GENERATORS AND RELATIONS FOR SPACE GROUPS

Transcription

1 GENERATORS AND RELATIONS FOR SPACE GROUPS Eric Lord Bangalore 2010

2 Contents Introduction Notation PART I. NON-CUBIC GROUPS Explanation of the figures Triclinic Monoclinic Orthorhombic Tetragonal.... Trigonal Hexagonal PART II. CUBIC GROUPS Explanation of the figures Cubic Appendix : matrix methods

3 GENERATORS AND RELATIONS FOR SPACE GROUPS Introduction In their book Generators and Relations for Discrete Groups Coxeter and Moser listed a set of algebraic relations satisfied by a minimal set of generators for seventeen abstract groups isomorphic to the seventeen wallpaper groups. The tabulation provided here does the same thing for the 230 crystallographic space groups. The list of generators given for each space group in the International Tables for Crystallography is extensive and, of course, highly redundant. What is given here is a minimal set of generators for each group and the algebraic relations they satisfy. Choosing a minimal set of generators for each of the space groups involves considerable arbitrariness. This arbitrariness is reduced by making use of the space-filling polyhedra known as asymmetric units. These are space-filling polyhedra that are related to each other in the tiling of 3D space that they produce, by the various Euclidean transformations of the group (rotations, reflections, inversions, rotary-inversions, glidereflections and screw transformations). For some of the space groups the choice of asymmetric unit is not unique. An asymmetric unit is essentially a Voronoi region surrounding a point of general position, and "some general positions are more general than others". For example, the number of facets of the unit is usually reduced if the point for which it is a Voronoi region is chosen to lie on the axis of a screw transformation or rotary-inversion, or on a glide plane. These and other considerations have occasionally led me to a different choice of asymmetric unit from the one proposed in the International Tables. The minimal sets of generators have been chosen from those that map an asymmetric unit to a contiguous asymmetric unit sharing a face. However the unit is chosen, the number of generators of the minimal set is unaffected that is an invariant, characteristic of the space group. For each group our tabulation gives: (i) the number assigned to the group in the International Tables for Crystallography; its Hermann-Mauguin symbol; {a list of the chosen generators a minimal set followed by additional (redundant) generators that extend the minimal set to a set that relates an asymmetric unit to all contiguous unit}; generators indicated in the H-M symbol, expressed in terms of the chosen set; (ii) a set of generating relations that define the abstract group; (iii) translations expressed in terms of the chosen generators; (iv) a particular realization of the generators in terms of Euclidean transformations; specified in terms of the image of a general point [x, y, z]. For trigonal and hexagonal groups we employ a hexagonal coordinate system (x and y axes at 2π/3, z axis perpendicular to them); (v) a simple diagram of the asymmetric unit. 1

4 Notation The type of a generator will frequently be indicated by the letter that denotes it: X, Y, Z: translations [x + 1, y, z], [x, y + 1, z], [x, y, z + 1]. W : a centering translation. W = [x + ½, y + ½, z + ½] for body-centred. The notations W 1 = [x, y + ½, z + ½], W 2 = [x + ½, y, z + ½], and W = W 3 = [x + ½, y + ½, z] will be used for the face-centred cases. The base-centering transformation W means [x + ½, y + ½, z] unless otherwise specified. T : the translation [x + 2 / 3, y + 1 / 3, z + 1 / 3 ] along an edge of the rhombohedral cell in rhombohedral groups. S : a screw transformation. A, B, C, D, N : glide reflections. The nature of a generator of a cyclic subgroup will be indicated by the letter used to denote it, according to the following scheme. To avoid constant repetition the generating relations for these cyclic subgroups will be omitted from the tabulation but they are always implied: I : an inversion I 2 = E. Unless otherwise stated, I = [ x, y, z ] R : a diad rotation R 2 = E. M : a reflection M 2 = E. F : fourfold rotation. F 4 = E. Unless otherwise stated, F = [ y, x, z]. F : 4 transformation. F 4 = E. Unless otherwise stated, F = [y, x, z ]. Q : [ y, x y, z] in trigonal or hexagonal groups; [z, x, y] in cubic groups. Q 3 = E. In the cubic groups X = QZQ 1, Y = QXQ 1, Z = QYQ 1. Q : A 3 transformation. [y, y x, z] in trigonal or hexagonal groups; [ z, x, y ] in cubic groups. Q 6 = E. Note that Q 2 = Q and Q 3 = I. H : sixfold rotation [x y, x, z]. H 6 = E. H : A 6 transformation [y x, x, z]. H 6 = E 2

5 Explanation of the Figures PART I. NON-CUBIC GROUPS The figure given with each of the non-cubic space groups represents an asymmetric unit viewed perpendicularly to the xy plane. The units are various kinds of prisms. The generators that relate a unit to a contiguous unit are indicated as follows: a diad axis (R) parallel to the xy plane. When this symbol occurs as an edge of the figure the axis is half-way between the upper and lower faces of the unit (typically at z = 0 and z = ½, so that the axis would then be at z = ¼). When it bisects the figure, it represents two diad axes, on the upper and lower faces. a mirror (M) perpendicular to the xy plane a glide plane perpendicular to the xy plane a c-glide plane a d-glide plane a 2 1 screw axis parallel to the xy plane (located half way between the upper and lower faces of the unit). The symbols and are multi-purpose symbols; their meaning is given by context: at the mid-point of an edge of the figure represents a diad axis perpendicular to the xy plane. In the centre of the figure it represents a screw axis perpendicular to the xy plane, of a type indicated by the overall symmetry of the figure. (Our figures do not distinguish between enantiomorphic groups P4 1 and P4 3 for example have the same figure. This causes no confusion because the name of the group makes the distinction!) At a vertex of the figure represents a rotation axis perpendicular to the xy plane: 3-fold, 4-fold or 6- fold according to whether the angle at the vertex is 120º, 90º or 60º. at the mid-point of an edge of the figure represents an inversion centre halfway between the upper and lower faces of the unit. In the centre of the figure it represents two inversion centres, on the upper and lower faces. At a vertex of the figure it represents the centre of a rotary-inversion half way between the upper and lower faces of theunit; the axis is perpendicular to the xy plane. The transformation is a 3, a 4 or a 6 according to whether the angle at the vertex is 60º, 90º or 120º. A 4 2 axis is produced by two generators a 2-fold rotation and a 4 2 screw transformation, with a common axis. A special symbol is used to denote this. It was needed for only three of the groups (77, 94 and 102). (6 2, 6 3 and 6 4 axes are also produced by two generators but, luckily, no situations arose requiring symbols for these.) When a figure is shaded in grey, this denotes the existence of two mirrors parallel to the xy plane, on the upper and lower faces of the unit. 3

6 Finally, the existence of a translation as one of the generators relating the unit to a contiguous unit is indicated by a parallel pair of unmarked (very pale grey) edges of the figure, or, for the Z translation, by the empty interior of the figure. All this may seem complicated at first. However, once one has become familiar with these conventions, the extreme simplicity of the figures will be appreciated. All of the information about a space group is encapsulated, in a coordinate-free manner, in the figure representing its asymmetric unit. In fact, the whole of the unit cell diagram for a space group, given in the International Tables (many of which look quite daunting), can be rapidly reconstructed from the figure we have assigned to it, once our notation has become familiar. All this, of course, applies only to the non-cubic groups. The asymmetric units of the cubic groups are more complicated and do not lend themselves to representation in a 2D figure. 4

7 Triclinic (1) P1 {X, Y, Z} YZY 1 Z 1 = ZXZ 1 X 1 = XYX 1 Y 1 = E (2) P 1 {X, Y, I 1, I 2 } XYX 1 Y 1 = (I 1 X) 2 = (I 1 Y) 2 = (I 2 X) 2 = (I 2 Y) 2 = E Z = I 2 I 1 I 1 = [ x, y, z ] I 2 = [ x, y, z + 1] Monoclinic (3) P2 { X, Z, R 1, R 2 } ZXZ 1 X 1 = XR 1 X 1 R 1 = (ZR 1 ) 2 = XR 2 X 1 R 2 = (ZR 2 ) 2 = E Y = R 2 R 1 R 1 = [x, y, z ] R 2 = [x, y + 1, z ] (4) P2 1 {X, Z, S} ZXZ 1 X 1 = XSXS 1 = ZSZS 1 = E Y = S 2 S = [ x + ½, y + ½, z + ½] (5) C2 {Z, R, S, R 2 = SRS 1 } RS 2 RS 2 = (ZR) 2 = ZSZS 1 = E X = R 2 R Y = S 2 W = SR R = [ x, y, z + ½] S = [ x + ½, y + ½, z + ½] R = [1 + x, y, z + ½] 5

8 (6) Pm {X, Z, M 1, M 2 } ZXZ 1 X 1 = M 1 XM 1 X 1 = M 2 XM 2 X 1 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = E Y = M 2 M 1 M 1 = [x, y, z] M 2 = [x, y + 1, z] (7) Pc {X, Y, C} XYX 1 Y 1 = CXC 1 X 1 = CYC 1 Y = E Z = C 2 C = [x, y, z + ½] (8) Cm {Z, M, A, M 2 = AMA 1 } MA 2 MA 2 = AZA 1 Z 1 = MZMZ 1 = E X = A 2 Y = M 2 M W 3 = AM M = [x, y, z] A = [x + ½, y + ½, z] M 2 = [x, y + 1, z] (9) Cc {W, C, W a = CWC 1 } WW a W 1 W 1 a = C 2 WC 2 W 1 = E 1 X = W W a Y = W W a Z = C 2 C = [x, y, z + ½] W a = [x + ½, y ½, z] (10) P2/m {Z, M 1, M 2, R 1, R 2 } M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = (R 1 Z) 2 = (R 2 Z) 2 = (M 1 R 1 ) 2 = (M 1 R 2 ) 2 = (M 2 R 1 ) 2 = (M 2 R 2 ) 2 = E X = R 2 R 1 Y = M 2 M 1 M 1 = [x, y, z] M 2 = [x, y + 1, z] R 1 = [ x, y, z ] R 2 = [ x + 1, y, z ] 6

9 (11) P2 1 /m {Z, M, I 1, I 2, M 2 = I 1 MI 1 } S = MI 1 M 2 I 2 MI 2 = (ZI 1 ) = (ZI 2 ) = ZMZ 1 M = ZM 2 Z 1 M 2 = E X = I 2 I 1 Y = M 2 M M = [x, y, z] I 1 = [ x, y, z ] I 2 =[ x + 1, y, z ] M 2 = [x, y + 1, z] (12) C2/m {Z, R, M, I, R 2 = IRI } (MR) 2 = (IR) 2 = MZMZ = (RZ) 2 = (IZ) 2 = E X = (IR) 2 Y = M 2 M W = IRM M = [x, y, z] R = [ x, y, z ] I = [ x + ½, y + ½, z ] R 2 = [ x + 1, y, x ] (13) P2/c {Y, R 1, R 2, C} CYC 1 Y = (R 1 C) 2 = (R 2 C) 2 = R 1 YR 1 Y 1 = R 2 YR 2 Y 1 = E X = R 2 R 1 Z = C 2 R 1 = [ x, y, z + ½] R 2 = [ x + 1, y, z + ½] C = [x, y + 1, z + ½] (14) P2 1 /c {X, S, C} (C 1 S) 2 = XSXS 1 = XCX 1 C 1 = E Y = S 2 Z = C 2 S = [ x + 1, y + ½, z + ½] C = [x, y + ½, z + ½] 7

10 (15) C2/c {R, C, S, R 2 = SRS 1 } (RC) 2 = (SC) 2 = (SC 1 ) 2 = RS 2 RS 2 = E X = R 2 R Y = S 2 Z = C 2 W = SR R = [ x, y, z ] S = [ x + ½, y + ½, z ] C = [x, y, z + ½] R 2 = [ x, y, z ] Orthorhombic (16) P222 {Z, R 1, R 2, R 3, R 4 } (R 1 R 4 ) 2 = (R 4 R 2 ) 2 = (R 2 R 3 ) 2 = (R 3 R 1 ) 2 = (R 1 Z) 2 = (R 2 Z) 2 = (R 3 Z) 2 = (R 4 Z) 2 X = R 2 R 1 Y = R 4 R 3 R 1 = [ x, y, z ] R 2 = [ x + 1, y, z ] R 3 = [x, y, z ] R 4 = [x, y + 1, z ] (17) P222 1 {X, Y, R 1, R 2 } S = R 2 R 1 (R 2 X) 2 = R 1 XR 1 X 1 = (R 1 Y) 2 = R 2 YR 2 Y 1 = E Z = S 2 R 1 = [x, y, z ] R 2 = [ x, y, z + ½] (18) P {Z, S, R, R 2 = SRS } S 2 = SR R 2 2 = RZRZ 1 = SZS 1 Z = E X = S 2 2 = R 2 R Y = S 2 R = [ x, y, z] S = [ x + ½, y + ½, z ] R 2 = [ x + 1, y, z] 8

11 (19) P {S 1, S 3, S 4 = S 3 S 1 1 S 3 } S 2 = S 3 S 1 S 2 1 S 2 4 = S 2 2 S 3 S 2 2 S 1 3 = S 2 3 S 1 S 2 3 S 1 1 = S 2 1 S 2 S 2 1 S 1 2 = E X = S 1 Y = S 2 Z = S 3 S 1 = [x + ½, y + ½, z ] S 3 = [ x, y, z + ½] S 4 = [x + ½, y ½, z ] (20) C222 1 {R 1, R 2, W, W 2 = R 1 WR 1 } S = R 2 R 1 (S 2 R 1 ) 2 = (S 2 R 2 ) 2 = WSWS 1 = W 2 SW 2 S 1 = W 2 2 WW 2 2 W 1 = W 2 W 2 W 2 W 1 2 = E 1 X = WW 2 Y = WW 2 Z = S 2 R 1 = [x, y, z ] R 2 = [ x, y, z + ½] (21) C222 {Z, R 1, R 2, R 3 } (R 1 R 2 ) 2 = (ZR 1 ) 2 = (ZR 2 ) 2 = ZR 3 Z 1 R 3 = (R 3 R 1 R 3 R 2 ) 2 = E X = (R 3 R 2 ) 2 Y = (R 3 R 1 ) 2 W 3 = R 3 R 2 R 1 R 1 = [x, y, z ] R 2 = [ x, y, z ] R 3 = [ x + ½, y + ½, z] (22) F222 {R 1, R 2, R 3, R 4, R 5 = R 3 R 1 R 3, R 6 = R 2 R 4 R 2 } (R 1 R 2 ) 2 = (R 2 R 5 ) 2 = (R 3 R 4 ) 2 = (R 3 R 6 ) 2 = (R 5 R 1 R 4 ) 2 = (R 4 R 6 R 1 ) 2 = R 1 R 4 R 5 R 6 = E X = (R 3 R 2 ) 2 Y = R 5 R 1 Z = R 6 R 4 W 1 = R 4 R 1 W 2 = R 3 R 4 R 2 W 3 = R 3 R 2 R 1 R 1 = [x, y, z + ½] R 2 = [ x, y, z + ½] R 3 = [ x + ½, y + ½, z] R 4 = [x, y + ½, z ] R 5 = [x, y + 1, z + ½] R 6 = [x, y + ½, z + 1] 9

12 (23) I222 {R 1, R 2, S, R 3 = SR 1 S, R 4 = SR 2 S } R 5 = R 1 R 2 R 3 2 = R 4 2 = R 5 2 = (R 1 R 2 ) 2 = (R 1 R 4 ) 2 = (R 2 R 3 ) 2 = (R 3 R 4 ) 2 = E X = R 3 R 1 Y= R 4 R 2 Z = S 2 W = R 3 R 4 R 5 R 1 = [ x, y, z ] R 2 = [x, y, z ] S = [ x + ½, y + ½, z + ½] R 3 = [ x + 1, y, z ] R 4 = [x, y + 1, z ] (24) I {R 1, R 2, R 3, R 4 = R 1 R 3 R 1, R 5 = R 2 R 4 R 2, R 6 = R 2 R 3 R 2 } S 1 = R 3 R 2 S 2 = R 3 R 1 S 3 = R 1 R 2 S 2 1 R 1 S 2 1 R 1 = S 2 2 R 2 S 2 2 R 2 = S 2 3 R 3 S 2 3 R 3 = E X = S 1 Y = S 2 Z = S 3 W = S 3 R 4 R 1 = [x, y, z + ½] R 2 = [ x, y, z ] R 3 = [ x + ½, y + ½, z] R 4 = [ x + ½, y ½, z] R 5 = [ x ½, y ½, z] [ x ½, y + ½, z] (25) Pmm2 {Z, M 1, M 2, M 3, M 4 } R = M 1 M 2 (M 1 M 2 ) 2 = (M 2 M 3 ) 2 = (M 3 M 4 ) 2 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = M 3 ZM 3 Z 1 = M 4 ZM 4 Z 1 = E X = M 4 M 2 Y = M 3 M 1 M 1 = [x, y, z] M 2 = [ x, y, z] M 3 = [x, y + 1, z] M 4 = [ x + 1, y, z] (26) Pmc2 1 {Y, M 1, M 2, C} S = M 1 C M 1 CM 1 C 1 = M 2 CM 2 C 1 = M 1 YM 1 Y 1 = M 2 YM 2 Y 1 = CYC 1 Y = E X =M 2 M 1 Z = C 2 M 1 = [ x, y, z] M 2 = [ x + 1, y, z] C = [x, y, z + ½] 10

13 (27) Pcc2 {X, C, R 1, R 2 } C 2 = R 1 C R 1 CR 1 C 1 = R 2 CR 2 C 1 = (R 1 X) 2 = (R 2 X) 2 = CXC 1 X = E Y = R 2 R 1 Z = C 2 C = [ x, y, z + ½] R 1 = [ x, y, z] R 2 = [ x, y + 1, z] (28) Pma2 {Z, M, R 1, R 2, M 2 = R 1 MR 1 } A = R 1 M MZMZ 1 = M 2 ZM 2 Z 1 = R 1 ZR 1 Z 1 = R 2 ZR 2 Z 1 = R 2 R 1 MR 1 R 2 M = E X = M 2 M Y = R 2 R 1 M = [ x, y, z] R 1 = [ x + ½, y, z] R 2 = [ x + ½, y + 1, z] M 2 = [ x + 1, y, z] (29) Pca2 1 {Y, C, A} S = A 1 C ACAC 1 = CYC 1 Y 1 = AYA 1 Y = E X = A 2 Z = C 2 A = [x + ½, y, z] C = [ x + ½, y, z + ½] (30) Pnc2 {X, C, R, R 2 = CRC 1 } N = CR (RX) 2 = CXC 1 X 1 = C 2 RC 2 R = E Y = R 2 R Z = C 2 R = [ x, y, z] C = [x, y + ½, z + ½] R 2 = [ x, y + 1, z] 11

14 (31) Pmn2 1 {Y, M, S, M 2 = SMS 1 } N = SM SYS 1 Y = S 2 MS 2 M = MYMY 1 = E X = M 2 M Z = S 2 M = [ x, y, z] S = [ x + ½, y + ½, z + ½] M 2 = [ x + 1, y, z] (32) Pba2 {Z, B, A} R = A 1 B AZA 1 Z 1 = BZB 1 Z 1 = (AB) 2 = (AB 1 ) 2 = E X = A 2 Y = B 2 B = [ x + ½, y + ½, z] A = [x + ½, y + ½, z] (33) Pna2 1 {A, S, A 2 = SA 1 S 1 } N = SA A 2 SA 2 S 1 = S 2 AS 2 A 1 = E X = A 2 1 Y = AA 2 Z = S 2 A = [x + ½, y + ½, z] S = [ x + ½, y + ½, z + ½] A = [x + ½, y ½, z] (34) Pnn2 {Z, N, R} N 2 = NR NZN 1 Z 1 = RZRZ 1 = E X = Z 1 2 N 2 Y = Z 1 N 2 R = [ x, y, z] N = [x + ½, y + ½, z + ½] 12

15 (35) Cmm2 {Z, M 1, M 2, R} (M 1 M 2 ) 2 = RZRZ 1 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = (M 2 RM 1 R) 2 = E X = (RM 1 ) 2 Y = (RM 2 ) 2 W = RM 1 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] R = [ x + ½, y + ½, z] (36) Cmc2 1 {M, C, B, M 2 = BMB 1 } MCMC 1 = BCBC 1 = B 2 MB 2 M = E X = BMB 1 M Y = B 2 Z = C 2 W = BM M = [ x, y, z] C = [x, y, z + ½] B = [ x + ½, y + ½, z] M 2 = [ x + 1, y, z] (37) Ccc2 {R 1, R 2, C, R 3 = CR 2 C 1 } C 2 = CR 1 CR 1 C 1 R 1 = C 2 R 2 C 2 R 2 = (R 1 R 2 R 3 ) 2 = E X = R 2 R 3 Y = R 3 R 1 R 2 R 1 Z = C 2 W = R 2 R 1 R 1 = [ x, y, z] R 2 = [ x + ½, y + ½, z] C = [x, y, z + ½] R 3 = [ x ½, y + ½, z] (38) Amm2 {M 1, M 2, M 3, C, M 4 = CM 2 C 1 } R = M 1 M 2 (M 1 M 2 ) 2 = (M 2 M 3 ) 2 = (M 3 M 4 ) 2 = (M 4 M 1 ) 2 = CM 1 C 1 M 1 = CM 3 C 1 M 3 = E X = M 3 M 1 Y = M 4 M 2 Z = C 2 W = CM 1 R M 1 = [ x, y, z] M 2 = [x, y, z] M 3 = [ x + 1, y, z] C = [x, y + ½, z + ½] M 4 = [x, y + 1, z] 13

16 (39) Abm2 {M, R 1, R 2, C, M 2 = R 1 MR 1 } B = R 1 M M 2 CMC 1 = M 2 C 1 MC = M 2 R 2 MR 2 = R 1 CR 1 C 1 = R 2 CR 2 C 1 = C 2 MC 2 M = E X = R 2 R 1 Y = B 2 Z = C 2 W 1 = MC M = [x, y, z] R 1 = [ x, y + ½, z] R 2 = [ x + 1, y + ½, z] C = [x, y + ½, z + ½] M 2 = [x, y + 1, z] (40) Ama2 {M, R, S, M 2 = RMR, R 2 = SRS 1 } A = MR M 2 SMS 1 = (MSMRS 1 ) 2 = S 2 RS 2 R = S 2 MS 2 M = (MR) 2 S(MR) 2 S 1 = E X = M 2 M Y = R 2 R Z = S 2 W 1 = SR R = [ x + ½, y, z] M = [ x, y, z] S = [ x + ½, y + ½, z + ½] M 2 = [ x + 1, y, z] R 2 = [ x + ½, y + 1, z] (41) Aba2 {B, A, C} R = A 1 C ACA 1 C 1 = BCBC 1 = (AB 1 ) 2 = (BA 1 ) 2 = E X = A 2 Y = B 2 Z = C 2 W 1 = BC A = [x + ½, y + ½, z] B = [ x + ½, y + ½, z] C = [ x + ½, y, z + ½] (42) Fmm2 {M 1, M 2, R, C, M 3 = RM 1 R} M 3 CMC 1 = (M 1 M 2 ) 2 = M 2 CM 2 C 1 = CRC 1 R = M 1 C 2 M 1 C 2 = RM 1 RCM 1 C 1 = E X = (RM 1 ) 2 Y = (RM 2 ) 2 Z = C 2 W 1 = RM 2 C W 2 = CM 1 W 3 = RM 1 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] R = [ x + ½, y + ½, z] C = [ x + ½, y, z + ½] M 3 = [ x + 1, y, z] 14

17 (43) Fdd2 {D 1, D 2, R 1 = D 1 2 D 1, S = D 2 D 1, R 2 = SR 1 S 1, D = SD 1 S 1 } (D 1 D 1 2 ) 2 = (D 2 D 1 ) 2 (D 1 D 2 ) 2 = D 2 1 D 2 2 D 2 1 D 2 2 = D 2 1 (D 2 D 1 D 2 )D 2 1 (D 2 D 1 D 2 ) 1 = D 2 2 (D 1 D 2 D 1 )D 2 2 (D 1 D 2 D 1 ) 1 = E X = D 3 2 (D 1 D 2 D 1 ) 1 Y = D 3 1 (D 2 D 1 D 2 ) 1 Z = S W 1 = D 1 W 2 = D 2 W 3 = (D 1 D 2 D 1 ) 1 2 D 2 D 1 D 1 = [ x + ¼, y + ¼, z + ¼] D 2 = [x + ¼, y + ¼, z + ¼] R 1 = [ x, y, z] S = [ x + ½, y, z + ½] R 2 = [ x + 1, y, z] D = [ x + ¾, y ¼, z] (44) Imm2 {M 1, M 2, S, M 3 = SM 1 S 1, M 4 = SM 2 S 1 } R = M 1 M 2 (M 1 M 2 ) 2 = S 2 M 1 S 2 M 1 = S 2 M 2 S 2 M 2 = (M 4 M 1 ) 2 = (M 3 M 2 ) 2 = E X = M 3 M 1 Y = M 4 M 2 Z = S 2 W = SM 1 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] S = [ x + ½, y + ½, z + ½] M 3 = [ x + 1, y, z] M 4 = [x, y + 1, z] (45) Iba2 {A, B, S} R = A 1 B (AB) 2 = (AB 1 ) 2 = SAS 1 A = SBS 1 B = (S 1 BSA) 2 = (S 1 ASB) 2 = E X = A 2 Y = B 2 Z = S 2 W = BSA A = [x + ½, y + ½, z] B = [ x + ½, y + ½, z] S = [ x + ½, y + ½, z + ½] (46) Ima2 {M, R, C, M 2 = RMR, R 2 = CRC 1 } A = RM CMC 1 M = C 2 RC 2 R = CM 2 C 1 M 2 = E X = M 2 M Y = R 2 R Z = C 2 W = CRM M = [ x, y, z] R = [ x + ½, y, z] C = [x, y + ½, z + ½] M 2 = [ x + 1, y, z] R 2 = [ x + ½, y + 1, z] 15

18 (47) Pmmm {M 1, M 2, M 3, M 4, M 5, M 6 } (M 1 M 3 ) 2 = (M 2 M 3 ) 2 = (M 3 M 4 ) 2 = (M 4 M 1 ) 2 = (M 1 M 5 ) 2 = (M 2 M 5 ) 2 = (M 3 M 5 ) 2 = (M 3 M 5 ) 2 = (M 4 M 5 ) 2 = (M 1 M 6 ) 2 = (M 2 M 6 ) 2 = (M 3 M 6 ) 2 = (M 4 M 6 ) 2 = (M 5 M 6 ) 2 = E X = M 2 M 1 Y = M 3 M 4 Z = M 6 M 5 M 1 = [ x, y, z] M 2 = [ x + 1, y, z] M 3 = [x, y + 1, z] M 4 = [x, y, z] M 5 = [x, y, z ] M 6 = [x, y, z + 1] (48) Pnnn { R 1, R 2, I 1, I 2, R 3 = I 1 R 1 I 2, R 4 = I 1 R 2 I 2 } N 1 = I 2 R 1 N 2 = I 2 R 2 N 3 = I 2 R 1 R 2 R 2 3 = R 2 4 = (R 1 R 2 ) 2 = I 1 I 2 R 1 R 2 I 2 I 1 R 1 R 2 = (N 1 N 2 N 3 ) 2 = (N 3 N 2 N 1 ) 2 = E X = R 3 R 1 Y = R 4 R 2 Z = I 2 I 1 (N 2 1 = YZ N 2 2 = ZX N 2 3 = XY) R 1 = [x, y, z ] R 2 = [ x, y, z ] I 1 = [ x + ½, y + ½, z ½] I 2 = [ x + ½, y + ½, z + ½] N 1 = [ x + ½, y + ½, z + ½] N 2 = [x + ½, y + ½, z + ½] N 3 = [x + ½, y + ½, z + ½] R 3 = [x + 1, y, z ] R 4 = [ x, y + 1, z ] (49) Pccm { R 1, R 2, R 3, R 4, M, M 2 = R 1 MR 1 } C 1 = R 1 M C 2 = R 2 M M 2 R 3 MR 3 = (R 1 R 2 ) 2 = (R 2 R 3 ) 2 = (R 3 R 4 ) 2 = (R 4 R 1 ) 2 = (R 1 R 2 M) 2 = (R 2 R 3 M) 2 = (R 3 R 4 M) 2 = (R 4 R 1 M) 2 = E X = R 3 R 1 Y = R 4 R 2 Z = C 2 1 = C 2 2 = M 2 M R 1 = [ x, y, z + ½] R 2 = [x, y, z + ½] R 3 = [ x + 1, y, z + ½] R 4 = [x, y + 1, z + ½] M = [x, y, z ] M 2 = [x, y, z + 1] 16

19 (50) Pban {Z, R 1, R 2, I} B = IR 1 A = IR 2 N = IR 1 R 2 (R 1 R 2 ) 2 = (AB 2 = (ABN) 2 = (IZ) 2 = (R 1 Z) 2 = (R 2 Z) 2 = E X = B 2 Y = A 2 R 1 = [ x, y, z ] R 2 = [x, y, z ] I = [ x + ½, y + ½, z ] (51) Pmma {M 1, M 2, M 3, R 1, R 2, M 4 = R 1 M 2 R 1 } A = R 1 M 2 (M 1 M 2 ) 2 = (M 2 M 3 ) 2 = (M 3 M 4 ) 2 = (M 4 M 1 ) 2 = (R 1 M 1 ) 2 = (R 1 M 3 ) 2 = (R 2 M 1 ) 2 = (R 2 M 3 ) 2 = M 4 R 2 M 2 R 2 = E X = M 4 M 2 Y = M 3 M 1 Z = R 2 R 1 M 1 = [x, y, z] M 2 = [ x, y, z] M 3 = [x, y + 1, z] R 1 = [ x + ½, y, z ] R 2 = [ x + ½, y, z + 1] M 4 = [ x + 1, y, z] (52) Pnna {R 1, R 2, I 1, I 2, R 3 = R 2 R 1 R 2, R 4 = I 1 R 2 I 1 } N 1 = R 3 I 1 N 2 = I 1 R 2 R 1 A = I 1 R 2 R 4 I 2 R 2 I 2 = R 3 R 4 R 1 R 4 = R 4 R 2 R 1 R 2 R 4 R 1 = (R 1 I 1 I 2 ) 2 = E X = R 4 R 2 Y = R 3 R 1 Z = I 2 I 1 R 1 = [x, y, z + ½] R 2 = [ x, y + ½, z] I 1 = [ x + ½, y + ½, z ] I 2 = [ x + ½, y + ½, z + 1] R 3 = [x, y + 1, z + ½] R 4 = [ x + 1, y + ½, z] (53) Pmna {Y, M, R 1, R 2, M 2 = R 1 MR 1 } N = R 1 R 2 YM A = R 1 M (MR 2 ) 2 = YMY 1 M = YR 1 Y 1 R 1 = (YR 2 ) 2 = (M 2 R 2 ) 2 = E X = M 2 M Z = (R 1 R 2 ) 2 M = [ x, y, z] R 1 = [ x + ½, y, z + ½] R 2 = [x, y + 1, z ] M 2 = [ x + 1, y, z] 17

20 (54) Pcca {R 1, R 2, R 3, C, R 4 = R 2 R 1 R 2 ] C 2 = R 3 C A = C 1 R 1 R 4 CR 1 C 1 = (R 1 R 3 R 4 ) 2 = C 2 R 2 C 2 R 2 = C 2 R 3 C 2 R 3 = E X = R 4 R 1 Y = R 2 R 3 Z = C 2 R 1 = [ x, y, z + ½] R 2 = [ x + ½, y, z] R 3 = [ x + ½, y + 1, z] C = [ x + ½, y, z + ½] R 4 = [ x, y + 1, z + ½] (55) Pbam {B, A, M 1, M 2 } M 1 AM 1 A 1 = M 2 AM 2 A 1 = M 1 BM 1 B 1 = M 2 BM 2 B 1 = (AB) 2 = (AB 1 ) 2 = E X = A 2 Y = B 2 Z = M 2 M 1 B = [ x + ½, y + ½, z] A = [x + ½, y + ½, z] M 1 = [x, y, z ] M 2 = [x, y, z + 1] (56) Pccn {C, R 1, R 2, I, I 2 = CIC] C 2 = CR N = C 1 R 2 CI I 2 2 = CR 1 C 1 R 1 = CR 2 C 1 R 2 = (IR 1 R 2 ) 2 = (IC 2 ) 2 = E X = I 2 I Y = R 2 R 1 Z = C 2 C = [ x + ½, y, z + ½] R 1 = [ x + ½, y ½, z] R 2 = [ x + ½, y + ½, z] I 2 = [ x + 1, y, z ] 18

21 (57) Pbcm {M, R, I 1, I 2, M 2 = I 1 MI 1, R 2 = I 1 RI 1 ] B = I 1 R C = I 1 M M 2 RMR = M 2 I 2 MI 2 = M 2 R 2 MR 2 = R 2 I 2 RI 2 = RR 2 M 2 M = E X = I 2 I 1 Y = B 2 Z = C 2 M = [x, y, z ] R = [x, y, z + ½] I 1 = [ x, y + ½, z + ½] I 2 = [ x + 1, y + ½, z + ½] M 2 = [x, y, z + 1] R 2 = [x, y + 1, z + ½] (58) Pnnm {M, S 1, S 2, M 2 = S 1 MS 1 1 } N 1 = S 2 M N 2 = S 1 M M 2 S 2 MS 1 2 = (S 1 1 S 2 ) 2 = (S 1 S 2 ) 2 = (S 1 S 2 M) 2 = (S 1 1 S 2 M) 2 = E 2 2 X = S 1 Y = S 2 Z = M 2 M M = [x, y, z ] S 1 = [x + ½, y + ½, z + ½] S 2 = [ x + ½, y + ½, z + ½] M 2 = [x, y, z + 1] (59) Pmmn {Z, M 1, M 2, I} N = IM 1 M 2 (M 1 M 2 ) 2 = (IM 1 IM 2 ) 2 = (IM 1 M 2 IM 1 IM 2 ) 2 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = (IZ) 2 = E X = (IM 1 ) 2 Y = (IM 2 ) 2 M 1 = [ x, y, z] M 2 = [x, y, z] I = [ x + ½, y + ½, z ] (60) Pbcn {B, S 1, S 2 } C = S 2 B N = S 2 B 1 S 1 (S 2 S 1 ) 2 = (S 2 S 1 1 ) 2 = BS 2 BS 1 2 = (BS 1 ) 2 = (BS 1 1 ) 2 = E 2 X = S 1 Y = B 2 2 Z = S 2 S 1 = [x + ½, y + ½, z ] S 2 = [ x + ½, y + ½, z + ½] B = [ x + ½, y + ½, z] 19

22 (61) Pbca {B, C, A} BCBC 1 = CACA 1 = ABAB 1 = (ABC) 2 = (CBA) 2 = E X = A 2 Y = B 2 Z = C 2 A = [x + ½, y, z + ½] B = [ x + ½, y + ½, z] C = [x, y + ½, z + ½] (62) Pnma {M, I, S, M 2 = IMI, I 2 = SIS } N = MS A = SI I 2 2 = M 2 SMS 1 = (I 2 M 2 M) 2 = MI 2 I 1 MI 1 I 2 ) = (IS 2 ) 2 = S 2 MS 2 M = E X = I 2 I Y = M 2 M Z = S 2 M = [x, y, z] I = [ x, y + ½, z + ½] S = [ x + ½, y + ½, z + ½] M 2 = [x, y +1, z] I 2 = [ x + 1, y + ½, z + ½] (63) Cmcm {M 1, M 2, R, I, M 3 = IM 2 I } C = M 2 R M 3 RM 2 R = (M 1 M 2 ) 2 = (M 1 M 3 ) 2 = (M 1 R) 2 = (IM 1 IR) 2 = E X = (IM 1 ) 2 Y = (RI) 2 Z = M 2 M 3 W = IM 1 R M 1 = [ x, y, z] M 2 = [x, y, z ] R = [x, y, z + ½] I = [ x + ½, y + ½, z + ½] M 3 = [x, y, z + 1] (64) Cmca {M, C, R, I, R 2 = IRI} A = CIM IRICRC = (RM) 2 = (IRIM) 2 = CMC 1 M = CIC 1 I = (C 2 RIR) 2 = E X = (IM) 2 Y = (IR) 2 Z = C 2 W = IMR M = [ x, y, z] C = [x, y + ½, z + ½] R = [x, y, z + ½] I = [ x + ½, y + ½, z + ½] R 2 = [x, y + 1, z + ½] 20

23 (65) Cmmm {M 1, M 2, M 3, M 4, R} (M 1 M 2 ) 2 = (RM 1 RM 2 ) 2 = (RM 3 ) 2 = (RM 4 ) 2 = (M 1 M 3 ) 2 = (M 2 M 3 ) 2 = (M 1 M 4 ) 2 = (M 2 M 4 ) 2 = E X = (RM 1 ) 2 Y = (RM 2 ) 2 Z = M 4 M 3 W = RM 1 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] M 3 = [x, y, z ] M 4 = [ x, y, z + 1] R = [ x + ½, y + ½, z] (66) Cccm {M, R 1, R 2, R 3, M 4 = R 1 MR 1 } C 1 = R 2 MR 1 R 2 C 2 = R 1 MR 1 R 2 M 4 R 2 MR 2 = (R 1 R 2 ) 2 = (MR 1 R 2 ) 2 = (MR 3 ) 2 = (MR 4 ) 2 = (R 3 R 1 R 3 R 2 ) 2 = E X = (R 3 R 1 ) 2 Y = (R 3 R 2 ) 2 Z = M 4 M W = R 3 R 1 R 2 M = [x, y, z ] R 1 = [ x, y, z + ½] R 2 = [x, y, z + ½] R 3 = [ x, y, z] M 4 = [x, y, z + 1] (67) Cmma {Z, M 1, M 2, R 1, R 2 } A = R 1 M 1 (M 1 M 2 ) 2 = (R 1 R 2 ) 2 = (M 1 R 2 ) 2 = (M 2 R 1 ) 2 = (ZR 1 ) 2 = (ZR 2 ) 2 = ZM 1 Z 1 M 1 = ZM 2 Z 1 M 2 = E X = (R 1 M 1 ) 2 Y = (M 2 R 2 ) 2 W = R 1 R 2 M 1 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] R 1 = [ x + ½, y, z ] R 2 = [x, y + ½, z ] (68) Ccca {R 1, R 2, R 3, C, R 4 = R 3 R 2 R 3 } C 2 = CR 3 A = CR 3 R 1 R 4 CR 2 C = (R 1 R 2 ) 2 = (R 1 R 4 ) 2 = CR 3 C 1 R 3 = (C 2 R 1 ) 2 = (C 2 R 2 ) 2 = E X = (R 3 R 1 ) 2 Y = (R 3 R 2 ) 2 Z = C 2 W = R 3 R 2 R 1 R 1 = [ x, y, z ] R 2 = [x, y, z ] R 3 = [ x + ½, y + ½, z] C = [x, y + ½, z + ½] R 4 = [x, y + 1, z ] 21

24 (69) Fmmm {M 1, M 2, M 3, R 1, R 2, M 4 = R 1 M 3 R 1 } M 4 R 2 M 3 R 2 = (M 2 M 3 ) 2 = (M 3 M 1 ) 2 = (M 1 M 2 ) 2 = (R 1 R 2 ) 2 = (R 1 M 2 ) 2 = (R 2 M 1 ) 2 = (R 1 R 2 M 3 ) 2 = E X = (R 1 M 1 ) 2 Y = (R 2 M 2 ) 2 Z = M 4 M 3 W = R 1 R 2 M 2 M 1 M 1 = [ x, y, z] M 2 = [x, y, z] M 3 = [x, y, z ] R 1 = [ x + ½, y, z + ½] R 2 = [x, y + ½, z + ½] M 4 = [x, y, z + 1] (70) Fddd {D 1, D 2, D 3, S = D 2 D 1, R 1 = D 1 2 D 1, R 2 = SR 1 S 1, R 3 = D 1 3 D 2, R 4 = D 3 D 1 1, D = SD 1 S 1 } SR 3 R 4 = (D 2 D 1 3 ) 2 = (D 3 D 1 1 ) 2 = (D 1 D 1 2 ) 2 = (D 1 D 2 D 1 3 ) 2 = (D 1 D 2 D 3 ) 2 = (D 2 D 1 D 3 ) 2 = D 2 2 D 2 3 D 2 2 D 2 3 = D 2 3 D 2 1 D 2 3 D 2 1 = D 2 1 D 2 2 D 2 1 D 2 2 = E X = (D 2 D 3 ) 2 Y = (D 3 D 1 ) 2 Z = (D 1 D 2 ) 2 2 W 1 = D 1 2 W 2 = D 2 2 W 3 = D 3 D 1 = [ x + ¼, y + ¼, z + ¼] D 2 = [x + ¼, y + ¼, z + ¼] D 3 = [x + ¼, y + ¼, z + ¼] R 1 = [ x, y, z] S = [ x + ½, y, z + ½] R 2 = [ x + 1, y, z] R 3 = [x, y, z ] R 4 = [ x + ½, y, z + ½] D = [ x + ¾, y ¼, z] (71) Immm {M 1, M 2, M 3, I, M 4 = IM 3 I } (M 1 M 2 ) 2 = (M 2 M 3 ) 2 = (M 3 M 1 ) 2 = (IM 1 IM 2 ) 2 = (IM 2 IM 3 ) 2 = (IM 3 IM 1 ) 2 = E X = (IM 1 ) 2 Y = (IM 2 ) 2 Z = M 4 M 3 W = IM 3 M 2 M 1 M 1 = [ x, y, z] M 2 = [x, y, z] M 3 = [x, y, z ] I = [ x + ½, y + ½, z + ½] M 4 = [x, y, z + 1] 22

25 (72) Ibam {M, R 1, R 2, I, M 2 = IMI } B = IR 2 A = IR 1 M 2 R 1 MR 1 = M 2 R 2 MR 2 = (R 1 R 2 ) 2 = (MR 1 R 2 ) 2 = (R 1 R 2 IR 1 IR 2 I) 2 = (R 1 IR 1 (IM) 2 ) 2 = (R 2 IR 2 (IM) 2 ) 2 = (IMR 2 IR 1 ) 2 =E X = A 2 Y = B 2 Z = M 2 M W = IMR 2 R 1 M = [x, y, z ] R 1 = [ x, y, z + ½] R 2 = [x, y, z + ½] I = [ x + ½, y + ½, z + ½] M 2 = [x, y, z + 1] (73) Ibca {R 1, R 2, C, I, R 3 = IR 1 I } B = IR 1 A = IR 2 R 3 R 2 R 1 R 2 = R 3 CR 1 C = CR 2 C 1 R 2 = BCBC 1 = CACA 1 = ABAB 1 = (ABC) 2 = (CBA) 2 = (CI) 2 = E X = A 2 Y = B 2 Z = C 2 W = IC 1 BA R 1 = [x, y, z + ½] R 2 = [ x, y + ½, z] C = [x, y + ½, z + ½] I = [ x + ½, y + ½, z + ½] R 3 = [x + 1, y, z + ½] (74) Imma {M 1, M 2, R 1, R 2, M 3 = R 1 M 1 R 1, M 4 = R 2 M 2 R 2 } A = R 1 M 1 (M 1 M 2 ) 2 = (M 2 M 3 ) 2 = (M 3 M 4 ) 2 = (M 4 M 1 ) 2 = (R 1 M 2 ) 2 = (R 1 M 4 ) 2 = (R 2 M 1 ) 2 = (R 2 M 3 ) 2 = E X = M 3 M 1 Y = M 4 M 2 Z = (R 1 R 2 ) 2 W = R 1 R 2 M 1 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] R 1 = [ x + ½, y, z + ½] R 2 = [x, y + ½, z ] M 3 = [ x + 1, y, z] M 4 = [x, y + 1, z] 23

26 Tetragonal (75) P4 {Z, R, F} ZFZ 1 F 1 = ZRZ 1 R = (RF 1 ) 4 = E X = RF 2 Y = FRF R = [ x + 1, y, z] (76) P4 1 {X, S, Y = SXS 1 ) XS 2 XS 2 = E Y = SXS 1 Z = S 4 S = [ y, x, z + ¼] (77) P4 2 {R 1, R 2, S, Y = SR 2 R 1 S 1 } R 1 SR 1 S 1 = R 2 R 1 S 2 R 2 R 1 S 2 = (R 2 R 1 SR 2 S 1 ) 2 = E X = R 2 R 1 Y = SR 2 R 1 S 1 Z = R 1 S 2 R 1 = [ x, y, z] R 2 = [ x + 1, y, z] S = [ y, x, z + ½] (78) P4 3 {X, S, Y = SX 1 S 1 } XS 2 XS 2 = E Y = SX 1 S 1 Z = S 4 S = [y, x, z + ¼] 24

27 (79) I4 {F, S, F 2 = SFS 1 } (FF 2 ) 2 = S 2 FS 2 F 1 = E X = F 2 F 1 Y = F 2 1 F Z = S 2 W = SF 2 S = [ x + ½, y + ½, z + ½] F 2 = [ y + 1, x, z] (80) I4 1 {R, S, R 2 = SRS 1, R 3 = S 1 RS, R 4 = S 2 RS 2 } R 1 R 2 R 3 R 4 = (RR 3 R 2 ) 2 = RS 4 RS 4 = E X = R 2 R Y = R 4 R Z = S 4 W = S 2 R R = [ x, y, z] S = [ y + ½, x, z + ¼] R 2 = [ x + 1, y, z] R 3 = [ x + 1, y + 1, z] R 4 = [ x, y + 1, z] (81) P 4 { F, Z, R} (R F ) 4 = (R F ) 2 (R F 1 ) 2 = RZRZ 1 = F Z F 1 Z = E X = R F 2 Y = R 2 F 2 R = [ x + 1, y, z] (82) I 4 { F, W, R, F 2 = W F W 1 } (R F ) 4 = (W F ) 4 = R(W F 2 ) 2 R(W F 2 ) 2 = F (W F 2 ) 2 F 1 (W F 2 ) 2 = (W F 2 ) 2 ( F 2 W) 2 = E X = R F 2 Y = F R F Z = (W F 2 ) 2 R = [ x + 1, y, z] F 2 = [y, x + 1, z ] 25

28 (83) P4/m {F, M 1, M 2, R} (RF) 4 = (M 1 F) 4 = (M 2 F) 4 = (M 1 R) 2 = (M 2 R) 2 = FM 1 F 1 M 1 = FM 2 F 1 M 2 = E X = RF 2 Y = R 2 F 2 Z = M 2 M 1 M 1 = [x, y, z ] M 2 = [x, y, z + 1] R = [ x + 1, y, z] (84) P4 2 /m { F, R, M 1, M 2 } (RM 1 ) 2 = (RM 2 ) 2 = ( F 2 M 1 ) 2 = ( F 2 M 2 ) 2 = M 1 M 2 F M 1 M 2 F 1 = (R F ) 4 = E X = R F 2 Y = F 1 R F 1 Z = M 2 M 1 F = [y, x, z + ½] R = [ x + 1, y, z] M 1 = [x, y, z ] M 2 = [x, y, z + 1] (85) P4/n {Z, F, I} N = IF 2 FZF 1 Z 1 = (IZ) 2 = (FI) 4 = E X = F 1 IFI Y = FIF 1 I I = [ x + ½, y + ½, z ] (86) P4 2 /n { F, I 1, I 2, F 2 = I 2 F I 1 } N = I 1 F 2 S = F I 2 R = I 1 I 2 S 2 (I 1 I 2 S 2 ) 2 = F I 1 I 2 F 1 I 1 I 2 = ( F 2 I 1 F I 2 F I 1) 2 = ( F 2 I 2 F I 1 F I 2 ) 2 = E X = I 2 F 1 I 1 F Y = I 1 F I 2 F 1 Z = I 2 I 1 F = [y, x, z + ½] I 1 = [ x + ½, y + ½, z ] I 2 = [ x + ½, y + ½, z + 1] F 2 = [y, x + 1, z + ½] 26

29 (87) I4/m {F, M, I, M 2 = IMI} (IF) 4 = MFMF 1 = M 2 FM 2 F 1 = (FIF 1 M 2 M) 2 = E X = IFIF 1 Y = IF 1 IF Z = M 2 M W = IMF 2 M = [x, y, z ] I = [ x + ½, y + ½, z + ½] M 2 = [x, y, z + 1] (88) I4 1 /a {S, F, F 2 = S F S } A = F S X = A 2 Y = F 2 F Z = S 4 W = F S 2 F S = [ y, x + ½, z + ¼] F = [y, x, z ] F 2 = [y, x + 1, z ] (89) P422 {Z, R 1, R 2, R 3 } F = R 3 R 1 (R 1 R 2 ) 2 = (R 2 R 3 ) 4 = (R 3 R 1 ) 4 = (R 2 R 3 R 1 ) 2 = (R 1 Z) 2 = (R 2 Z) 2 = (R 3 Z) 2 = E X = R 2 R 3 R 1 R 3 Y = R 3 R 2 R 3 R 1 R 1 = [x, y, z ] R 2 = [ x, y, z ] R 3 = [y, x, z ] (90) P {Z, F, R} S = RF (RZ) 2 = (RF) 4 = FZF 1 Z = (F 1 RFR) 2 = E X = (RF 1 ) 2 Y = S 2 R = [ y + ½, x + ½, z ] 27

30 (91) P {X, R 1, R 2, Y = R 2 X 1 R 2 } S = R 2 R 1 (R 1 X) 2 = (XR 2 ) 2 (X 1 R 2 ) 2 = (X(R 2 R 1 ) 2 R 2 ) 2 = E Y = R 2 X 1 R 2 Z = S 2 R 1 = [ x, y, z ] R 2 = [ y, x, z + ¼] (92) P {S 1, S 2, S 3 = S 2 1 S 1 2 S } R = S 2 S 1 S 2 2 S 2 3 = (S 1 S 1 2 ) 2 = (S 1 S 2 ) 2 = (S 1 S 1 3 ) 2 = (S 1 S 3 ) 2 = S 4 1 S 2 2 S 4 1 S 2 2 = E X = S 1 S S 1 Y = S 2 Z = S 1 S 1 = [ y, x, z + ¼] S 2 = [ x + ½, y + ½, z + ½] S 3 = [ x + ½, y ½, z + ½] (93) P {R 1, R 2, R 3, R 4, R 5, R 6 = R 2 R 5 R 1 } S = R 5 R 2 (R 2 R 5 R 1 ) 2 = (R 1 R 2 ) 2 = (R 2 R 3 ) 2 = (R 3 R 4 ) 2 = (R 4 R 1 ) 2 = (R 3 R 4 R 5 ) 2 = (R 3 R 4 R 6 ) 2 = (R 3 R 6 R 5 ) 2 = R 4 R 2 R 5 R 2 R 5 R 2 R 4 R 5 R 2 R 5 = E X = R 3 R 1 Y = R 4 R 2 Z = R 5 R 6 R 1 = [ x, y, z ] R 2 = [x, y, z ] R 3 = [ x + 1, y, z ] R 4 = [x, y + 1, z ] R 5 = [y, x, z + ½] R 6 = [y, x, z ½] (94) P {R 1, R 2, R 3, S} (R 1 R 2 ) 2 = (R 3 R 2 R 3 R 1 ) 2 = SR 3 S 1 R 3 = R 1 R 3 R 2 S 2 R 1 R 3 R 2 S 2 = R 1 R 2 R 3 S 2 R 1 R 2 R 3 S 2 = E X = R 3 R 2 R 1 Y = R 1 R 3 R 2 Z = S 2 R 3 R 1 = [y, x, z ] R 2 = [ y, x, z ] R 3 = [ x + 1, y, z] S = [ y + ½, x ½, z + ½] 28

31 (95) P (X, R 1, R 2, Y = R 2 XR 2 } S = R 2 R 1 (R 1 X) 2 = (X 1 R 2 ) 2 (XR 2 ) 2 = (X 1 (R 2 R 1 ) 2 R 2 ) 2 = E Y = R 2 XR 2 Z = (R 2 R 1 ) 2 R 1 = [ x, y, z ] R 2 = [y, x, z + ¼] (96) P {S 1, S 2, S 3 = S 2 1 S 1 2 S } R = S 2 S 1 S 2 2 S 2 3 = (S 1 S 1 2 ) 2 = (S 1 S 2 ) 2 = (S 1 S 1 3 ) 2 = (S 1 S 3 ) 2 = S 4 1 S 2 2 S 4 1 S 2 2 = E X = S 1 S S 1 Y = S 2 Z = S 1 S 1 = [y, x, z + ¼] S 2 = [ x + ½, y + ½, z + ½] S 3 = [ x + ½, y ½, z + ½] (97) I422 {F, R 1, R 2, F 2 = R 1 F 1 R 1 } (FF 2 ) 2 = (FR 2 ) 2 = (FR 1 R 2 R 1 ) 2 = E X = (R 1 F 1 ) 2 Y = (R 1 F) 2 Z = (R 2 R 1 ) 2 W = R 2 R 1 F 2 R 1 = [ y + ½, x + ½, z + ½] R 2 = [y, x, z ] F 2 = [y, x + 1, z ] (98) I {R 1, R 2, R 3, R 4 = SR 3 S 1, R 5 = S 2 R 3 S 2, R 6 = S 1 R 3 S} S = R 2 R 1 R 3 R 4 R 5 R 6 = (R 1 R 4 ) 2 = (R 1 R 6 ) 2 = (R 4 R 3 R 6 ) 2 = E X = R 3 R 4 Y = R 3 R 6 Z = S 4 W = R 3 S 2 R 1 = [ y, x, z ] R 2 = [x, y, z + ¼] R 3 = [ x + ½, y + ½, z] R 4 = [ x ½, y + ½, z] R 5 = [ x ½, y ½, z] R 6 = [ x + ½, y ½, z] 29

32 (99) P4mm {Z, M 1, M 2, M 3 } F = M 1 M 3 (M 1 M 2 ) 2 = (M 2 M 3 ) 4 = (M 3 M 1 ) 4 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = M 3 ZM 3 Z 1 = E X = M 3 M 2 M 3 M 1 Y = M 3 M 1 M 3 M 2 M 1 = [ x, y, z] M 2 = [x, y, z] M 3 = [ x + ½, y + ½, z] (100) P4bm {Z, M, F} B = MF (MFMF 1 ) 2 = ZMZ 1 M = ZFZ 1 F 1 = E X = (MF 1 ) 2 Y = B 2 M = [ y + ½, x + ½, z] F = [ y, x, z] (101) P4 2 cm {M, R, C} S = CM CRC 1 R = (MCMC 1 ) 2 = (RCMC 1 RM) 2 = E X = RC 1 RMCM Y = MRC 1 RMC Z = C 2 M = [y, x, z] R = [ x + 1, y, z] C = [x, y, z + ½] (102) P4 2 nm {M 1, M 2, R, S} (M 1 M 2 ) 2 = SRS 1 R = (RM 2 RM 1 ) 2 = RM 1 M 2 S 2 RM 1 M 2 S 2 = M 2 RM 1 S 2 M 2 RM 1 S 2 = E X = RM 1 M 2 Y = M 1 RM 2 Z = RS 2 M 1 = [y, x, z] M 2 = [ x, y, z] R = [ x + 1, y, z] S = [ y + ½, x + ½, z + ½] 30

33 (103) P4cc {F 1, F 2, C} CF 1 C 1 F 1 = CF 2 C 1 F 2 = F 1 F 2 F 2 1 F 2 F 1 F 2 2 = E 1 X = F 2 F 1 Y = F 1 2 F 1 Z = C 2 F 1 = [ y, x, z] F 2 = [ y + 1, x, z] C = [y, x, z + ½] 2 (104) P4nc {F 1, F 2, C} N = CF 1 CF 1 C 1 F 2 = CF 2 C 1 F 1 = F 1 F 2 F 2 1 F 2 F 1 F 2 2 = E 1 X = F 2 F 1 Y = F 1 2 F 1 Z = C 2 F 1 = [ y, x, z] F 2 = [ y + 1, x, z] C = [ y + ½, x + ½, z + ½] (105) P4 2 mc {M 1, M 2, C, M 3 = CM 1 C 1, M 4 = CM 2 C 1 } S = CM 2 (M 1 M 2 ) 2 = (M 1 CM 1 C 1 ) 2 = (M 2 CM 2 C 1 ) 2 = M 1 C 2 M 1 C 2 = M 2 C 2 M 2 C 2 = E X = M 4 M 1 Y = M 3 M 2 Z = C 2 M 1 = [ x + 1, y, z ] M 2 = [x, y, z ] C = [y, z, z + ½] M 3 = [x, y + 1, z ] M 4 = [ x, y, z ] (106) P4 2 bc {A, C, B = CA 1 C 1 } C 2 AC 2 A 1 = A 2 CA 2 C 2 = E X = A 2 Y = B 2 Z = C 2 A = [ x + ½, y + ½, z] C = [ y + ½, x + ½, z + ½] B = [x + ½, y + ½, z + ½] 31

34 (107) I4mm {M 1, M 2, C, M 3 = CM 1 C 1 } F = M 2 M 1 (M 2 M 1 ) 4 = (CM 1 C 1 M 1 ) 2 = CM 2 C 1 M 2 = C 2 M 1 C 2 M 1 = E X = CM 1 C 1 M 2 M 1 M 2 Y = M 2 CM 1 C 1 M 2 M 1 Z = C 2 W = CM 1 M 2 M 1 M 1 = [x, y, z] M 2 = [y, x, z] C = [ y + ½, x + ½, z + ½] M 3 = [ x, y, z] (108) I4cm {F, C, M} CMC 1 M = FCFC 1 = (FMF 1 M) 2 = E X = (MF 1 ) 2 Y = (MF) 2 Z = C 2 W = F 2 CM C = [y, x, z + ½] M = [ y + ½, x + ½, z] (109) I4 1 md {M, S, M 2 = SMS 1, M 3 = S 2 MS 2, M 4 = S 1 MS } D = MS (MSMS 1 ) 2 = S 4 MS 4 M = E X = S 2 MS 2 M Y = S 1 MS 2 MS 1 Z = S 4 W = D 2 M = [ x, y, z] S = [ y + ½, x, z + ¼] M 2 = [x, y, z] M 3 = [ x + 1, y, z] M 4 = [x, y + 1, z] (110) I4 1 cd {A, S, B = S 1 AS} (AB) 2 = (AB 1 ) 2 = S 4 AS 4 A 1 = E X = A 2 Y = B 2 Z = S 4 W = AB 1 S 2 = ASA 1 S A = [x + ½, y + ½, z] S = [y + ½, x, z + ¼] B = [ x + ½, y + ½, z] 32

35 (111) P 4 2m {Z, R 1, R 2, M} F = MR 1 (R 1 R 2 ) 2 = (R 1 M) 4 = (R 2 M) 4 = (R 1 Z) 2 = (R 2 Z) 2 = MZMZ 1 = E X = R 1 MR 2 M Y = R 2 MR 1 M R 1 = [ x + 1, y, z ] R 2 = [x, y, z ] M = [y, x, z] (112) P 4 2c {R 1, R 2, C, R 3 = CR 2 C 1, R 4 = CR 1 C 1 } F = R 2 C (C 2 R 1 ) 2 = (C 2 R 2 ) 2 = (R 1 C) 4 = (R 2 C) 4 = E X = R 1 R 3 Y = R 4 R 2 Z = C 2 R 1 = [ x + 1, y, z + ½] R 2 = [x, y, z + ½] C = [y, x, z + ½] R 3 = [ x, y, z + ½] R 4 = [ x + 1, y, z + ½] (113) P m { F, Z, M} S = M F 1 (M F M F 1 ) 2 = MZMZ 1 = Z F 1 Z F = E X = (M F ) 2 Y = (M F 1 ) 2 M = [ y + ½, x + ½, z] (114) P c { F, C, F 2 = C F C} S = C F (C F C F 1 ) 2 = F C 2 F 1 C 2 = E X = (C F ) 2 Y = (C F 1 ) 2 Z = C 2 C = [ y + ½, x + ½, z + ½] F 2 = [ y, x + 1, z] 33

36 (115) P 4 m2 {Z, M 1, M 2, R} F = M 1 R (M 1 M 2 ) 2 = (M 1 R) 4 = (M 2 R) 4 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = (RZ) 2 = E X = M 2 RM 1 R Y = RM 2 RM 1 M 1 = [x, y, z] M 2 = [ x + 1, y, z] R = [y, x, z ] (116) P 4 c2 { F, R, C} RCRC 1 = F 2 C F 2 C 1 = ( F R) 4 = ( F C) 2 = E X = F 2 R Y = F R F Z = C 2 R = [ x + 1, y, z] C = [ x, y, z + ½] (117) F 4 b2 {Z, F, R, } B = R F 1 ( F R F 1 R) 2 = F Z F 1 Z = (RZ) 2 = E X = (R F ) 2 Y = B 2 R = [ y + ½, x + ½, z ] (118) P 4 n2 { F, R 1, R 2, F 2 = R 2 F 1 R 1 } N = F R 2 ( F R 1 R 2 ) 4 = ( F R 1 F 1 R 2 ) 2 = F R 2 R 1 F 1 R 2 R 1 = F R 1 R 2 F 1 R 1 R 2 = E X = R 2 F R 1 F Y = F 1 R 2 F R 1 F 2 Z = R 2 R 1 R 1 = [ y + ½, x + ½, z ½] R 2 = [ y + ½, x + ½, z + ½] F 2 = [ y, x + 1, z] 34

37 (119) I 4 m2 {M, R 1, R 2, R 3 = R 2 R 1 R 2, M 2 = R 1 MR 1 } F = R 1 M M 2 R 3 MR 3 = (MM 2 ) 2 = (R 1 M) 4 = (R 3 M) 4 = (R 3 M) 4 = (R 2 M) 4 = (R 2 M 2 ) 4 = M(R 2 R 1 ) 2 M(R 1 R 2 ) 2 = E X = MR 2 M 2 R 2 Y = M 2 R 2 MR 2 Z = R 3 R 1 W = R 2 MR 1 M M = [ x, y, z] R 1 = [y, x, z ] R 2 = [ y + ½, x + ½, z + ½] R 3 = [y, x, z + 1] M 2 = [x, y, z] (120) I 4 c2 { F, R 1, R 2, R 3 = R 2 R 1 R 2 } C = R 1 F 1 R 3 F R 1 F = ( F R 1 ) 2 = ( F R 3 ) 2 = ( F R 2 F 1 R 2 ) 2 = E X = (R 2 F ) 2 Y = (R 2 F 1 ) 2 Z = R 1 R 3 W = R 1 R 2 F 2 F = [ y, x, z + ½] R 1 = [y, x, z ] R 2 = [ y + ½, x + ½, z + ½] R 3 = [y, x, z + 1] (121) I 4 2m {M, R, C, R 2 = CRC} F = MR (C 2 R) 2 = (MR) 4 = (RC) 4 = (CMRM) 4 = MC 2 MC 2 = MRMC 2 MRMC 2 = E X = R 2 MRM Y = R 2 MR 2 R Z = C 2 W = CRMR M = [y, x, z] R = [x, y, z ] C = [ y + ½, x + ½, z + ½] R 2 = [ x + 1, y, z ] (122) I 4 2d { F, R, R 2 = F R F 1, F 2 = RR 2 F R 2 R} D = R F F 2 R F R F 1 R F 2 R F 1 R F R = E X = ( F R F ) 2 Y = (R F 2 ) 2 Z = (R F R F ) 2 W = D 2 F = [y, x, z + ¼] R = [x, y + ½, z ] R 2 = [x, y + ½, z + ½] F 2 = [y + 1, x, z + ¼] 35

38 (123) P4/mmm {M 1, M 2, M 3, M 4, M 5 } F = M 1 M 2 (M 2 M 3 ) 4 = (M 3 M 1 ) 2 = (M 1 M 2 ) 4 = (M 1 M 4 ) 2 = (M 2 M 4 ) 2 = (M 3 M 4 ) 2 = (M 1 M 5 ) 2 = (M 2 M 5 ) 2 = (M 3 M 5 ) 2 = E X = M 2 M 3 M 2 M 1 Y = M 3 M 2 M 1 M 2 Z = M 5 M 4 M 1 = [ x, y, z] M 2 = [y, x, z] M 3 = [x, y + 1, z] M 4 = [x, y, z ½] M 5 = [x, y, z + 1] (124) P4/mcc {M, R 1, R 2, R 3, M 2 = R 1 MR 1 } F = R 1 R 3 C 1 = R 1 M C 2 = R 3 M M 2 R 2 MR 2 = M 2 R 3 MR 3 = (R 1 R 2 ) 2 = (R 1 MR 1 R 3 ) 4 = (R 3 R 1 ) 4 = (R 3 R 2 ) 4 = (R 1 R 2 M) 2 = (R 1 MR 1 MR 1 ) 2 = (R 2 MR 1 MR 1 ) 2 = (R 3 M R 1 MR 1 ) 2 = E X = R 3 R 2 R 3 R 1 Y = R 2 R 3 R 1 R 3 Z = M 2 M M = [x, y, z ] R 1 = [ x, y, z + ½] R 2 = [x, y + 1, z + ½] R 3 = [y, x, z + ½] M 2 = [x, y, z + 1] (125) P4/nbm {Z, M, R 1, R 2 } F = R 1 R 2 N = MFR 1 B = MF (R 1 R 2 ) 4 = (R 1 M) 4 = (R 2 M) 2 = MZMZ 1 = (R 1 Z) 2 = (R 2 Z) 2 = E X = (MR 2 R 1 ) 2 Y = B 2 M = [ y + ½, x + ½, z] R 1 = [ x, y, z ] R 2 = [y, x, z ] (126) P4/nnc {R 1, R 2, C, R 3 = CR 1 C} F = R 1 R 2 N 1 = CRF 1 N 2 = CF (R 1 R 2 ) 4 = (R 1 C) 4 = (R 2 C) 2 = (R 1 C 2 ) 2 = E X = (R 2 CR 1 ) 2 Y = R 1 CFC R 2 Z = C 2 R 1 = [ x, y, z ] R 2 = [y, x, z ] C = [ y + ½, x + ½, z + ½] R 3 = [x, y + 1, z ] 36

39 (127) P4/mbm {F, M 1, M 2, M 3 } B = M 2 F (M 2 M 3 ) 2 = (M 1 M 2 ) 2 = M 1 FM 1 F 1 = (M 2 FM 2 F 1 ) 2 = (FM 3 ) 4 = FM 3 F 1 M 3 = E X = (M 2 F 1 ) 2 Y = (M 2 F) 2 Z = M 3 M 1 M 1 = [x, y, z ] M 2 = [ y + ½, x + ½, z] M 3 = [x, y, z + 1] (128) P4/mnc {F, M, R, M 2 = RMR } N = RMF C = RM (FRF 1 R) 2 = FMF 1 M = FRMRF 1 RMR = E X = (RF 1 ) 2 Y = (RF) 2 Z = M 2 M M = [x, y, z ] R = [ y + ½, x + ½, z + ½] M 2 = [x, y, z + 1] (129) P4/nmm {Z, M 1, M 2, R} F = M 2 M 1 N = R 1 FM 2 RM 2 RR 2 MR 2 = (RM 1 ) 2 = (RM 2 ) 4 = (R 2 M 1 ) 2 = (M 1 RM 2 R) 2 = (M 1 M 2 ) 4 = (R RM 2 R) 4 = (R 2 RM 2 R) 4 = (M 2 RM 2 R) 4 = R 2 RM 2 RR 2 M 2 = E X = M 1 RM 2 RM 1 M 2 Y = M 1 M 2 M 1 RM 2 R M 1 = [y, x, z] M 2 = [ x, y, z] R = [ y + ½, x + ½, z ] R 2 = [ y + ½, x + ½, z + 1] (130) P4/ncc {F, R, C} N = RF 2 C (RC 2 ) 2 = FC 2 F 1 C 2 = F 2 CF 2 C 1 = F 2 (RFRF 1 R)F 2 (RF 1 RFR) = E X = (RF) 2 Y = (FR) 2 Z = C 2 = C 2 2 R = [y + ½, x + ½, z + ½] C = [y, x, z + ½] 37

40 (131) P4 2 /mmc {M 1, M 2, M 3, R, M 4 = RM 3 R} C = RM 3 (M 1 R) 4 = (M 2 R) 4 = (M 1 M 2 ) 2 = (M 1 M 3 ) 2 = (M 2 M 3 ) 2 = (M 1 M 4 ) 2 = (M 2 M 4 ) 2 = (RM 4 ) 2 = (M 1 M 2 M 3 ) 2 = (M 1 M 2 M 4 ) 2 = E X = RM 2 RM 1 Y = RM 1 RM 2 Z = C 2 M 1 = [ x, y, z] M 2 = [x, y, z] M 3 = [x, y, z ] R = [y + ½, x + ½, z + ½] M 4 = [x, y, z + 1] (132) P4 2 /mcm {M 1, M 2, R 1, R 2, M 3 = R 1 M 2 R 1 } C = R 1 M 2 S = M 1 C M 3 R 2 M 2 R 2 = (R 1 R 2 ) 2 = (M 1 M 2 ) 2 = (M 1 M 3 ) 2 = (R 1 M 1 ) 4 = (R 2 M 1 ) 4 = (R 1 R 2 M 2 ) 2 = E X = R 2 M 1 R 1 M 1 Y = R 1 M 1 R 2 M 1 Z = C 2 M 1 = [y, x, z] M 2 = [x, y, z ] R 1 = [x, y, z + ½] R 2 = [ x + 1, y, z + ½] M 3 = [x, y, z + 1] (133) P4 2 /nbc {R 1, R 2, C, R 3 = CR 2 C} N = C 1 R 2 R 1 R 2 B = C 1 R 2 R 1 (CR 2 C) 2 = (R 1 C) 2 = (R 2 C 2 ) 2 = E X = C 2 R 1 R 2 R 1 Y = C 2 R 1 R 3 R 1 Z = C 2 R 1 = [y, x, z ] R 2 = [x, y, z + ½] C = [ y + ½, x + ½, z + ½] R 3 = [x, y + 1, z + ½] (134) P4 2 /nnm {M, R 1, R 2, R 3, R 4 = R 3 R 1 R 2 } N 1 = R 3 R 1 MR 1 N 2 = R 1 R 3 M (R 1 R 4 ) 2 = (R 1 M) 4 = (R 4 M) 4 = (R 2 M) 2 = (R 3 M) 2 = (R 1 R 2 R 3 ) 2 = (MR 1 MR 2 R 3 ) 2 = E X = R 4 MR 1 M Y = MR 4 MR 1 Z = R 3 R 2 M = [y, x, z] R 1 = [x, y, z + ½] R 2 = [ y + ½, x + ½, z ] R 3 = [ y + ½, x + ½, z + 1] R 4 = [ x + 1, y, z + ½] 38

41 (135) P4 2 /mbc { F, M, R, M 2 = RMR} S = F 1 M B = R F 1 C = RM M 2 F M F 1 = ( F 2 M) 2 = F C 2 F 1 C 2 = F 2 (R F R F 1 R) F 2 (R F 1 R F R) = (RC 2 ) 2 = E X = (R F ) 2 Y = B 2 Z = C 2 M = [x, y, z ] R = [ y + ½, x + ½, z + ½] M 2 = [x, y, z + 1] (136) P4 2 /mnm { F, M 1, M 2, M 3 = F M 2 F 1 S = F 1 M 2 N = M 1 F 1 M 2 (M 1 M 2 ) 2 = (M 1 M 3 ) 2 = ( F M 1 F ) 2 = ( F 2 M 2 ) 2 = ( F 2 M 3 ) 2 = F 2 (M 1 F M 1 F 1 M 1 ) F 2 (M 1 F 1 M 1 F M 1 ) = (M 1 M 2 M 3 ) 2 = (M 1 F M 2 F M 2 ) 2 = E X = (M 1 F ) 2 Y = (M 1 F 1 ) 2 Z = M 3 M 2 F = [y, x, z + ½] M 1 = [ y + ½, x + ½, z] M 2 = [x, y, z ] M 3 = [x, y, z + 1] (137) P4 2 /nmc {R, M, C, M 2 = CMC 1 } S = MC 1 N = RMCM (RM) 4 = (RC) 2 = (MM 2 ) 2 = (C 2 MR) 4 = MC 2 MC 2 = E X = (MCR) 2 Y = (RMC) 2 Z = C 2 R = [ y + ½, x + ½, z + ½] M = [ x, y, z] C = [y, x, z + ½] M 2 = [x, y, z] (138) P4 2 /ncm {R, C, M, R 2 = CRC, M 2 = CMC 1 } S = CM N = MR 1 R 2 (R 1 R 2 ) 2 = (RM) 2 = (R 2 M 2 ) 2 = (MM 2 ) 2 = (RC 2 ) 2 = MC 2 MC 2 = (MRC 2 ) 2 = E X = (MC 1 R) 2 Y = (RMC) 2 Z = C 2 R = [y, x, z + ½] C = [x, y, z + ½] M = [ y + ½, x + ½, z] R 2 = [ y, x, z + ½] M 2 = [ y + ½, x ½, z] 39

42 (139) I4/mmm {M 1, M 2, M 3, R} F = M 2 M 1 (RM 1 ) 4 = (RM 2 ) 2 = (M 1 M 2 ) 4 = (M 1 M 3 ) 2 = (M 2 M 3 ) 2 = (M 1 RM 2 M 1 M 3 RM 2 ) 2 = E X = (RM 2 M 1 ) 2 Y = (M 2 M 1 R) 2 Z = (RM 3 ) 2 W = RM 1 M 2 M 1 M 3 M 1 = [x, y, z] M 2 = [y, x, z] M 3 = [x, y, z ] R = [ y + ½, x + ½, z + ½] (140) I4/mcm {M 1, M 2, R 1, R 2, M 3 = R 1 M 1 R 1 } F = R 1 R 2 C = R 1 R 2 M M 3 R 2 M 1 R 2 = (R 2 M 2 ) 2 = (R 1 R 2 ) 4 = (M 1 M 2 ) 2 = E X = M 2 R 2 R 1 R 2 M 2 R 1 Y = M 2 R 1 M 2 R 2 R 1 R 2 Z = M 3 M 1 W = R 2 R 1 R 2 R 1 M 1 M 1 = [x, y, z ] M 2 = [ y + ½, x + ½, z] R 1 = [ x, y, z + ½] R 2 = [y, x, z + ½] M 3 = [x, y, z + 1] (141) I4 1 /amd { R 1, R 2, M, M 2 = SMS 1, M 3 = S 2 MS 2, M 4 = S 1 MS} S = R 2 R 1 A = R 2 SM D = AR 2 (R 1 M) 2 = (R 1 M 3 ) 2 = (R 2 M) 4 = (R 2 M 2 ) 4 = (R 2 M 3 ) 4 = (R 2 M 4 ) 4 = (MM 2 ) 2 = E X = A 2 Y = R 2 A 2 R 2 Z = S 4 W = D 2 R 1 = [x, y + ½, z ] R 2 = [y, x, z + ¼] M = [ x, y, z] M 2 = [x, y, z] M 3 = [ x + 1, y, z] M 4 = [x, y + 1, z] (142) I4 1 /acd {R 1, R 2, B, A 1 = R 2 BR 2 } S = R 2 R 1 A = R 2 BS C = B 1 S 2 D = BS S 4 C 2 = (BR 1 ) 2 = (BR 2 ) 4 = B(R 2 R 1 R 2 )B 1 (R 2 R 1 R 2 ) = AC 2 A 1 C 2 = CA 2 C 1 A 2 = BA 2 B 1 A 2 = (R 2 C) 4 = E X = A 2 Y = B 2 Z = C 2 W = D 2 R 1 = [x, y + ½, z ] R 2 = [y, x, z + ¼] B = [ x + ½, y + ½, z] A 1 = [x + ½, y + ½, z] 40

43 Trigonal (143) P3 {Z, Q 1, Q 2 } (Q 1 Q 2 ) 3 = ZQ 1 Z 1 Q 1 1 = ZQ 2 Z 1 Q 1 2 = E X = Q Q 2 Y = Q 2 Q 1 Q 1 = [ y + 1, x y, z] Q 2 = [ y + 1, x y + 1, z] (144) P3 1 {X, S, Y = SXS 1, XY } XYX 1 Y 1 = S 3 XS 3 X 1 = E Z = S 3 S = [ y, x y, z + 1 / 3 ] (145) P3 2 {X, S, Y = S 1 XS XY } XYX 1 Y 1 = S 3 XS 3 X 1 = E Z = S 3 S = [y x, x, z + 1 / 3 ] (146) R3 {S, Q, Q 2 = SQS 1, Q 3 = S 1 QS} QS 3 Q 1 S 3 = Q 2 QQ 2 1 Q 1 = (QQ 2 ) 3 = (QS) 2 QS 2 = E X = Q 2 Q 1 Y = QQ 2 Q Z = S 3 T = SQ 1 S = [ 2 / 3 y, 1 / 3 + x y, 1 / 3 + z] Q 2 = [ y + 1, x y, z] Q 3 = [ y + 1, x y + 1, z] 41

44 (147) P 3 {Z, Q, I} ( Q I) 3 = (ZI) 2 = Q Z Q 1 Z = E X = Q I Q 2 Y = Q 1 IQ 2 Q = [y, y x, 1 / 3 z] I = [1 x, 1 y, 1 / 3 z] (148) R 3 { Q, S, I = S Q S} (S Q S) 2 = (IQ ) 3 = (S Q 1 ) 2 = (S 2 Q ) 2 = E X = I Q 1 IQ Y = I Q I Q 1 Z = S 3 T = IQ 1 S S = [ 2 / 3 y, 1 / 3 + x y, 1 / 3 + z] I = [1 x, 1 y, z] (149) P312 {Z, R 1, R 2, R 3 } Q = R 2 R 1 (R 2 R 3 ) 3 = (R 3 R 1 ) 3 = (R 1 R 2 ) 3 = (R 1 Z) 2 = (R 2 Z) 2 = (R 3 Z) 2 = E X = R 3 R 1 R 3 R 2 Y = R 3 R 2 R 3 R 1 R 1 = [x, x y, z] R 2 = [y x, y, z] R 3 = [1 y, 1 x, z] (150) P321 {Q, R 1, R 2, Q 2 = R 1 Q 1 R 1 } Q 2 R 2 QR 2 = (QQ 2 ) 3 = E X = Q 1 Q 2 Y = Q 2 Q 1 Z = R 2 R 1 Q = [1 y, x y, z] R 1 = [y, x, z] R 2 = [y, x, 1 z] Q 2 = [1 y, 1 + x y, z] 42

45 (151) P {X, R 1, R 2, Y = R 2 R 1 XR 1 R 2, XY} (R 2 X) 2 = (X 1 R 1 ) 2 (XR 1 ) 2 = (XR 1 R 2 R 1 R 2 R 1 ) 2 = E Y = R 2 R 1 XR 1 R 2 Z = (R 2 R 1 ) 3 R 1 = [x, x y, z] R 2 = [ x + y, y, 1 / 3 z] (152) P {X, R 1, R 2, Y = R 1 XR 1, XY} (R 1 X) 2 (R 1 X 1 ) 2 = X(R 1 R 2 ) 2 X 1 (R 1 R 2 ) 2 = E Y = R 1 XR 1 Z = (R 2 R 1 ) 3 R 1 = [y, x, z] R 2 = [ x, x + y, 1 / 3 z] (153) P {X, R 1, R 2, Y = R 1 R 2 XR 2 R 1, XY } (R 1 X) 2 = (R 2 X) 2 (R 2 X 1 ) 2 = (XR 2 R 1 R 2 R 1 R 2 ) 2 = E Y = R 1 R 2 XR 2 R 1 Z = (R 2 R 1 ) 3 R 1 = [ x + y, y, z] R 2 = [x, x y, 1 / 3 z] (154) P {X, R 1, R 2, Y = R 2 XR 2, XY } (R 2 X) 2 (R 2 X 1 ) 2 = X(R 2 R 1 ) 2 X 1 (R 2 R 1 ) 2 = E Y = R 2 XR 2 Z = (R 2 R 1 ) 3 R 1 = [ x, x + y, z] R 2 = [y, x, 1 / 3 z] 43

46 (155) R32 {Q R 1, R 2, Q 2 = R 2 Q 1 R 2, Q 3 = R 1 Q 2 1 R 1 } QR 1 Q 1 R 1 = (QQ 2 ) 3 = (QQ 2 Q)R 1 R 2 R 1 R 2 R 1 (QQ 2 Q) 1 R 1 R 2 R 1 R 2 R 1 = E X = Q 2 Q 1 Y = QQ 2 Q Z = (R 2 R 1 ) 3 R 1 = [y, x, z] R 2 = [ 2 / 3 x, 1 / 3 x + y, 1 / 3 z] Q 2 = [1 y, x y, z] Q 3 = [1 y, 1 + x y, z] (156) P3m1 {Z, M 1, M 2, M 3 } Q = M 2 M 1 (M 2 M 3 ) 3 = (M 3 M 1 ) 2 = (M 1 M 2 ) 2 = M i ZM i Z 1 = E X = M 1 M 3 M 1 M 2 Y = M 2 M 3 M 2 M 1 M 1 = [x, x y, z] M 2 = [ x + y, y, z] M 3 = [1 y, 1 x, z] (157) P31m {Z, M, Q} (QMQ 1 M) 3 = QZQ 1 Z 1 = MZMZ 1 = E X = QMQ 1 MQ Y = (Q 1 M) 2 M = [x y, y, z] Q = [1 y, x y, z] (158) P3c1 {C, Q 1, Q 2 } B = (Q 1 Q 2 ) 1 (Q 1 Q 2 ) 3 = Q 1 CQ 1 C 1 = Q 2 CQ 2 C 1 = E 1 1 X = Q 1 Q 2 Y = Q 2 Q 1 Z = C 2 C = [1 y, 1 x, ½ + z] Q 1 = [1 y, x y, z] Q 2 = [1 y, 1 + x y, z] 44

47 (159) P31c {C, Q, Q 2 = CQ 1 C 1 } (QQ 2 ) 3 = C 2 QC 2 Q 1 = E X = Q 1 Q 2 Y = Q 2 Q 1 Z = C 2 C = [y, x, ½ + z] Q = [1 y, x y, z] Q 2 = [1 y, 1 + x y, z] (160) R3m {M, S, M 2 = SMS 1, M 3 = S 1 MS} Q = S 2 MS 2 M MS 3 MS 3 = (M 2 M) 3 = E X = MSMS 1 MS 1 MS Y = SXS 1 Z = S 3 T = SMS 1 MS M = [x, x y, z] S = [ 2 / 3 y, 1 / 3 + x y, 1 / 3 + z] M 2 = [ y, x, z] M 3 = [ x + y, y, z] (161) R3c {S, C, S 2 = CSC 1, Q = S 1 S 2, Q 2 = S 2 S 1 } (Q 1 Q 2 ) 3 = S 3 C 2 = E X = Q 1 Q 2 Y = Q 2 Q 1 Z = C 2 T = S 2 2 S 1 S = [ 2 / 3 y, 1 / 3 + x y, ½ + z] C = [x, x y, ½ + z] S 2 = [ 2 / 3 x + y, 1 / 3 x, ½ + z] Q = [1 y, x y, z] Q 2 = [1 y, 1 + x y, z] (162) P 3 1m {Z, M, R 1, R 2 } Q = MR 2 (R 1 R 2 ) 3 = (MR 1 ) 2 = (MR 2 ) 6 = (ZR 1 ) 2 = (ZR 2 ) 2 = MZMZ 1 = E X = R 1 R 2 (MR 2 ) 2 Y = R 2 MR 2 R 1 R 2 M M = [x y, y, z] R 1 = [1 x + y, y, z] R 2 = [x, x y, z] 45

48 (163) P 3 1c {C, R 1, R 2, R 3 = CR 1 C} Q = C 1 R 1 R 3 C 1 R 2 C = (R 2 R 3 ) 3 = (R 3 R 1 ) 3 = (R 1 R 2 ) 3 = (CR 2 ) 2 = (R 1 C) 6 = E X = R 2 R 3 R 2 R 1 Y = R 2 R 1 R 2 R 3 Z = C 2 C = [y, x, ½ + z] R 1 = [ x + y, y, ½ z] R 2 = [ y, x, ½ z] R 3 = [x, x y, ½ z] (164) P 3 m1 { M 1, M 2, R 1, R 2, M 3 = R 1 M 1 R 1 } Q = M 1 R M 3 R 2 M 1 R 2 = (M 2 M 3 ) 3 = (M 3 M 1 ) 3 = (M 1 M 2 ) 3 = (M 1 R 1 ) 6 = (M 2 R 1 ) 2 = (M 2 R 2 ) 2 = E X = M 1 M 2 M 1 M 3 Y = M 2 M 3 M 2 M 1 Z = R 2 R 1 M 1 = [x, x y, z] M 2 = [1 y, 1 x, z] R = [y, x, z] R 2 = [y, x, 1 z] M 3 = [ x + y, y, z] (165) P 3 c1 {R, C, Q} Q = RQ 1 CQ (QRQ 1 R) 3 = (RQ 1 CQ) 2 Q = RCRC 1 = CQC 1 RQR = C 2 QRQC 2 (QRQ) 1 = E X = (Q 1 R) 2 Y = (RQ 1 ) 2 Z = C 2 R = [y, x, ½ z] C = [1 y, 1 x, ½ + z] Q = [1 y, x y, z] (166) R 3 m {M, R 1, R 2, M 2 = R 1 MR 1, M 3 = R 2 MR 2 } Q = MR 1 (MR 2 R 1 R 2 ) 2 = (MR 1 R 2 R 1 ) 2 = (MR 2 ) 6 = (MR 1 ) 6 = E X = MM 3 MM 2 Y = M 3 M 2 M 3 M Z = (R 2 R 1 ) 3 T = R 2 MR 1 M M = [x, x y, z] R 1 = [y, x, z] R 2 = [ 1 / 3 + x y, 2 / 3 y, 1 / 3 z] M 2 = [ x + y, y, z] M 3 = [1 y, 1 x, z] 46

49 (167) R 3 c { Q, I, R 1, R 2, } ( Q I) 6 = ( Q 2 IQ I Q I) 2 = Q R 2 Q 1 R 1 = (I(R 2 R 1 ) 3 ) 2 = Q 2 IQIR 2 R 1 Q 2 IQ 1 I R 1 R 2 = E X = I Q 1 IQ Y = I Q I Q 1 Z = (R 2 R 1 ) 3 T = R 2 R 1 Q 2 I = [1 x, 1 y, z] R 1 = [ 2 / 3 x, 1 / 3 + y x, 1 / 6 z] R 2 = [ 1 / 3 + x y, 2 / 3 y, 1 / 6 z] 47

50 Hexagonal (168) P6 {Z, H, Q} QZQ 1 Z 1 = HZH 1 Z 1 = (Q 1 H) 2 = E X = QH 2 Y = (QH 2 Q) 1 Q = [1 y, x y, z] (169) P6 1 {X, S, Y = S 2 XS 2 XY = SXS 1 } S 3 XS 3 X = X(SXS 1 )X 1 (SXS 1 ) 1 = E Y = S 2 XS 2 Z = S 6 S = [x y, x, z + 1 / 6 ] (170) P6 5 {X, S, Y = S 2 XS 2 XY = SXS 1 } S 3 XS 3 X = X(SXS 1 )X 1 (SXS 1 ) 1 = E Y = S 2 XS 2 Z = S 6 S = [x y, x, z 1 / 6 ] (171) P6 2 {R, S, R 2 = SRS 1, R 3 = S 1 RS} (RR 3 S) 2 = (RR 2 R 3 ) 2 = S 3 RSRS 3 RS 1 R = E X = R 2 R Y = RR 3 Z = S 3 R = [1 x, 1 y, z] S = [ x + y + 1, x + 1, z + 1 / 3 ] R 2 = [2 x, 1 y, z] R 3 = [1 x, y, z] 48

51 (172) P6 4 {R, S, R 2 = SRS 1, R 3 = S 1 RS } (RR 3 S) 2 = (RR 2 R 3 ) 2 = S 3 RSRS 3 RS 1 R = E X = R 3 R Y = RR 2 Z = S 3 R = [1 x, 1 y, z] S = [ y + 1, x y, z + 1 / 3 ] R 2 = [1 x, y, z] R 3 = [2 x, 1 y, z] (173) P6 3 {Q, S, Q 2 = SQS 1 } (QQ 2 ) 3 = S 2 QS 2 Q 1 = E X = Q 1 Q 2 Y = Q 2 Q 1 Z = S 2 Q = [1 y, x y, z] S = [1 x, 1 y, ½ + z] Q 2 = [1 y, 1 + x y, z] (174) P 6 {Z, H 1, H 2 } ( H 1 H 2 ) 3 = ( H 4 1 H 2 ) 6 = ( H 4 2 H 1 ) 6 = H 3 1 H 3 2 = H 1 Z H 1 1 Z = H 2 Z H 1 2 Z = H 2 1 ( H 1 H 2 ) 1 H 2 ( H 2 H 1 ) 1 = E 1 X = H 1 H 2 Y = H 1 2 H 1 H 1 = [1 + y x, 1 x, z] H 2 = [y x, 1 x, z] (175) P6/m {Z, H, R} (R H ) 6 = (R H 2 ) 6 = (R H 3 ) 2 = ( H 4 R H R) 6 = H Z H 1 Z = H 3 R H 3 R = = ZRZ 1 R = ( H 2 R( H 1 R) 2 ) 2 = E X = H R H 1 R Y = R H 1 R H H = [1 + y x, 1 x, z] R = [1 x, 1 y, z] 49

52 (176) P6 3 /m { H 1, H 2, I, I 2 = H 3 1 I H 3 1 } H 3 1 H 3 2 = ( H 1 H 2 ) 3 = ( H 1 H 2 2 ) 6 = H 2 1 I 2 I H 1 1 I 2 I = H 2 2 I 2 I H 1 2 I 2 I = E 1 X = H 1 H 2 Y = H 1 2 H 1 Z = I 2 I H 1 = [1 + y x, 1 x, ½ z] H 2 = [y x, 1 x, ½ z] I = [1 x, 1 y, z] I 2 = [1 x, 1 y, 1 z] (177) P622 {Z, R 1, R 2, R 3 } H = R 1 R 2 (R 2 R 3 ) 3 = (R 3 R 1 ) 2 = (R 1 R 2 ) 6 = (R 1 Z) 2 = (R 2 Z) 2 = (R 3 Z) 2 = E X = R 2 R 3 H 2 Y = R 1 XR 1 R 1 = [y, x, z] R 2 = [x, x y, z] R 3 = [1 x, 1 y, z] (178) P {X, R 1, R 2, Y = S 1 XS, XY} S = R 2 R 1 (R 1 X) 2 = (XR 2 ) 2 (X 1 R 2 ) 2 = SR 2 XSR 2 SX 1 = E Y = S 1 XS Z = S 6 R 1 = [x y, y, z] R 2 = [x, x y, 1 / 6 z] (179) P {X, R 1, R 2, Y = S 1 XS, XY} S = R 1 R 2 (R 1 X) 2 = (XR 2 ) 2 (X 1 R 2 ) 2 = SR 2 XSR 2 SX 1 = E Y = S 1 XS Z = S 6 R 1 = [x y, y, 1 / 6 z] R 2 = [x, x y, z] 50

53 (180) P {R 1, R 2, R 3, R 4 = R 1 R 3 R 1, R 5 = R 2 R 4 R 2 } (R 2 R 3 ) 2 = (R 1 R 5 ) 2 = ((R 2 R 1 ) 3 R 3 ) 2 = (R 3 R 4 R 5 ) 2 = E X = R 5 R 3 Y = R 3 R 4 Z = (R 2 R 1 ) 3 R 1 = [x, x y, z] R 2 = [1 y, 1 x, 1 / 3 z] R 3 = [1 x, 1 y, z] R 4 = [1 x, y, z] R 5 = [2 x, 1 y, z] (181) P {R 1, R 2, R 3, R 4 = R 1 R 3 R 1, R 5 = R 2 R 4 R 2 } (R 2 R 3 ) 2 = (R 1 R 5 ) 2 = ((R 2 R 1 ) 3 R 3 ) 2 = (R 3 R 4 R 5 ) 2 = E X = R 5 R 3 Y = R 3 R 4 Z = (R 1 R 2 ) 3 R 1 = [x, x y, 1 / 3 z] R 2 = [1 y, 1 x, z] R 3 = [1 x, 1 y, z] R 4 = [1 x, y, z] R 5 = [2 x, 1 y, z] (182) P { R 1, R 2, R 3, R 4, R 5 = R 1 R 4 R 1 } (R 2 R 3 ) 3 = (R 3 R 1 ) 3 = (R 1 R 2 ) 3 = R 5 R 4 R 2 R 1 R 3 R 1 R 4 R 5 R 1 R 3 R 1 R 2 = E X = R 3 R 1 R 3 R 2 Y = R 1 R 2 R 3 R 2 Z = R 5 R 4 R 1 = [1 y, 1 x, ½ z] R 2 = [y x, y, ½ z] R 3 = [x, x y, ½ z] R 4 = [y, x, z] R 5 = [y, x, ½ z] (183) P6mm {Z, M 1, M 2, M 3 } H = M 1 M 2 (M 2 M 3 ) 3 = (M 3 M 1 ) 2 = (M 1 M 2 ) 6 = M 1 ZM 1 Z 1 = M 2 ZM 2 Z 1 = M 3 ZM 3 Z 1 = E X = M 2 M 3 (M 2 M 1 ) 2 Y = (M 3 M 1 M 2 ) 2 M 1 = [y, x, z] M 2 = [x, x y, z] M 3 = [1 y, 1 x, z] 51

54 (184) P6cc {H, C, R} C 2 = HC RCRC 1 = (RH) 3 = (H 2 RHRHR) 2 = R(C 2 H)R(C 2 H) 1 = E X = RC 2 1 RC 2 Y = RC 1 HRH 1 C Z = C 2 C = [y, x, ½ + z] R = [1 x, 1 y, z] (185) P6 3 cm {Q, C, M} MCMC 1 = (QMQ 1 M) 3 = C 2 (QMQ)C 2 (QMQ) 1 = E X = (Q 1 M) 2 Y = (MQ 1 ) 2 Z = C 2 Q = [1 y, x y, z] C = [1 y, 1 x, ½ + z] M = [y, x, z] (186) P6 3 mc {M 1, M 2, C, M 3 = CM 2 C 1 } (M 1 M 2 ) 3 = M 1 CM 1 C 1 = (M 2 CM 2 C 1 ) 3 = M 2 C 2 M 2 C 1 = E X = M 1 M 2 M 3 Y = M 3 M 2 M 3 M 1 Z = C 2 M 1 = [1 y, 1 x, z] M 2 = [x, x y, z] C = [y, x, ½ + z] M 3 = [1 x, 1 y, z] (187) P 6 m2 {M 1, M 2, M 3, M 4, M 5 } Q = M 3 M 2 M 4 (M 1 M 2 ) 3 = (M 2 M 3 ) 3 = (M 3 M 1 ) 3 = (M 1 M 4 ) 2 = (M 2 M 4 ) 2 = (M 3 M 4 ) 2 = (M 1 M 5 ) 2 = (M 2 M 5 ) 2 = (M 3 M 5 ) 2 = E X = M 1 M 2 M 1 M 3 Y = M 3 M 2 M 3 M 1 Z = M 5 M 4 M 1 = [1 y, 1 x, z] M 2 = [x, x y, z] M 3 = [ x + y, y, z] M 4 = [x, y, z] M 5 = [x, y, 1 z] 52

55 (188) P 6 c2 {M, R 1, R 2, R 3, M 2 = R 1 MR 1 } H = MR 1 R 3 C = MR 1 M 2 R 2 MR 2 = M 2 R 3 MR 3 = (R 1 R 2 ) 3 = (R 2 R 3 ) 3 = (R 3 R 1 ) 3 = R 1 MR 1 = R 2 MR 2 = R 3 MR 3 = E R 1 MR 1 = R 2 MR 2 = R 3 MR 3 X = R 1 R 2 R 1 R 3 Y = R 3 R 2 R 3 R 1 Z = M 2 M 1 M = [x, y, z] M 2 = [x, y, 1 z] R 1 = [x, x y, ½ z] R 2 = [1 y, 1 x, ½ z] R 3 = [ x + y, y, ½ z] (189) P 6 2m {M, M 1, M 2, Q} H = QMQ 1 M 1 (QMQ 1 M 1 ) 6 = (QMQ 1 M 2 ) 6 = (MM 1 ) 2 = (MM 2 ) 2 = QM 1 Q 1 M 1 = QM 2 Q 1 M 2 = (MQMQ 1 ) 3 = E X = (Q 1 M) 2 Y = (MQ 1 ) 2 Z = M 2 M 1 Q = [1 y, x y, z] M = [y, x, z] M 1 = [x, y, z] M 2 = [x, y, 1 z] (190) P 6 2c {M, R, Q} H = QRQ 1 RM C = RM QMQ 1 M = (QRQ 1 R) 3 = Q(RMR)Q 1 (RMR) = E X = (Q 1 R) 2 Y = (RQ 1 ) 2 Z = C 2 M = [x, y, z] R = [y, x, ½ z] Q = [1 y, x y, z] (191) P6/mmm {M 1, M 2, M 3, M 4, M 5 } H = M 1 M 2 (M 2 M 3 ) 3 = (M 3 M 1 ) 2 = (M 1 M 2 ) 6 = (M 4 M 1 ) 2 = (M 4 M 2 ) 2 = (M 4 M 3 ) 2 = (M 5 M 1 ) 2 = (M 5 M 2 ) 2 = (M 5 M 3 ) 2 = E X = M 2 M 3 H 2 Y = M 1 XM 1 Z = M 5 M 4 M 1 = [y, x, z] M 2 = [x, x y, z] M 3 = [1 y, 1 x, z] M 4 = [x, y, z] M 5 = [x, y, 1 z] 53

56 (192) P6/mcc {M, R 1, R 2, R 3, M 2 = R 1 MR 1 } H = R 1 R 2 C 1 = M 1 R 2 R 1 R 2 C 2 = M 1 R 1 (R 2 R 3 ) 3 = (R 3 R 1 ) 2 = (R 1 R 2 ) 6 = MR 2 R 3 MR 3 R 2 = MR 3 R 1 MR 1 R 3 = MR 1 R 2 MR 2 R 1 = E X = R 2 R 3 H 2 Y = R 1 XR 1 Z = M 2 M M = [x, y, z] M 2 = [x, y, 1 z] R 1 = [y, x, ½ z] R 2 = [x, x y, ½ z] R 3 = [1 x, 1 y, ½ z] (193) P6 3 /mcm {M 1, M 2, R 1, R 2, M 3 = R 1 M 2 R 1 } C = R 1 M 2 M 3 R 2 M 1 R 2 = (R 1 R 2 ) 3 = (M 1 R 1 ) 6 = (M 1 R 2 ) 2 = (M 1 M 2 ) 2 = (M 1 M 3 ) 2 = M 1 C 2 M 1 C 2 = (R 2 C 2 ) 2 = E X = (R 2 R 1 M 1 ) 2 Y = (M 1 R 2 R 1 ) 2 Z = C 2 M 1 = [y, x, z] M 2 = [x, y, z] M 2 = [x, y, 1 z] R 1 = [x, x y, ½ z] R 2 = [1 y, 1 x, ½ z] (194) P6 3 /mmc {M 1, M 2, M 3, R, M 4 = RM 3 R} C = RM 3 (M 1 M 2 ) 3 = (M 1 R) 6 = (M 2 R) 2 = (M 3 M 1 ) 2 = (M 3 M 2 ) 2 = E X = (M 2 M 1 R) 2 Y = (RM 2 M 1 ) 2 Z = C 2 M 1 = [x, x y, z] M 2 = [1 y, 1 x, z] M 3 = [x, y, z] R = [y, x, ½ z] M 4 = [x, y, 1 z] 54

57 PART II. CUBIC GROUPS As in part I, our tabulation gives: (i) the number assigned to the group in the International Tables for Crystallography; its Hermann-Mauguin symbol; {a list of the chosen generators a minimal set followed by additional (redundant) generators that extend the minimal set to a set that relates an asymmetric unit to all contiguous unit}; generators indicated in the H-M symbol, expressed in terms of the chosen set; (ii) a set of generating relations that are sufficient to define the abstract group; (iii) translations expressed in terms of the chosen generators; (iv) a particular realization of the generators in terms of Euclidean transformations; specified in terms of the image of a general point [x, y, z]. (v) a diagram of the asymmetric unit. Explanation of the Figures Each figure illustrates an asymmetric unit. The cube outlined in grey is an eighth of a unit cell, with the axes like this: Twofold axes are indicated in red, threefold axes in green and fourfold axes in blue. Centres of 3 and 4 transformations are indicated by and, respectively. Mirror faces of the units are unmarked. 55

58 (195) P23 {Q, R 1, R 2, R 3 = R 1 R 2 } K 1 = QR 1 L 1 = Q 1 R 1 K 2 = QR 2 L 2 = Q 1 R 2 (R 1 R 2 ) 2 = (QR 3 ) 3 = (L 1 K 2 ) 2 = (K 1 K 2 ) 3 = (L 1 L 2 ) 3 = (K 1 L 2 1 ) 3 = K 1 3 L 2 3 = QR 3 K 2 2 L 1 K 1 = K 1 2 L 1 2 K 1 L 1 2 K 1 2 L 1 = K 2 2 L 2 2 K 2 L 2 2 K 2 2 L 2 = E Z = QR 3 L 1 R 1 = [x, 1 y, z] R 2 = [1 x, y, z] R 3 = [1 x, 1 y, z] (196) F23 {Q, Q 2, R} (Q 1 Q 2 ) 2 = (QQ 2 ) 3 = (QR) 3 = (Q 2 R) 3 = E Z = (Q 2 RQ 1 ) 2 W = RQQ 2 Q Q 2 = [ z, x, y] R = [½ x, ½ y, z] (197) I23 {S 1, S 2, R = S 1 S 2 } Q = S 1 2 S 1 K = QR = S 1 2 S 2 1 S 2 L = Q 1 R = S 1 1 S 2 2 S 1 (S 1 S 2 ) 2 = (S 1 2 S 1 ) 3 = (S 2 1 S 2 2 ) 2 = (S 2 1 S 2 2 ) 3 = S 3 1 S 3 2 S 3 1 S 3 2 = K 2 L 2 KL 2 K 2 L = E Z = (S 1 1 S 2 2 ) 2 W = S 3 1 Z S 1 = [½ + y, ½ z, ½ x] S 2 = [½ + z, ½ x, ½ y] R = [1 x, y, z] Axes of S 1 = 3 2 (λ,λ 1 / 3, 2 / 3 λ)[ 1 / 6, 1 / 6, 1 / 6 ] and S 2 = 3 2 (λ, 2 / 3 λ, λ 1 / 3 )[ 1 / 6, 1 / 6, 1 / 6 ] marked in turquoise. 56

59 (198) P2 1 3 {Q, S 1, S 2, Q 2 = S 1 Q, Q 3 = S 2 Q, Q 4 = QS 1 } S 1 S 2 2 S 1 1 S 2 2 = S 2 S 2 1 S 1 2 S 2 1 = QS 2 2 Q 1 S 2 1 = E 2 Z = S 2 S 1 = [½ + x, ½ y, z] S 2 = [½ x, 1 y, ½ + z] Q 2 = [½ + z, ½ x, y] Q 3 = [½ z, 1 x, ½ + y] Q 4 = [ z, ½ + x, ½ y] S 1 and S 2 are the the 2-fold screw transformations indicated in red. The two blue shaded facets are related through the threefold rotation Q 3 = 3(λ, 1 λ, ½ λ). The two yellow shaded facets are similarly related through Q 4 = 3(λ, λ + ½, λ). (Axes of Q 3 and Q 4 have been omitted in the figure.) (199) I2 1 3 {Q, R 1, R 2, S, R 3 = QR 2 Q 1 } S 2 = R 3 R 1 K 1 = QR 1 L 1 = Q 1 R 1 K 2 = QR 2 L 2 = Q 1 R 2 L 1 K 2 1 K 2 L 2 2 = K 3 2 S 3 = K 1 L 1 R 2 R 1 = (K 2 L 1 ) 2 S 2 = K 2 2 L 2 L 1 = (L 1 K 2 ) 2 (K 2 L 2 ) 2 = L 1 K 1 L 2 1 K 2 2 = (K 1 K 2 2 ) 2 = S 2 2 R 2 S 2 2 R 2 = K 2 1 L 2 1 K 1 L 2 1 K 2 1 L 1 = K 2 2 L 2 2 K 2 L 2 2 K 2 2 L 2 = E 2 Z = S 2 W = K S 2 R 1 = [½ x, y, z] R 2 = [1 x, ½ y, z] S = [z, 1 x, ½ y] R 3 = [x, 1 y, ½ z] The two shaded triangular facets are related through S = 3 1 (λ, 5 / 6 λ, λ 1 / 6 )[ 1 / 6, 1 / 6, 1 / 6 ] (200) Pm 3 {Q, M 1, M 2, M 3 = Q 1 M 1 Q} Q = (M 2 Q) 3 Q (M 1 M 2 ) 2 = (M 2 M 3 ) 2 = (M 1 M 2 M 3 ) 2 = (M 1 QM 1 Q 1 ) 2 = (M 2 QM 2 Q 1 ) 2 = (M 2 Q) 6 = E Z = QM 1 Q 1 M 2 M 1 = [x, 1 y, z] M 2 = [x, y, z] M 3 = [1 x, y, z] 57

60 (201) Pn 3 { Q 1, Q 2, R = Q 2 1 Q 1 } N = R Q 1 3 Q = Q 1 2 Q 2 = Q 2 2 K = QR L = Q 1 R K 2 = Q 2 R L 2 = Q 2 1 R ( Q 1 RQ 1 1 R) 2 = RQ 1 1 Q 2 RQ 2 1 Q 1 = ( Q 1 3 RQ 1 R) 3 = K 2 L 2 KL 2 K 2 L = K 2 2 L 2 2 K 2 L 2 2 K 2 2 L 2 = E Z = ( Q 1 2 RQ 1 1 R) 2 Q 1 = [½ z, ½ x, ½ y] Q 2 = [½ + z, ½ + x, ½ y] R = [1 x, y, z] (202) Fm 3 {Q, M, R} Q = (MQ) 3 Q (RQ) 3 = (RM) 2 = (MQ) 6 = (MQMQ 1 ) 2 = E Z = (Q 1 RQM) 2 W = R(QM) 2 Q R = [½ x, ½ y, z] M = [x, y, z] (203) Fd 3 { Q 1, Q 2 } D = Q 1 Q 2 2 Q 1 2 ( Q 1 Q 1 2 ) 2 = (Q 2 1 Q 2 2 ) 2 = ( Q 2 1 Q 2 2 ) 3 = E Z = Q Q 2 W = D 2 Q 1 = [¼ z, ¼ x, ¼ y] Q 2 = [¼ + z, ¼ x, ¼ + y] 58

61 (204) Im 3 { Q, M, M 2 = Q M Q 1 } I = Q 3 Q = Q 4 (QM) 6 = (MQMQ 1 ) 2 = (IMIQMQ 1 ) 2 = E Z = (M Q 3 ) 2 W = QZQ 1 (MQ) 3 Q = [½ z, ½ x, ½ y] M = [x, y, z] M 2 = [1 + x, y, z] (205) Pa 3 { Q 1, Q 2, Q = ( Q 1 Q 2 ) 1 } A = ( Q 2 2 Q 1 ) 1 Q = Q 3 2 ( Q 1 Q 1 2 ) 2 Q Q 1 6 = Q 2 6 = (Q 1 Q 2 ) 3 = (Q 1 2 Q 2 ) 2 = ( Q 1 2 Q 2 2 ) 2 = ( Q 1 Q 2 1 ) 2 Q 1 3 Q 2 3 = Q 1 2 (Q 2 1 Q 1 Q 2 1 )(Q 1 1 Q 2 Q 1 1 )Q 2 2 (Q 1 1 Q 2 Q 1 1 )(Q 2 1 Q 1 Q 2 1 ) = E Z = ( Q 2 1 Q 1 ) 2 Q 1 = [½ + z, x, ½ y] Q 2 = [z, ½ x, ½ + y] The asymmetric unit has six facets, related in pairs through Q, Q 1 = 3 (½ λ, ½ + λ, λ; ½, ½, 0) and Q 2 = 3 ( λ, ½ λ, λ; 0, ½, 0) (206) Ia 3 { Q, R, I 2 = ( Q 2 R) 2 Q 1 R } A = Q 2 RQ Q = Q 2 I = Q 3 K = Q R L = Q 1 R K 6 = L 6 = (KL) 2 K 3 L 3 = QI 2 Q 1 I 2 = Q RILKL = Q(RI) 2 Q 1 (RI 2 ) 2 = K 2 L 2 KL 2 K 2 L = E Z = (KL) 2 W = I 2 I R = [x, ½ y, z] I 2 = [½ x, ½ y, ½ z] 59

62 (207) P432 {F, R 1, R 2 } Q = F 1 R 2 K = QR 1 L = Q 1 R 1 (R 1 R 2 ) 4 = (R 2 F) 3 = (QR 2 ) 4 = (F 2 R 1 ) 2 = (F 2 R 2 ) 4 = (QFR 1 ) 2 = (FQR 1 ) 4 = (FR 1 F 1 R 1 ) 2 = (FR 1 FR 2 R 1 R 2 ) 2 = K 2 L 2 KL 2 K 2 L = (KL 2 KL 3 ) 2 = E Y = F 2 R 2 R 1 R 2 F = [1 + y, x, z] R 1 = [1 x, y, z] R 2 = [1 x, z, y] (208) P {R 1, R 2, R 3, R 4 } Q = R 2 R 1 Q 2 = R 4 R 3 R 5 = R 1 R 3 R 6 = R 2 R 4 (R 1 R 2 ) 3 = (R 3 R 4 ) 3 = (R 1 R 3 ) 2 = (R 2 R 4 ) 2 = R 1 R 3 R 2 R 4 = E X = R 2 (R 1 R 4 ) 3 R 1 R 1 = [½ z, ½ y, ½ x] R 2 = [½ x, ½ z, ½ y] R 3 = [½ + z, ½ y, ½ + x] R 4 = [½ x, ½ + z, ½ + y] (209) F432 {F, R 1, R 2 } Q = R 1 F 1 (R 1 R 2 ) 2 = (FR 1 ) 3 = (FR 2 ) 3 = (R 1 F 2 ) 4 = (R 2 F 2 ) 4 = (R 1 R 2 F) 4 = E Y = (R 1 R 2 F 2 ) 2 W = R 2 F 2 R 1 F 2 F = [x, z, y] R 1 = [y, x, z] R 2 = [½ y, ½ x, z] 60

63 (210) F {Q, R 1, R 2, R 3 = Q 1 R 2 } S = R 3 R 1 (QR 1 ) 3 = (QR 2 ) 2 = (QR 2 R 1 Q(R 2 R 1 ) 3 ) 2 = ((R 2 R 1 ) 3 Q) 2 = E Z = S 4 W = R 2 R 1 R 2 QR 1 Q 1 R 1 = [x, y, z] R 2 = [¼ z, ¼ y, ¼ x] R 3 = [¼ y, ¼ x, ¼ z] (211) I432 {R 1, R 2, R 3 } Q = R 2 R 1 F = QR 3 Q (R 2 R 1 ) 3 = (QR 3 ) 4 = (FR 1 F 1 R 1 ) 2 = (Q(FR 1 ) 2 ) 2 = (R 1 FQF) 4 = E Z = (R 1 R 2 R 1 R 3 ) 2 W = Q 1 R 1 R 3 F 2 R 1 = [½ z, ½ y, ½ x] R 2 = [½ x, ½ z, ½ y] R 3 = [y, x, z] (212) P {R 1, R 2, R 3, S, Q 2 = R 3 R 1 } Q = R 2 R 1 G = QR 3 J = Q 1 R 3 (R 1 R 2 ) 3 = (R 1 R 3 ) 3 = S 4 Q 2 S 2 Q 1 = S 4 (Q 1 2 Q) 2 = S 2 G 2 = S 4 Q 1 (R 2 R 3 ) 2 Q = (QS 4 Q 1 )S 2 (QS 4 Q 1 )S 2 = (Q 1 S 4 Q)S 2 (Q 1 S 4 Q)S 2 = (G 2 J 3 ) 2 = E Z = S 4 R 1 = [¼ z, ¼ y, ¼ x] R 2 = [¼ x, ¼ z, ¼ y] R 3 = [¾ x, ¼ + z, ¼ + y] S = [ ¼ + y, ¾ x, ¼ + z] Q 2 = [½ + z, ½ x, y] The two shaded triangular facets are related through the fourfold screw transformation S = 4 1 (¼, ½, z)[0, 0, ¼], indicated in blue 61

64 (213) P {R 1, R 2, R 3, S, Q 2 = R 1 R 3 } Q = R 1 R 2 G = QR 3 J = Q 1 R 3 (R 1 R 2 ) 3 = (R 1 R 3 ) 3 = S 4 Q 1 2 S 2 Q = S 4 (QQ 1 2 ) 2 = S 2 J 2 = S 4 Q(R 2 R 3 ) 2 Q 1 = (QS 4 Q 1 )S 2 (QS 4 Q 1 )S 2 = (Q 1 S 4 Q)S 2 (Q 1 S 4 Q)S 2 = (G 2 J 3 ) 2 = E Z = S 4 R 1 = [¼ x, ¼ z, ¼ y] R 2 = [¼ z, ¼ y, ¼ x] R 3 = [¼ + z, ¾ y, ¼ + x] S = [¾ y, ¼ + x, ¼ + z] Q 2 = [ z, ½ x, ½ + y] The two shaded triangular facets are related through the fourfold screw transformation S = 4 1 (½, ¼, z)[0, 0, ¼], indicated in blue (214) I {R 1, R 2, R 3, R 4, R 5 = R 2 R 3 R 2, Q = R 1 R 5, S = R 2 R 3 } (R 1 R 2 ) 2 = (R 1 R 3 ) 3 = (R 1 R 4 ) 3 (R 4 R 3 ) 3 = S 4 Q 1 (R 4 R 2 ) 4 Q = (S 3 R 1 S 2 R 1 ) 2 = E Z = (R 2 R 4 ) 4 W = S 4 Q(R 1 R 4 ) 3 Q 1 R 1 = [¼ x, ¼ z, ¼ y] R 2 = [x, y, ½ z] R 3 = [ ¼ + z, ¼ y, ¼ + x] R 4 = [¼ + y, ¼ + x, ¼ z] R 5 = [¼ z, ¼ y, ¼ x] S = [ ¼ + z, ¼ + y, ¼ x] Shaded facets related through S = 4 1 (0, y, ½)[0, ¼, 0] 62

65 (215) P 4 3m {M 1, M 2, R, M 3 = RM 1 R} F = M 1 R Q = M 1 M 2 K = QR L = Q 1 R (M 1 M 2 ) 3 = (M 3 M 2 ) 3 = (M 1 R) 4 = (M 2 R) 4 = (M 3 R) 4 = K 2 L 2 KL 2 K 2 L = E Y = F 2 M 2 RM 2 M 1 = [y, x, z] M 2 = [x, z, y] R = [1 x, y, z] M 3 = [1 y, 1 x, z] F = [y, 1 x, z] (216) F 4 3m {M 1, M 2, M 3, M 4 } F = M 1 M 2 M 3 Q = M 3 M 1 (M 1 M 2 ) 2 =(M 2 M 3 ) 3 = (M 3 M 1 ) 3 = (M 1 M 4 ) 3 = (M 2 M 4 ) 3 = (M 3 M 4 ) 2 = (M 1 M 2 M 3 ) 4 = (M 2 M 3 M 4 ) 4 = (M 3 M 1 M 4 ) 4 = (M 1 M 2 M 4 ) 4 = E X = M 4 M 1 M 2 M 4 M 3 M 1 M 2 M 3 W = M 4 M 1 M 2 M 3 M 2 M 1 M 1 = [x, z, y] M 2 = [x, z, y] M 3 = [y, x, z] M 4 = [½ y, ½ x, z] (217) I 4 3m { F, M, M 2 = F 1 M F } Q = MM 2 G = F M J = F 1 M K = Q F 2 L = Q 1 F 2 (MM 2 ) 3 = ( F 2 M) 4 = QG 3 Q 1 J 3 = J 3 G 3 J 3 G 3 = M( F Q) 2 M( F Q) 2 = K 2 L 2 KL 2 K 2 L = E Z = ( F Q) 2 W = ZJ 3 F = [½ z, ½ y, ½ + x] M = [y, x, z] M 2 = [x, z, y] 63

66 (218) P 4 3n {Q, F 1, F 2 } N = F F 2 2 K 1 = Q F 1 L 1 = Q F 1 K 2 = Q F 2 L 2 = Q 1 2 F 1 ( F 2 1 F 2 2 ) 2 = (Q F 2 1 F 2 2 ) 3 = (Q F 1 ) 4 = (Q 1 F 2 ) 4 = ( F 1 Q 1 ) 2 ( F 2 Q) 2 = (L 1 K 2 ) 2 = (K 1 K 2 ) 3 = (L 1 L 2 ) 3 = (K 1 L 1 2 ) 3 = K 3 1 L 3 2 = K 2 1 L 2 1 K 1 L 2 1 K 2 1 L 1 = K 2 2 L 2 2 K 2 L 2 2 K 2 2 L 2 = E Z = (Q 1 F 1 ) 2 F 1 = [½ + z, ½ y, ½ x] F 2 = [½ + y, ½ x, ½ z] 1 (219) F 4 3c { F 1, F 2 } Q = F 2 F 1 C = F 2 F 1 1 F 2 2 F 1 ( F 2 F 1 1 ) 3 = ( F 1 F 2 ) 3 = ( F 2 1 F 1 2 ) 4 = ( F 2 2 F 1 1 ) 4 = E Z = C 2 W = (F 1 2 F 2 1 ) 2 F 1 = [½ x, z, y] F 2 = [y, ½ x, z] (220) I 4 3d {Q, F 1, F 2, S, R = Q F 2 2 Q 1 } D = R F 2 2 K 1 = Q F 1 L 1 = Q F 1 K 2 = Q F 2 L 2 = Q 1 2 F 2 ( F 1 F 1 2 ) 3 = (Q F 1 ) 4 = (Q 1 F 2 ) 4 = ( F 1 S) 4 = ( F 2 S) 4 = K 3 1 L 3 2 = K 3 2 Q 1 L 3 1 Q = K 2 1 L 2 1 K 1 L 2 1 K 2 1 L 1 = K 2 2 L 2 2 K 2 L 2 2 K 2 2 L 2 = E Z = (L 1 K 1 ) 2 W = D 2 F 1 = [¼ + z, ¾ y, ¼ x] F 2 = [¼ + y, ¾ x, ¼ z] S = [z, 1 x, ½ y] R = [x, 1 y, ½ z] Shaded triangular facets related through Shaded triangular facets related through S = 3 1 (λ, 5 / 6 λ, λ 1 / 6 )[ 1 / 6, 1 / 6, 1 / 6 ] S = 3 1 (λ, 5 / 6 λ, λ 1 / 6 )[ 1 / 6, 1 / 6, 1 / 6 ]??? 64