Cubic Space Groups

The 36 cubic crystallographic space groups are different from the remaining 194 space groups in that they each have body diagonal 3-fold axes arising from their tetrahedral and octahedral point groups. These body diagonal 3-fold axes make their orbifolds a less understood topology problem in that the Siefert fibered spaces approach of lifting from a base Euclidean 2-orbifold is inapplicable since fibration along the required orthogonal projections become tangled together by the 3-fold axes. On the other hand, many aspects of the cubic groups orbifolds are more straightforward than for the simpler space groups derived from cyclic and dihedral point groups.

Each cubic group has an index-4 rhombohedral based trigonal subgroup and an index-3 orthorhombic or tetragonal subgroup. There are only seven rhombohedral space groups: R3 (3), R32 (32), Rm and Rc (m), R3m and R3c (3m), and R () where the symbol in parentheses is the corresponding trigonal point group. Thus it seemed a reasonable approach to order the cubic groups in columns according to their index-4 rhombohedral subgroups and in rows according to their index-2 cubic subgroups. After some additional partitioning of Rm and Rc columns we arrived at the subgroup graph shown in Fig 6.1. We thank John H. Conway of Princeton for an e-mail exchange explaining his "odd-subroutine approach" to the group classification problem which he has applied to a number of group classification problems including the crystallographic space groups.

Fig. 6.1 uses the subgroup, group normalizer, and lattice complex information given in the International Tables for Crystallography, Volume A (ITCr) (Hahn, 1995). Each box contains the cubic space group symbol in the upper right subbox, the index-3 orthorhombic or tetrahedral subgroup in the middle, and their respective ITCr sequence numbers on the bottom line. The upper left subbox contains the simplest lattice complex of the cubic space group. Boldface type identifies group normalizers, and group normalizer basins are identified by bold solid lines leading down from cubic (but not orthorhombic) group normalizers.

The seven rhombohedral trigonal subgroups of the cubic groups are shown in the bottom row of the figure with their space group symbols and simplest lattice complex in the top row of each box. The index-3 subgroups (monoclinic/triclinic) of the rhombohedral groups are indicated on the middle line and the respective ITCr numbers on the bottom line of each box. The divider strip between the cubic and rhombohedral groups gives the point groups for all the groups involved in each column, with the cubic/orthorhombic (or tetragonal) to the left and the rhombohedral/monoclinic (or triclinic) to the right.

Figure 6.1. Cubic, Rhombohedral, and Certain Orthorhombic/Tetragonal Space Groups and Subgroups.

Two boxes in a row that are not separated by a space belong to a specific column. To minimize clutter in the drawing we use the convention that whenever a subgroup connection line goes to the midline separating adjacent boxes, both boxes are involved in the subgroup relation. If that line goes to another pair of adjoined boxes, the right goes to the right and the left to the left EXCEPT when there is a loop in the subgroup relation line, which indicates a right-left interchange. All solid lines join order-2 subgroups and pertain to the cubic, orthorhombic/tetragonal, rhombohedral, and monoclinic/triclinic sets of groups individually. We were previously unaware of that generalization, which suggests some classification possibilities. However, as detailed below, the orthorhombic space group set has some duplication.

The dashed lines leading to a dashed box two levels further down is an index-4 subgroup relationship. Note that each dashed box is a repeat of the regular box three levels up in the same column, reflecting the Bravais lattice repetition I, P, F, I, P which occurs in each column as denoted by the initial letters in the space group symbols. This relationship only holds for the cubic space groups in the figure and not for the orthorhombics/tetragonals. Since Fig. 6.1 is meant to be used mainly for orbifold applications, it does not include explicit information on how many unit cells are required for each space group/subgroup relationship.

Note that for order-2 subgroups, there are two independent cubic space group families, one starting at Imm and ending at F23, and the second starting at Iad and ending at P2₁3. The ending space groups are the only two cases of space groups without order-2 subgroups. The two series are sometimes called the A and B cubic space group families, respectively. They show quite different topological properties as indicated by their lattice complex examples in the upper left subboxes. The orthorhombic analogues do not have this property since there is a repetition of I4₁/acd in the third and fourth rows, a repetition of I4₁22 in the fourth and fifth rows, and a repetition of I422 in the second and fourth rows, which merge the orthorhombic/tetragonal set into one family.

The underlying topological spaces for the 36 cubic Euclidean 3-orbifolds listed in order along the seven rows of Fig. 6.1 are:

Row 1: D
Row 2: S, D, D, DP, D, PP, DP
Row 3: S, S, D, D, D, RP3, PP, D, PP, D, PP
Row 4: S, S, D, S, PP, PP, D, PP
Row 5: S, S, PP, PP
Row 6: S, S, S, PP
Row 7: S

with S, D, DP, PP and RP3 denoting S³, D³, D³ with single RP² suspension, double RP² suspension, and RP³, respectively. The covering 3-Euclidean orbifolds corresponding to the orthorhombic / tetragonal space groups in the same boxes have the same topological underlying spaces as their cubic counterparts except for the first two S³ entries in the 6th row which become the lens spaces L(4,1) and L(4,-1), and the single space group P2₁2₁2₁ in the 7th row which produces a Euclidean 3-manifold with no singular set. The list of 10 (or 13) space groups without singular sets are listed below. The corresponding arbitrary notation used by Wolf (1984) is given in parentheses.

#1 P1 (G1); #4 P2₁ (G2); #144 P3₁ (G3) [enantiomer: #145 P3₂ (G3)]; #76 P4₁ (G4) [enantiomer: #78 P4₃ (G4)]; #169 P6₁ (G5) [enantiomer: #173 P6₅ (G5)]; #19 P2₁2₁2₁ (G6); #7 Pb{Pc} (B1); #9 Bb{Cc} (B2); #29 Pca2₁ (B3); #33 Pna2₁ (B4)

Every Euclidean 3-orbifold has Euclidean 3-manifold covers derived from these space groups through subgroup relationships. Note that Table 6.1 contains G1, G6, and B2.

Color Group Representations of Group/Subgroup Pairs

Crystallographic color groups and the concept of antisymmetry are discussed in Appendix A. In the bicolor space groups, there are 1191 out of the total of 1651 group members that have mixed symmetry and antisymmetry group elements. In each of these 1191 groups, half of the geometrical symmetry elements have one color, say black, which represents regular crystallographic symmetry, and the remaining half have a second color, say white, which represents antisymmetry. Thus, the bicolor space groups are often called the black and white space groups.

In going from a space group to an index-2 normal subgroup in Fig 6.1, half of the symmetry operators are lost. The operators lost are the antisymmetry operators for a particular black and white space group which represent the group / normal subgroup space-group pair. If the subgroup has the same Bravais lattice as the parent group, antisymmetry is denoted only by (*)'s attached to generator element of the bicolor space group symbol. If the Bravais lattice changes, then the color space group also has a bicolor Bravais lattice denoted by a subscript attached to the Bravais lattice symbol.

To illustrate this use of the bicolor space groups, Tables 6.1 and 6.2 list all the cubic index-2 transformations of Fig. 6.1 using both the regular crystallographic space group symbols and the bicolor symbols. Table 6.1 gives the transformations that preserve the Bravais lattice. In these cases, there must be a change in the crystallographic point group to an index-2 subgroup. For example 229 Imm connects to 220 I3m via the black-white space group Im*m while changing the point group from mm for Imm to 3m for I3m. Table 6.2 gives all the remaining transformations which preserve the point group but change the Bravais Lattice. For example 229 Imm connects to 221 Pmm via the black-white space group I_Pmm. Such tables provide a structured computational mechanism for constructing group-subgroup graphs such as Fig. 6.1. The color group names also provide convenient labels for the links of related transformations in critical net orbifold graphs as illustrated in Fig. 6.2.

Primitive Cubic and NaCl Family

The general notes following Fig. 5.1 also apply to Fig. 6.2.

The symbol along each group-to-subgroup arrow denotes the bicolor (Shubnikov) space group symbol describing, by use of bold face characters, which generators in the bicolor space group are antisymmetry operators. Antisymmetry, which is discussed in Appendix A, characterizes the required reduction of symmetry needed to form the subgroup. The cubic space group symbol Pmm, which is shorthand for P4/m2/m, tells us the group is generated using a primitive Bravais lattice, P, a mirror, m, perpendicular to the 4-axis along [010], a axis along [-11-1], and a mirror, m, perpendicular to a 2-fold axis along [0-11]. Since only even order axes can be antisymmetric, an odd-order (i.e., 3 or ) axis is never antisymmetric; thus only the Bravais lattice P and the two m's generators in Pmm can become antisymmetric. Any time a bicolor symbol character is antisymmetric (bold), the corresponding character in the subgroup symbol is reduced or eliminated. For example Pmm becomes P432, while Pmm becomes Pm as shown in Fig. 6.2. The symbol P_F denotes a colored Bravais lattice. Note that all index-2 subgroups have either the same point group or the same Bravais lattice as the parent group as described above. The colored Bravais lattice symbol specifies the change in Bravais lattice.

Other details associated with Fig. 6.2.

• Silvered 3-ball underlying topological spaces are present in orbifolds 221, 229, 200, 215, 226, 225, 202 and 216.
• S³ underlying spaces are present in orbifolds 207, 195, 209 and 196.
• A doubly suspended RP² (projective plane) underlying topological space is present in orbifold 219
• Orbifold 229 at the top of the figure is related to 221 by an index-4 relationship that is not a bicolor group. All other subgroups shown have index-2 relationships to their parent groups.
• The coordination and multiplicity vectors at the upper left hold for all the orbifolds except that a factor of eight in multiplicity must be used for any orbifold containing a lattice complex with subscript 2.

Figure 6.2. Primitive Cubic and NaCl Critical Nets Superimposed onto Cubic Space Group Orbifolds.

• 12 space groups in Fig. 4.2 starting with Imm with critical net peaks on lattice complex I,
• 5 space groups in Fig. 5.1 starting with Fmm with critical net peaks on lattice complex F,
• 5 space groups in Fig. 5.2 starting with Fdm with critical net peaks on lattice complex D, and
• 12 space groups in Fig. 6.2 starting with Pmm with critical net peaks on lattice complex P.

Appendix A. Subgroup Families from Color Groups

5. Lattice Complexes on Critical Nets on Orbifolds

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Page last revised: June 25, 1996