12)(34) (12)(34) (13) (24) RP, SP, QPQ, QRQ, QSQP, QSPSQ, R, SQ, PQP, QRP, QRQ, PSRP, PRPQ 68 (12) (13)(24) (12) (12)(34) QSRSQ 168 (12)(34) (13) (24) (13)(24) QP, SP, RPR, RQR, RSRP, RSPSR, S, RQ, PQP, QSQ, PSRP, PRPQ, 69 (12) (12) (13)(24) (12)(34) RSQSR 169 (12)(34) (13) (13)(24) (24) PRQRP QR, SP, QPQ, QSQ, PRQP, PQPQP, S, RQ, PRP, QSQ, PSQP, PQPQ, 70 (12) (13)(24) (13)(24) (12) PQSQP 170 (12)(34) (13)(24) (13) (24) PQRQP QS, RP, QPQ, QRQ, PSQP, PQPQP, R, PQP, QSP, SRP, PRPSP, PRQRP, 71 (12) (13)(24) (12) (13)(24) PQRQP 171 (12)(34) (13) (24) (14)(23) PSRSP RS, QP, RPR, RQR, PSRP, PRPRP, S, PQP, QRP, RSP, PRPSP, PRSRP, 72 (12) (12) (13)(24) (13)(24) PRQRP 172 (12)(34) (13) (14)(23) (24) PSQSP QPQ, QSQ, PRQ, SRQ, RPR, RSR, S, QRQ, QPR, PSQ, QPQSQ, QPSPQ, 73 (12) (13)(24) (14)(23) (12) QRQR 173 (12)(34) (14)(23) (13) (24) QSRSQ QPQ, QRQ, PSQ, RSQ, SPS, SRS, PQ, PR, PSP, SQPS, SRPS, SPSPS 74 (12) (13)(24) (12) (14)(23) QSQS 174 (12)(34) (12)(34) (12)(34) (13) 75 (12) (12) (13)(24) (14)(23) RPR, RQR, PSR, QSR, SPS, SQS, RSRS 175 (12)(34) (12)(34) (13) (12)(34) PS, PQ, PRP, RQPR, RSPR, RPRPR Q, SPS, SRS, PSQS, RSQS, SQPQS, PR, PS, PQP, QRPQ, QSPQ, QPQPQ 76 (12) (34) (12) (13)(24) SQRQS, SQSQSQS 176 (12)(34) (13) (12)(34) (12)(34) Q, RPR, RSR, PRQR, SRQR, RQPQR, SR, PQ, PSP, PRPR, PRQR, PRSRP 77 (12) (34) (13)(24) (12) RQSQR, RQRQRQR 177 (12)(34) (12)(34) (13)(24) (13) R, SP, QPQ, QSQ, PRP, QRQP, SR, PQ, PRP, PSPR, PSQR, PSRSP 78 (12) (13)(24) (34) (12) QRPRQ, QRSRQ 178 (12)(34) (12)(34) (13) (13)(24) Q, R, SPS, SRQS, PSQS, SQPQS, PR, SQ, PQP, PSPQ, PSRQ, PSQSP 79 (12) (34) (34) (13)(24) SQSQSQS, SQSRSQS 179 (12)(34) (13) (12)(34) (13)(24) Q, S, RPR, RSQR, PRQR, RQPQR, PQ, PSP, SRP, RSR, PRPR, PRQR 80 (12) (34) (13)(24) (34) RQRQRQR, RQRSRQR 180 (12)(34) (12)(34) (14)(23) (13) R, S, QPQ, PRP, PSP, QSRQ, QRQP, PQ, PRP, RSP, SRS, PSPS, PSQS 81 (12) (13)(24) (34) (34) QRPRQ 181 (12)(34) (12)(34) (13) (14)(23) S, RPR, RSQR, PRQR, QRQR, RQPQR, PR, PQP, QSP, SQS, PSPS, PSRS 82 (12) (12)(34) (13)(24) (34) RQRSRQR 182 (12)(34) (13) (12)(34) (14)(23) R, SPS, SRQS, PSQS, QSQS, SQPQS, SQ, PR, PSP, PQPQ, PQRQ, PQSQP 83 (12) (12)(34) (34) (13)(24) SQSRSQS 183 (12)(34) (13)(24) (12)(34) (13) Q, SPS, SRQS, PSQS, RSQS, SQPQS, PS, RQ, PRP, PQPQ, PQSQ, PQRQP 84 (12) (34) (12)(34) (13)(24) SQSQSQS 184 (12)(34) (13)(24) (13) (12)(34) S, RP, QPQ, PSP, QSRQ, QRQP, PS, RQ, PQP, PRPQ, PRSQ, PRQRP 85 (12) (13)(24) (12)(34) (34) QRPRQ 185 (12)(34) (13) (13)(24) (12)(34) R, SP, QPQ, PRP, QSRQ, QRQP, RQ, SQ, PSP, PRQP, PQPQ, PQSQP 86 (12) (13)(24) (34) (12)(34) QRPRQ 186 (12)(34) (13)(24) (13)(24) (13) Q, RPR, RSQR, PRQR, SRQR, RQPQR, RQ, SQ, PRP, PSQP, PQPQ, PQRQP 87 (12) (34) (13)(24) (12)(34) RQRQRQR 187 (12)(34) (13)(24) (13) (13)(24) S, QR, QPQ, PSP, QSQP, QSRP, RQ, SQ, PQP, PSRP, PRPQ, PRQRP 88 (12) (13)(24) (13)(24) (34) QSPSQ 188 (12)(34) (13) (13)(24) (13)(24) R, QS, QPQ, PRP, QRQP, QRSP, PSP, QRP, RQP, SRP, PQPRP, 89 (12) (13)(24) (34) (13)(24) QRPRQ 189 (12)(34) (13)(24) (14)(23) (13) PQSQP Q, RS, RPR, PRQR, RQPQR, RQSRQR, PRP, QSP, RSP, SQP, PQPSP, PQRQP 90 (12) (34) (13)(24) (13)(24) RQRQRQR 190 (12)(34) (13)(24) (13) (14)(23) S, RPR, RQP, QSR, RQRSR, RQSQR, SQS, SPQ, SPR, SRP, SPSRS, SRQRS 91 (12) (13)(24) (14)(23) (34) RSPSR 191 (12)(34) (13) (13)(24) (14)(23) R, QPQ, PSQ, SRQ, QRPRQ, QRQSQ, PR, PSP, SQP, QSQ, PQPQ, PQRQ 92 (12) (13)(24) (34) (14)(23) QSRSQ 192 (12)(34) (14)(23) (12)(34) (13) Q, RPR, RQS, PSR, RQPQR, RQRSR, PS, PRP, RQP, QRQ, PQPQ, PQSQ 93 (12) (34) (13)(24) (14)(23) RSQSR 193 (12)(34) (14)(23) (13) (12)(34) SPS, SRQS, PSQS, QSQS, RSQS, PS, PQP, QRP, RQR, PRSR, PRPRPP 94 (12) (12)(34) (12)(34) (13)(24) SQPQS 194 (12)(34) (13) (14)(23) (12)(34) RPR, RSQR, PRQR, QRQR, SRQR, PSP, QRP, RQP, SQP, PQPRP, PRSRP 95 (12) (12)(34) (13)(24) (12)(34) RQPQR 195 (12)(34) (14)(23) (13)(24) (13) 96 (12) (13)(24) (12)(34) (12)(34) RP, SP, QPQ, QSRQ, QRQP, QRPRQ 196 (12)(34) (14)(23) (13) (13)(24) PRP, QSP, RQP, SQP, PQPSP, PSRSP
12 97 (12) (13)(24) (13)(24) (12)(34) QR, SP, QPQ, QSQP, QSRP, QSPSQ 197 (12)(34) (13) (14)(23) (13)(24) RQR, RPQ, RPRSR, RPS, PSR, RSQSR 98 (12) (13)(24) (12)(34) (13)(24) QS, RP, QPQ, QRQP, QRSP, QRPRQ 198 (12)(34) (14)(23) (14)(23) (13) RQ, PSP, SQP, QSQ, PRQP, PQPQ RS, SPS, SQPQS, SQSRQS, SRPRQS, SQ, PRP, RQP, QRQ, PSQP, PQPQ 99 (12) (12)(34) (13)(24) (13)(24) SRQRQS 199 (12)(34) (14)(23) (13) (14)(23) 100 (12) (12)(34) (13)(24) (14)(23) SPS, SQR, SRP, SRQ, SQPQS, SQSRS 200 (12)(34) (13) (14)(23) (14)(23) SR, PQP, QRP, RQR, PSRP, PRPR
Table 4. Colorings that will give rise to index 4 subgroups of H, where (H) ≅ D4.
generators for generators for no P Q R S the subgroup no P Q R S the subgroup fixing color 1 fixing color 1 P, PSP, PQPQP, PQRQP, PQSQP, QR, RPR, RSPSR, RSRSR, RQPQR, 1 (1) (12) (13) (12)(34) PRPRP, PRQRP, PRSPSRP, 122 (12) (34) (12) (13) RQSQR, RSQPQSR, RSQRQSR, PRSQSRP, PRSRSRP RSQSQSR SRP, PSP, PRPRP, PRQRP, P, RPR, RQR, RSR, QPS, QRQ, QSQ, PRSQSRP, PRSRSRP, PQPQP, 2 (1) (12) (12)(34) (13) SQS, SRS 123 (12) (12) (34) (13) PQRQP, PQSQP P, RPR, RQR, RSR, QPQ, QRS, QSQ, SR, SQS, SPRPS, SRPRS, SRQRS, 3 (1) (12)(34) (12) (13) SPS, SQS 124 (12) (13) (14) (12) SPSQPS, SPQPQPS, SPQRQPS P, PQP, PRP, PSPSP, PSRSP, RP, SPS, SQS, SRS, PQP, PSP, QPQ, PSQPQSP, PSQSRQSP, 4 (1) (12) (13) (13)(24) 125 (12) (13) (12) (14) QRQ, QSQ PSQRPRQSP, PSQRQRQSP P, PQP, PSP, PRPRP, PRQRP, QP, SPS, SQS, SRS, PRP, PSP, RPR, PRSQSRP, PRSRPSRP, 5 (1) (12) (13)(24) (13) 126 (12) (12) (13) (14) RQR, RSR PRSPQPSRP, PRSPSPSRP P, PQP, PSP, PRQRP, PRSPRP, RP, SPS, SQS, SRS, PQP, PSQ, QPQ, PRPRQPRP, PRPQPQPRP, 6 (1) (13)(24) (12) (13) 127 (12) (13) (12) (23) QRQ PRPQSQPRP SQS, QPS, SRQPS, SPSPS, P, SPS, SQS, SRS, QPR, QRQ, QSR, SPQPQPS, SPRPRPS, SPRQRPS, 7 (1) (12) (13) (14)(23) RQR 128 (12) (12)(34) (13)(24) (13) SPRSRPS P, PQP, PSP, PRPRP, PRQRP, PSP, PRPRP, PRSRP, PRQS, 8 (1) (12) (14)(23) (13) PRSPSRP, PRSRSRP, PRSQPQSRP, 129 (12) (12)(34) (13) (13)(24) PRQPQRP, PRQRQRP, PRQSQP, PRSQRQSRP, PRSQSQSRP PRQSRSQRP P, RPR, RQR, RSR, QPS, QRS, QSQ, SPS, RQS, SQPQS, SQSQS, SQRPRS, 9 (1) (14)(23) (12) (13) SQS 130 (12) (12)(34) (14)(23) (13) SQRQRS, SRSRS P, PQP, PRP, PSPSP, PSRSP, RPR, SQR, RQPQR, RQRQR, PSQPQSP, PSQSQSP, PSQRPRQSP, 10 (1) (12) (13) (24) 131 (12) (12)(34) (13) (14)(23) RQSPSR, RQSQSQR, RSQSR, RSRSR PSQRQRQSP, PSQRSRQSP QSR, QRQ, QPQPQ, QPRPQ, P, PQP, PRPRP, PRSRP, PSPQRP, QPSPQ, QSPSQ, QSQSQ, 11 (1) (13) (24) (12) PSQSP, PSRQRP, PRQSQRP 132 (12) (13) (12)(34) (14)(23) QSRPRSQ, QSRSRSQ P, PQP, PRP, PSPSP, PSQSP, QPQ, PQRQ, QRPQRQ, QRPRQ, PSRPRSP, PSRSQRSP, 12 (1) (24) (12) (13) 133 (12) (13)(24) (12)(34) (13) QRSRQ, QRQRPQRQ, QRQSQRQ PSRQPQRSP, PSRQRQRSP P, R, S, QRPQ, QPQSQ, QPSPQ, RS, SPS, SRQRS, SQPQS, SQRQS, 13 (1) (12) (34) (23) QSPSQ, QSRSQ 134 (12) (13)(24) (13) (12)(34) SQSPRS, SQSQSQS, SQSRSQS QRQ, QSQ, QPRS, QPQPQ, QPSPQ, P, R, S, QPQ, QSQ, QRQPRQ, QPRPRPQ, QPRQRPQ, 14 (1) (12) (23) (34) QRSPRQ, QRPRPRQ 135 (12) (13) (13)(24) (12)(34) QPRSPSRPQ, QPRSRSRPQ P, SPS, SQS, SRS, QPQ, QRQ, QSQ, RP, QPQ, QSQ, SRQ, PSP, QRQP, 15 (1) (23) (12) (13) RPR, RQR, RSR 136 (12) (14)(23) (12)(34) (13) QRPRQ P, R, S, QPQ, QRPSQ, QRQSQ, PS, SRS, SPS, SQPS, SPRPS, SQPQS, 16 (1) (12) (23) (24) QRSRQ, QSRSQ 137 (12) (14)(23) (13) (12)(34) SQRQS S, SPS, SQS, SRPRS, SRQRS, SP, RPR, RQR, QSR, PQP, RSRP, SRSQSRS, SRSRSRS, SRSPQPSRS, 17 (12) (13) (12)(34) (1) 138 (12) (13) (14)(23) (12)(34) RSPSR SRSPRPSRS, SRSPSPSRS R, SP, PQP, PRP, QPQ, QRQ, RQ, RPR, RQSQR, RQPQR, RSQSR, 18 (12) (13) (1) (12)(34) QSRSQ, QSPSQ, QSQSQ 139 (12) (13)(24) (13)(24) (13) RSRPSR, RSPQPSR, RSPSPSR Q, SP, PQP, PRP, RPR, RQR, RQ, SQ, QPQ, PRP, PQSP, PQPQP, 19 (12) (1) (13) (12)(34) RSQSR, RSPSR, RSRSR 140 (12) (13)(24) (13) (13)(24) PQRQP S, SPS, SQS, SRSRS, SRPQPRS, RQ, SQ, QPQ, PQP, PRSP, PRPRP, SRPRPRS, SRPSPRS, SRQPQRS, 20 (12) (13) (13)(24) (1) 141 (12) (13) (13)(24) (13)(24) PRQRP SRQRQRS, SRQSQRS RQ, RPR, RQPQR, RQSQR, R, PQP, PRP, SQS, QPQ, QRQ, RSQPSR, RSRSR, RSPRPSR, 21 (12) (13) (1) (13)(24) PSRSP, PSPSP, PSQSP 142 (12) (14)(23) (14)(23) (13) RSPSPSR Q, QPQ, QRQ, QSQSQ, QSRSQ, QS, QPQ, QRQ, PQR, PRP, PSP, QSPQPSQ, QSPSRPSQ, 22 (12) (1) (13) (13)(24) 143 (12) (14)(23) (13) (14)(23) RPR, RSR QSPRPRPSQ, QSPRQRPSQ S, SPS, SQS, SRQRS, SRSRS, RS, RPR, RQR, PQP, PRQ, PSQ, SRPRQPRS, SRPSPRS, 23 (12) (13) (14)(23) (1) 144 (12) (13) (14)(23) (14)(23) QPQ SRPQPQPRS, SRPQSQPRS R, SPS, SQS, SRS, PQP, PRP, PSQ, RPR, RSR, RQP, RQSQR, QRQR, 24 (12) (13) (1) (14)(23) QPQ, QRQ 145 (12) (13)(24) (14)(23) (13) SRQR, RQRPRQR Q, SPS, SQS, SRS, PQR, PRP, PSR, RQ, SPS, SRS, SQP, QPQ, QSQ, 25 (12) (1) (13) (14)(23) RPR 146 (12) (13)(24) (13) (14)(23) SQRQS, SQSQS S, SPS, SQS, SRQRS, SRSRS, SPS, SQS, SRP, SRQRS, QSRS, RSRS, SRPRQPRS, SRPSQPRS, 26 (12) (12)(34) (13) (1) 147 (12) (13) (13)(24) (14)(23) SRSPSRS SRPQPQPRS RPR, RQR, RSR, SP, PQP, PRQ, SR, QPQ, QRQ, PSQ, RPR, QSQR, 27 (12) (12)(34) (1) (13) QPQ, QSQ 148 (12) (14)(23) (13) (13)(24) QSRSQ Q, QPQ, QSQ, QRQPRQ, QRSRQ, SQ, RPR, RQR, PSR, QPQ, RSRQ, QRPSRPRQ, QRPRPRPRQ, 28 (12) (1) (12)(34) (13) 149 (12) (13) (14)(23) (13)(24) RSQSR QRPRQRPRQ S, SRS, SQPQS, SQRQS, SQSQS, P, S, QPR, QSRQ, RQRQ, RSR, SPRQPS, SPSPS, SPQPQPS, 29 (12) (13)(24) (13) (1) 150 (23) (12)(34) (13)(24) (24) QRPRQ SPQSQPS R, RSR, RQPQR, RQSQR, RPQPR, S, RQ, PQP, PRP, QPQ, QSQ, RPRPR, RPSPR, RQRPRQR, 30 (12) (13)(24) (1) (13) 151 (12) (13) (13) (24) PSPSP, PSQSP, PSRSP RQRQRQR, RQRSRQR Q, QSQ, QRPRQ, QRQRQ, QRSRQ, SQ, SQPQS, SQRQS, SPQRS, SPRPS, QPQPQ, QPSRPQ, QPRPRPQ, 31 (12) (1) (13)(24) (13) 152 (12) (13) (24) (13) SPSRS, SRPRS QPRQRPQ S, QPQ, QRQ, QSQ, PQR, PRP, PSP, Q, QPQ, QSRQ, QRQRQ, QRPSPRQ, 32 (12) (13)(24) (14) (1) RPR, RSR 153 (12) (24) (13) (13) QRPQPQPRQ, QRPQSQPRQ,
13 QRPRQPRQ
R, SP, QPQ, QSQ, PQP, PRP, S, QR, QPQ, PQP, PRP, PSP, 33 (12) (13)(24) (1) (14) QRPRQ, QRQRQ, QRSRQ 154 (12) (13) (13) (34) QSPSQ, QSQSQ, QSRSQ Q, RPR, RQR, RSR, PQS, PRS, PSP, RS, RPR, RQRP, RQRQ, RQPQR, 34 (12) (1) (13)(24) (14) SPS 155 (12) (13) (34) (13) RQSQR, RQRSRQR QP, SP, PRP, RPR, RQR, RSPSR, RS, SPS, SQPQS, SQRQS, SQSQS, 35 (12) (12) (13) (12)(34) RSQSR, RSRSR 156 (12) (34) (13) (13) SRQPRS, SRPRPRS, SRPSPRS S, SQS, SPSPSPS, SPSQSPS, RP, SPS, SQS, SRQ, PQP, PSP, SPSRSPS, SPRPS, SRQPS, 36 (12) (12) (12)(34) (13) SRPRS, SRSRS 157 (12) (13) (14) (23) SPQPQPS, SPQSQPS SPS, RS, SQRPRQS, SQRQRQS, Q, SPS, SQS, SRS, PQR, PRP, PSP, SQRSRQS, SQPQS, SQSQS, SRPRS, 37 (12) (12)(34) (12) (13) 158 (12) (23) (13) (14) RPR, RSR SRQRS PQ, SR, PRP, RPR, RQR, PSPSP, R, QPQ, QRQ, QSQ, SPS, SQS, SRP, 38 (12) (12) (13) (13)(24) PSQSP, PSRSP 159 (12) (13) (24) (14) SRQRS, SRSRS S, SPS, SQPQS, SQRQS, SRSQS, RS, RPR, RQR, PQP, PRP, PSQ, SQSQSQS, SQSPQPSQS, 39 (12) (12) (13)(24) (13) QPQ, QRQ 160 (12) (13) (14) (34) SQSPRPSQS, SQSPSPSQS PR, RSR, RQRPRQR, RQRQRQR, SPS, RQS, SRQRS, SRSRS, SQPQS, RQRSRQR, RQPQR, RQSQR, 40 (12) (13)(24) (12) (13) 161 (12) (13) (34) (14) SQSQS, SQRPRS, SQRQRQS RPQPR, RPSPR Q, QSQ, QRPRQ, QPSRQ, QP, SPS, SQS, SRS, PRP, PSR, RPR, QRSQSRQ, QRSRSRQ, QRQPQRQ, 41 (12) (12) (13) (14)(23) RQR 162 (12) (34) (13) (14) QRQRQRQ, QRQSQRQ SPS, SQS, SRQ, PSRS, RSRS, SRPRS, R, S, PQ, PSPSP, PSRSP, PSQPQSP, 42 (12) (12) (14)(23) (13) SRSQSRS 163 (12) (13) (23) (34) PSQRP, PSQSQSP SRQ, QSQ, QPQPQ, QPRPQ, RP, SP, QPQ, QSQ, PQP, QRPRQ, QPSPQ, QRPRQ, QRQRQ, 43 (12) (14)(23) (12) (13) QRQRQ, QRSRQ 164 (12) (24) (34) (13) QRSPSRQ, QRSRSRQ RP, SP, QPQ, QRQ, PQP, QSPSQ, SPS, SRS, SQPQS, SQR, PSQS, 44 (12) (13) (12) (12)(34) QSQSQ, QSRSQ 165 (12) (24) (13) (34) QSQS, SQSRSQS PQ, QRQ, QSRPRSQ, QSRQRSQ, Q, R, SPS, PQP, PRP, PSP, SQPQS, QSRSRSQ, QSPSQ, QSQSQ, 45 (12) (13) (12)(34) (12) 166 (12) (34) (34) (13) SQRS, SQSQS QPRPQ, QPSPQ QS, SPS, SRS, SQPQS, SQRQRQS, SP, RPR, RQR, RSR, PQP, PRP, SQRSPRQS, SQRPQPRQS, 46 (12) (12)(34) (13) (12) QPQ, QRQ, QSQ 167 (12) (34) (13) (34) SQRPRPRQS PR, RQR, RSPSR, RSQSR, RSRSR, R, SP, PQP, PRP, QPQ, QRSQ, RPSPR, RPQPQPR, RPQRQPR, 47 (12) (13) (12) (13)(24) 168 (12) (13) (34) (34) QRPRQ, QRQRQ RPQSQPR SQP, SPS, SRS, SQRPRQS, Q, R, S, PQP, PRP, PSQPSP, SQRQRQS, SQRSRQS, SQSQS, 48 (12) (13) (13)(24) (12) 169 (12) (34) (34) (23) PSRPSP, PSPSPSP SQPQPQS, SQPRPQS PS, RQ, PRP, QPQ, QSQ, PQPQP, Q, S, RPR, RSR, RQSP, RQPQR, 49 (12) (13)(24) (13) (12) PQRQP, PQSQP 170 (12) (34) (13) (24) RQRQR, RQSQSQR, RQSRSQR SPS, SRS, SQP, SQRQS, QSQS, RS, RPR, RQR, PQP, PRP, PSP, 50 (12) (13) (12) (14)(23) RSQS, SQSPSQS 171 (12) (13) (13) (14) QPQ, QRQ, QSQ PS, SQS, SRPRS, SRQRS, SRSRS, RQ, SPS, SQS, SRS, PQP, PRP, PSP, SPRPS, SPQPQPS, SPQRQPS, 51 (12) (13) (14)(23) (12) 172 (12) (13) (14) (13) QPQ, QSQ SPQSQPS SP, QPQ, QRQ, QSQ, PQR, PRP, RS, RPR, RQR, PQP, PRQ, PSP, 52 (12) (14)(23) (13) (12) RPR, RSR 173 (12) (14) (13) (13) QPQ, QSQ QR, RPR, RSPSR, RSQSR, RSRSR, P, Q, S, RSQR, RPQPR, RPRPR, 53 (12) (13) (13) (12)(34) RQSPQR, RQPQPQR, RQPRPQR 174 (23) (24) (14) (1) RPSPR, RQPQR, RQRQR QS, QPQ, QRPRQ, QRQRQ, QRSP, P, R, S, QPQ, QRQ, QSQPSQ, 54 (12) (13) (12)(34) (13) QRSR, QRSQSRQ 175 (23) (12) (1) (24) QSRSQ, QSPRPSQ, QSPSPSQ RS, SPS, SQPQS, SQRQS, SQSQS, P, Q, R, SQS, SRS, SPQPS, SPSPS, SRQRS, SRPQPRS, SRPRPRS, 55 (12) (12)(34) (13) (13) 176 (23) (1) (24) (13) SPRPRPS, SPRQRPS, SPRSRPS SRPSPRS RQ, SQ, PQP, PRP, QPQ, PSPSP, P, R, S, QPQ, QSQ, QRPRQ, 56 (12) (13) (13) (13)(24) PSQSP, PSRSP 177 (23) (12) (24) (1) QRSQRQ, QRQPQRQ, QRQRQRQ SQS, PS, SRPRS, SRQRS, SRSPSRS, P, R, S, QRPQ, QSQ, QPSQPQ, 57 (12) (13) (13)(24) (13) SRSQPS, SRSRPS 178 (23) (12) (23) (34) QPQPQPQ, QPQRQPQ RQ, SQ, PRP, PSP, QPQ, PQPQP, P, R, S, QRQ, QSQ, QPRPQ, 58 (12) (13)(24) (13) (13) PQRQP, PQSQP 179 (23) (12) (34) (24) QPSQPQ, QPQPQPQ, QPQRQPQ RQ, SPS, SQS, SRS, PQP, PRP, PSQ, P, R, S, QPQ, QRQ, QSQSQ, 59 (12) (13) (13) (14)(23) QPQ 180 (23) (12) (24) (34) QSRPSQ, QSPQPSQ, QSPSPSQ QS, SPQPQPS, SPQRPS, SPQSQPS, P, R, S, QPQ, QRQ, QSQPSQ, 60 (12) (13) (14)(23) (13) SPSPS, SRQS, SQPQS 181 (23) (12) (23) (24) QSRPSQ, QSPSPSQ SR, QPQ, QRQ, QSQ, PQR, PRP, P, R, S, QPQ, QSQ, QRQPRQ, 61 (12) (14)(23) (13) (13) PSP, RPR 182 (23) (12) (24) (23) QRSRQ, QRPRPRQ, QRPSPRQ RQP, PRP, PSP, PQSPSQP, S, QP, RQR, RSR, PRP, PSP, PQSQSQP, PQSRSQP, PQPQP, 62 (12) (13) (14) (12)(34) 183 (12)(34) (12) (13) (1) RPQPR, RPRPR, RPSPR PQRQRQP, PQRSRQP R, RSR, RPQPR, RPRPR, RPSPR, QP, RP, PSP, SPS, SQS, SRPRS, RQRQR, RQSPQR, RQPQPQR, 63 (12) (13) (12)(34) (14) SRQRS, SRSRS 184 (12)(34) (12) (1) (13) RQPRPQR QPQ, QRQ, QSR, QSQPQSQ, Q, RP, SQS, SRS, PQP, PSP, SPQPS, QSQRQSQ, QSQSQSQ, QSPSQ, 64 (12) (12)(34) (13) (14) 185 (12)(34) (1) (12) (13) SPRPS, SPSPS QSRPRSQ, QSRSRSQ SQ, RPR, RQR, PSR, QPQ, QRQ, S, SRS, SPQPS, SPRPS, SPSPS, 65 (12) (13) (14) (13)(24) RSQSR, RSRSR 186 (12)(34) (13) (12) (1) SQRPQS, SQSPQS, SQPQPQS R, RSR, RPQPR, RPRPR, RPSPR, RQ, QPQ, QSQ, SPS, SQS, SRP, RQRPQR, RQSQR, RQPQPQR, 66 (12) (13) (13)(24) (14) SRQRS, SRSRS 187 (12)(34) (13) (1) (12) RQPSPQR SPQ, QSQ, QPRPQ, QRSPQ, Q, SP, RQR, RSR, PQP, PRP, QPQPQPQ, QPQRQPQ, QPQSQPQ, 67 (12) (13)(24) (13) (14) 188 (12)(34) (1) (13) (12) RPQPR, RPRPR, RPSPR QPSPSPQ SQP, SPS, SQSPSQS, SQSQSQS, S, PQP, PRP, PSP, QPR, QRQ, QSR, SQSRSQS, SQRQS, SQPQPQS, 68 (12) (13) (14) (14)(23) 189 (12)(34) (13) (14) (1) RQR SQPRS QS, SRS, SPQPS, SPSPS, SQPQS, R, PQP, PSP, PRP, QPS, QRQ, QSQ, SQRQS, SPRPRPS, SPRQRPS, 69 (12) (13) (14)(23) (14) 190 (12)(34) (13) (1) (14) SQS, SRS SPRSRPS QRP, QPQ, QRPSQ, QRQPQRQ, Q, PRP, PSP, PQP, RPS, RQR, RSR, QRQRQRQ, QRQSQRQ, QRSRQ, 70 (12) (14)(23) (13) (14) 191 (12)(34) (1) (13) (14) SQS, SRS QRPRPRQ QP, QRQ, QSQ, QSQ, QPRPQ, R, PS, PQP, PRSQ, PRPRP, PRQRP, QSQSQ, QSRSQ, QSPQPSQ, 71 (12) (13) (24) (12)(34) PRSPSRP, PRSRSRP 192 (12)(34) (12) (12) (13) QSPRPSQ S, SPS, SQS, SRPRS, SRSQRS, QP, SP, PRP, PRP, PRQRP, PRSRP, SRQRPQRS, SRQPQPQRS, 72 (12) (13) (12)(34) (24) 193 (12)(34) (12) (13) (12) PRPQPRP, PRPSPRP SRQPSPQRS Q, S, RQP, PSQP, PQPQP, PRPRP, SR, PQP, PRP, PSP, QPR, QRQ, 73 (12) (12)(34) (13) (24) PRQRP, PRSRP 194 (12)(34) (13) (12) (12) QSQ, RQR
14 R, RSR, RPRPRPR, RPRQRPR, PSP, PSPQPSP, PSPRPSP, 74 (12) (13) (24) (13)(24) RPRSPR, RPQPR, RQPQR, RQRQR, 195 (12)(34) (13) (13) (12) PSPSPSP, PSQPRSP, PSQRQSP, RQSQR PSQSQSP, PSRQRSP, PSRSRSP S, RPR, RQR, RSR, PQP, PRQ, PSP, QS, RP, QRQ, PQP, PSP, QPQPQ, 75 (12) (13) (13)(24) (24) QPQ, QSQ 196 (12)(34) (13) (12) (13) QPRPQ, QPSPQ S, RQ, PQSP, PRP, QPQ, QSQ, RS, QP, RQR, PRP, PSP, RPQPR, 76 (12) (13)(24) (13) (24) PQPQP, PQRQP 197 (12)(34) (12) (13) (13) RPRPR, RPSPR SRQ, QSQ, QRQRQ, QPSRQ, R, SPS, SQS, SRP, SRQRS, SSQSRS, QRPQPRQ, QRPRPRQ, QRPSPRQ, 77 (12) (13) (24) (14)(23) SRSPSRS, SRSRSRS 198 (12)(34) (12) (13) (14) QRSRSRQ S, SPS, SQS, SRQRS, SRPQSRS, SQP, PSP, PQRQP, PRSQP, 78 (12) (13) (14)(23) (24) SRPRPRS, SRPSPRS, SRSPSRS, 199 (12)(34) (14) (12) (13) PQPQPQP, PQPRPQP, PQPSPQP, SRSRSRS PQSQSQP S, SPS, SRQS, SQPQSQS, SQPRSQS, Q, R, PQP, PRS, PSP, SQPS, SPRPS, 79 (12) (14)(23) (13) (24) SQPSPQS, SQSPSQS 200 (12)(34) (34) (23) (13) SPSPS R, RSR, RPQPR, RPRPR, RPSPR, S, RQR, RSR, RPSP, RPSQ, RPQPR, RQPQR, RQRPRQR, RQRQRQR, 80 (12) (13) (34) (12)(34) 201 (12)(34) (12) (13) (24) RPRPR, RPSRSPR RQRSQR SQS, SQPQPQS, SQPRPQS, S, RP, QPQ, PQP, PSP, QRSQ, SQPSQS, SQRQPRQS, SQRSPRQS, 81 (12) (13) (12)(34) (34) QRPRQ, QRQRQ 202 (12)(34) (12) (13) (34) SQRPRPRQS S, SRS, SPSPS, SPQS, SPRPRPS, RQR, RQPQPQR, RQPRQR, 82 (12) (12)(34) (13) (34) SPRQSRPS, SPRQPQRPS, 203 (12)(34) (12) (34) (13) RQPSPQR, RQSQPSQR, SPRQRQRPS RQSRPSQR, RQSPSPSQR PRP, PRSRP, PRPQPRP, PRPRPRP, R, RQR, RSPSR, RSQSR, RSRPRSR, PRPSPQRP, PRPSQSPRP, 83 (12) (13) (34) (13)(24) RSRSQRSR, RPQRSR, RSRQRQRSR 204 (12)(34) (13) (12) (24) PRPSRQRP, PRQSQRP S, SQS, SRPRS, SRQRS, SRSPSRS, R, RSR, RPQPR, RPSPR, RQRQR, 84 (12) (13) (13)(24) (34) SRSQPS, SRSRPS, SPSPS 205 (12)(34) (13) (24) (12) RPRPQR, RPRSQR, RPRQRPR QRQ, QSPR, RQSQ, QSPSQ, Q, S, PRP, PSR, RPQP, RQR, 85 (12) (13)(24) (13) (34) QSRSQ, QSQPQSQ, QSQSRQSQ 206 (12)(34) (24) (13) (23) PQRQP, PQSQP RP, RQR, RPSR, RPQPSQPR, R, SPS, SQS, SRQ, SRPRS, PSRS, RPQRQPR, RPQPQPQPR, 86 (12) (13) (34) (14)(23) SRSQSRS, SRSRSRS 207 (12)(34) (13) (12) (34) RPQPRPQPR S, RPR, RQR, QSR, PQP, PSP, RQ, PQP, PRP, PSP, QPS, QSQ, 87 (12) (13) (14)(23) (34) RSPSR, RSRP 208 (12)(34) (13) (13) (14) SQS, SRS S, QPQ, QRQ, RSQ, PRP, PSP, SQ, PQP, PRP, PSP, QPR, QRQ, 88 (12) (14)(23) (13) (34) QSPSQ, QSQP 209 (12)(34) (13) (14) (13) RQR, RSR R, RSR, RQR, RPRPRPR, RPRQRPR, SR, PQP, PRP, QSP, RQR, PSPR, RPRSRPR, RPSQPR, RPQPQPR, 89 (12) (13) (24) (1) 210 (12)(34) (14) (13) (13) PSRSP RPQRQPR SP, SQS, SRS, SPQRPS, SPQSQPS, R, S, PQP, PRP, QPQ, QRQ, QSQ, SPQPQPQPS, SPQPRPQPS, 90 (12) (13) (1) (24) PSPSP, PSQSP, PSRSP 211 (12)(34) (13) (14) (12) SPQPSPQPS Q, S, PQP, PRP, RPR, RQR, RSR, Q, R, SP, PQPRP, PQRQP, PQSQP, 91 (12) (1) (13) (24) PSPSP, PSQSP, PSRSP 212 (12)(34) (23) (24) (12) PRQRP, PRSRP R, S, QPQ, QSQ, PQP, PRP, PSP, Q, R, PQS, PSP, SPRP, SRS, PRQRP, 92 (12) (13) (34) (1) QRPRQ, QRQRQ, QRSRQ 213 (12)(34) (23) (24) (13) PRSRP S, SQS, SRPRS, SRSRS, SRQPQRS, Q, R, PRS, PSP, SPQP, SQS, 93 (12) (13) (1) (34) SRQSRQRS, SPRQRS, SRQRQRQRS 214 (12)(34) (23) (24) (14) PQRQP, PQSQP Q, S, RPR, RQR, PQP, PRP, PSP, SP, SQS, SPQPS, SPRPS, SRSRS, 94 (12) (1) (13) (34) RSPSR, RSQSR, RSRSR 215 (12)(34) (12) (13) (12)(34) SRQPRS, SRPRPRS, SRPSPRS Q, S, RPR, RSR, PQP, PRP, PSP, RP, RQR, RPQPR, RSRSR, RPSPR, 95 (12) (34) (13) (1) RQPQR, RQRQR, RQSQR 216 (12)(34) (12) (12)(34) (13) RSQPSR, RSPRPSR, RSPSPSR Q, R, SPS, SRS, PQP, PRP, PSP, QP, RP, PSP, SRS, SPQS, SPRPS, 96 (12) (34) (1) (13) SQPQS, SQRQS, SQSQS 217 (12)(34) (12)(34) (12) (13) SPSPS Q, R, SPS, SQS, SRPRS, SRSRS, RS, RQR, RPSP, RPSQ, RPQPR, 97 (12) (1) (34) (13) PQRS, SRQRQRS, SRQSQRS 218 (12)(34) (12) (13) (13)(24) RPRPR, RPSRSPR S, SPS, SRS, SQSPSQS, SQSQSQS, RS, RQR, RPRP, RPRQ, RPQPR, SQSRSQS, SQRQS, SQPQPQS, 98 (12) (13) (14) (1) 219 (12)(34) (12) (13)(24) (13) RPSPR, RPRSRPR SQPRPQS, SQPSPQS QR, RPR, RSRPRSR, RSRQRSR, QS, QRQ, QPQP, QPQR, QPRPQ, RSRSRSR, RSPSR, RSQSR, RQPQR, 99 (12) (13) (1) (14) 220 (12)(34) (13)(24) (12) (13) QPSPQ, QPQSQPQ RQSQR Q, SPS, SQS, SRS, PQP, PRP, PSP, Q, R, PQP, PRS, SPSP, SQSP, PSRSP 100 (12) (1) (13) (14) RPR, RQR, RSR 221 (12)(34) (34) (23) (13)(24) Q, SPS, SQS, SRS, PQP, PRP, PSR, Q, S, PQP, PSR, RPRP, RQRP, 101 (12) (1) (13) (23) RPR, RQR 222 (12)(34) (34) (13)(24) (23) PRSRP QP, RP, PSP, SPS, SQPQS, SQRS, R, S, PRP, PSQ, QPQP, QRQP, 102 (12) (12)(34) (12)(34) (13) SQSQS 223 (12)(34) (13)(24) (34) (23) PQSQP QP, SP, PRP, RPR, RQSR, RQPQR, Q, R, SP, PRP, PQRPQP, PQSQP, 103 (12) (12)(34) (13) (12)(34) RQRQR 224 (12)(34) (23) (34) (12)(34) PQPQPQP, PQPSPQP RP, SP, PQP, QPQ, QRSQ, QRPRQ, Q, RP, S, PSP, PQRPQP, PQSPQP, 104 (12) (13) (12)(34) (12)(34) QRQRQ 225 (12)(34) (23) (12)(34) (34) PQPQPQP Q, QSQ, QRPRQ, QPQPQ, QPRPQ, QP, R, S, PSP, PRQPRP, PRSPRP, QPSPQ, QRQPQRQ, QRQRQRQ, 105 (12) (34) (13) (12)(34) 226 (12)(34) (12)(34) (23) (34) PRPRPRP QRQSRQ Q, RPR, RQR, RSR, PQP, PRS, PSP, Q, R, PRP, PSQP, SPQP, SQS, SRQP 106 (12) (34) (12)(34) (13) SPS, SQS 227 (12)(34) (24) (34) (13)(24) QS, SPS, SRPRS, SRQRS, SRSRS, Q, S, PQR, PSP, RPRP, RSRP, SQRQS, SQPQPQS, SQPRPQS, 107 (12) (12)(34) (34) (13) 228 (12)(34) (24) (13)(24) (34) PRQRP SQPSPQS RPR, RSPSR, RSRSR, RSQRQSR, R, S, PRQP, PSP, QPQP, QRQ, RSQSQSR, RQPQSR, PQSR, 108 (12) (34) (13) (13)(24) 229 (12)(34) (13)(24) (24) (34) QSQP RSQPSPQSR Q, QSQ, QRPRQ, QRQRQ, QRSRQ, 109 (12) (34) (13)(24) (13) QPRPQ, QPSPQ, QPQPQPQ, 230 (12)(34) (24) (34) (14)(23) Q, R, PQS, PRP, SPSP, SRSP, PSQSP QPQRQPQ, QPQSQPQ R, RSR, RQPQR, RQSQR, Q, S, PRQP, PSP, RPQP, RQR, RQRPRQR, RQRQPR, RQRSPR, 110 (12) (13)(24) (34) (13) 231 (12)(34) (24) (14)(23) (34) RSQP RPRPR PS, PRP, PQPQP, PQRQP, R, S, PRQ, PSP, QPQP, QSQP, 111 (12) (34) (14) (13)(24) PQSPSQP, PQSQPP, PQSR 232 (12)(34) (14)(23) (24) (34) PQRQP Q, QPQ, QSQ, QRQPRQ, Q, R, SP, PQPRP, PQRQP, PQSRP, QRPRSRQ, QRPSPRQ, QRSPSRQ, 112 (12) (34) (13)(24) (14) 233 (12)(34) (23) (24) (12)(34) PRQRP QRSQSRQ R, QPQ, QSQ, SRQ, PRP, PSP, Q, RP, S, PQPSP, PQRSP, PQSQP, 113 (12) (13)(24) (34) (14) QRQP, QRPRQ 234 (12)(34) (23) (12)(34) (24) PSQSP P, Q, RQS, RSPR, SPS, SRS, RPQPR, QP, R, S, PRPSP, PRQSP, PRSRP, 114 (12) (23) (12)(34) (13) RPRPR 235 (12)(34) (12)(34) (23) (24) PSRSP R, QPQ, QRQ, QSQ, PQS, PRP, PSP, Q, R, PRS, PSQP, SPQP, SQS, 115 (12) (13) (24) (12) SPS, SRS 236 (12)(34) (24) (23) (13)(24) PQRQP PR, RQR, RSPSR, RSQSR, RSRSR, Q, S, PRQP, PSR, RPQP, RQR, 116 (12) (13) (12) (24) RPSQPR, RPQPQPR, RPQRQPR 237 (12)(34) (24) (13)(24) (23) PQSQP
15 PQ, QRQ, QSPSQ, QSQSQ, QSRSQ, R, S, PRQP, PSQ, QPQP, QRQ, QPSPQ, QPRPRPQ, QPRQRPQ, 117 (12) (12) (13) (24) 238 (12)(34) (13)(24) (24) (23) PQSQP QPRSRPQ S, SRQS, SQPQS, SQSQS, SRPRS, Q, R, PQS, PRP, PSP, SRPS, SPQPS, 118 (12) (13) (34) (12) SRQRS, SPSRS, SRSQSRS 239 (12)(34) (24) (23) (14)(23) SPSPS S, RP, PQP, PSP, SQS, SQPQS, Q, S, PQR, PSRP, RPRP, RSR, 119 (12) (13) (12) (34) SQRQS, SQSPSQS, SQSRSQS 240 (12)(34) (24) (14)(23) (23) PRQRP S, QP, PRP, PSP, SRS, SRPRS, R, S, PRQ, PSQP, QPQP, QSQ, 120 (12) (12) (13) (34) SRQRS, SRSPSRS, SRSQSRS 241 (12)(34) (14)(23) (24) (23) PQRQP PS, SQS, SPQPS, SPRPS, SQRPRQS, 121 (12) (34) (13) (12) SQRQS, SQRSRQS, SQRQPQRQS, SQRQSQRQS
Table 5. Colorings that will give rise to index 4 subgroups of H, where (H) ≅ S4.
index PQ QR RS PR PS QS generators 2 (1) (1) (12) (1) (12) (12) QP, RP, SRSP 3 (123) (132) (23) (1) (23) (13) RP, SP, QRPQ 4 (234) (143) (13) (123) (23) (34) QP, SP, RSRP
Table 6. Low index subgroups of a tetrahedron Kleinian group K realized by the tetrahedron t10.
Number of Subgroups Number of Subgroups of of Tetrahedron Tetrahedron Groups Coxeter tetrahedron Kleinian Groups Index Index Index Index Index Index 2 3 4 2 3 4 t1 = [3, 5, 2, 3, 2, 2] 3 1 1 1 1 0 t2 = [5, 3, 5, 2, 2, 2] 1 0 0 0 0 0 t3 = [3, 5, 3, 2, 2, 3] 1 0 0 0 0 0 t4 = [3, 5, 3, 2, 2, 2] 1 0 0 0 0 0 t5 = [3, 4, 3, 2, 2, 3] 1 0 0 0 0 0 t6 = [3, 5, 3, 2, 2, 4] 1 0 0 0 0 0 t7 = [4, 3, 5, 2, 2, 2] 3 0 1 1 0 0 t8 = [4, 3, 4, 2, 2, 3] 3 1 4 1 1 3 t9 = [5, 3, 5, 2, 2, 3] 1 0 0 0 0 0 t10 = [3, 3, 6, 2, 2, 2]; 1 3 1 2 1 1 1 t11 = [4, 4, 3, 2, 2, 2]; 1 7 1 8 2 1 4 t12 = [3, 3, 3, 3, 2, 2]; 1 1 1 5 0 1 1 t13 = [4, 3, 6, 2, 2, 2]; 1 7 2 9 3 2 4 t14 = [3, 4, 2, 4, 2, 2]; 1 7 1 10 3 1 6 t15 = [3, 6, 3, 2, 2, 2]; 2 3 2 5 1 3 0 t16 = [5, 3, 6, 2, 2, 2]; 1 3 0 1 1 0 0 t17 = [4, 3, 3, 3, 2, 2]; 1 1 1 3 0 1 4 t18 = [3, 6, 2, 3, 2, 2]; 2 3 4 3 1 4 2 t19 = [4, 4, 4, 2, 2, 2]; 3 15 0 35 7 0 29 t20 = [6, 3, 6, 2, 2, 2]; 2 7 3 49 3 3 3 t21 = [3, 4, 4, 2, 2, 3]; 1 3 1 4 1 1 5 t22 = [5, 3, 3, 3, 2, 2]; 1 1 1 1 0 1 1 t23 = [3, 3, 6, 2, 2, 3]; 2 1 1 3 0 1 4 t24 = [3, 3, 3, 3, 2, 3]; 2 1 4 4 0 4 5 t25 = [4, 4, 2, 4, 2, 2]; 2 15 0 35 7 0 31 t26 = [6, 3, 3, 3, 2, 2]; 3 1 1 51 0 1 4 t27 = [4, 3, 6, 2, 2, 3]; 2 3 4 5 1 4 5 t28 = [4, 4, 4, 2, 2, 3]; 2 7 1 14 3 1 2 t29 = [5, 3, 6, 2, 2, 3]; 2 1 0 0 0 0 0
16 Number of Subgroups Number of Subgroups of of Tetrahedron Tetrahedron Groups Coxeter tetrahedron Kleinian Groups Index Index Index Index Index Index 2 3 4 2 3 4 t30 = [6, 3, 6, 2, 2, 3]; 4 3 5 6 1 5 6 t31 = [4, 4, 4, 2, 2, 4]; 4 15 0 35 7 0 49 t32 = [3, 3, 3, 3, 3, 3]; 4 1 13 86 0 13 6 Table 7. The number of index 2, 3, 4 subgroups of tetrahedron and tetrahedron Kleinian groups generated by Coxeter tetrahedra of finite volume (up to conjugacy).
17