GENERATORS AND RELATIONS FOR SPACE GROUPS

GENERATORS AND RELATIONS

FOR SPACE GROUPS

Eric Lord

Bangalore 2010

Contents

Introduction . . . . 1

Notation . . . . 2

PART I. NON-CUBIC GROUPS

Explanation of the figures . . . . 3

Triclinic . . . . 5

Monoclinic . . . . 5

Orthorhombic . . . . 8

Tetragonal . . . .

Trigonal . . . . 41

Hexagonal . . . . 48

PART II. CUBIC GROUPS

Explanation of the figures . . . . 55

Cubic . . . . 56

Appendix : matrix methods . . . . 69

1

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, glide-

reflections 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.

2

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

W1 = [x, y + ½, z + ½], W2 = [x + ½, y, z + ½], and W = W3 = [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 I2 = E. Unless otherwise stated, I = [ x , y , z ]

R : a diad rotation R2 = E.

M : a reflection M2 = E.

F : fourfold rotation. F4 = E. Unless otherwise stated, F = [ y , x, z].

F : 4 transformation. F4 = E. Unless otherwise stated, F = [y, x , z ].

Q : [−y, x − y, z] in trigonal or hexagonal groups; [z, x, y] in cubic groups. Q3 = 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. Q6 = E. Note that Q

−2 = Q and Q

3 = I.

H : sixfold rotation [x − y, x, z]. H6 = E.

H : A 6 transformation [y − x, −x, −z]. H6 = E

3

PART I. NON-CUBIC GROUPS

Explanation of the Figures

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 21 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 − P41 and P43 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 42 axis is produced by two generators – a 2-fold rotation and a 42 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). (62, 63 and 64 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.

4

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.

5

Triclinic

(1) P1 {X, Y, Z}

YZY−1

Z−1

= ZXZ−1

X−1

= XYX−1

Y−1

= E

(2) P 1 {X, Y, I1, I2}

XYX−1

Y−1

= (I1X)2 = (I1Y)

2 = (I2X)

2 = (I2Y)

2 = E

Z = I2I1

I1 = [ x , y , z ] I2 = [ x , y , z + 1]

Monoclinic

(3) P2 { X, Z, R1, R2}

ZXZ−1

X−1

= XR1X−1

R1 = (ZR1)2 = XR2X

−1R2 = (ZR2)

2 = E

Y = R2R1

R1 = [x, y , z ] R2 = [x, y + 1, z ]

(4) P21 {X, Z, S}

ZXZ−1

X−1

= XSXS−1

= ZSZS−1

= E

Y = S2

S = [ x + ½, y + ½, z + ½]

(5) C2 {Z, R, S, R2 = SRS−1

}

RS2RS

−2 = (ZR)

2 = ZSZS

−1 = E

X = R2R Y = S2 W = SR

R = [ x , y, z + ½] S = [ x + ½, y + ½, z + ½] R = [1 + x , y, z + ½]

6

(6) Pm {X, Z, M1, M2}

ZXZ−1

X−1

= M1XM1X−1

= M2XM2X−1

= M1ZM1Z−1

= M2ZM2Z−1

= E

Y = M2M1

M1 = [x, y , z] M2 = [x, y + 1 , z]

(7) Pc {X, Y, C}

XYX−1

Y−1

= CXC−1

X−1

= CYC−1

Y = E

Z = C2

C = [x, y , z + ½]

(8) Cm {Z, M, A, M2 = AMA−1

}

MA2MA

−2 = AZA

−1Z

−1 = MZMZ

−1 = E

X = A2 Y = M2M W3 = AM

M = [x, y , z] A = [x + ½, y + ½ , z] M2 = [x, y + 1, z]

(9) Cc {W, C, Wa = CWC−1

}

WWaW−1

Wa−1

= C2WC

−2W

−1 = E

X = W Wa Y = W Wa−1

Z = C2

C = [x, y , z + ½] Wa = [x + ½, y − ½, z]

(10) P2/m {Z, M1, M2, R1, R2}

M1ZM1Z−1

= M2ZM2Z−1

= (R1Z)2 = (R2Z)

2 = (M1R1)

2 = (M1R2)

2 = (M2R1)

2 = (M2R2)

2 = E

X = R2R1 Y = M2M1

M1 = [x, y , z] M2 = [x, y + 1, z] R1 = [ x , y, z ] R2 = [ x + 1, y, z ]

7

(11) P21/m {Z, M, I1, I2, M2 = I1MI1} S = MI1

M2I2MI2 = (ZI1) = (ZI2) = ZMZ−1

M = ZM2Z−1

M2 = E

X = I2I1 Y = M2M

M = [x, y , z] I1 = [ x , y , z ] I2 =[ x + 1, y , z ] M2 = [x, y + 1, z]

(12) C2/m {Z, R, M, I, R2 = IRI }

(MR)2 = (IR)

2 = MZMZ = (RZ)

2 = (IZ)

2 = E

X = (IR)2 Y = M2M W = IRM

M = [x, y , z] R = [ x , y, z ] I = [ x + ½, y + ½, z ] R2 = [ x + 1, y, x ]

(13) P2/c {Y, R1, R2, C}

CYC−1

Y = (R1C)2 = (R2C)

2 = R1YR1Y

−1 = R2YR2Y

−1 = E

X = R2R1 Z = C2

R1 = [ x , y, z + ½] R2 = [ x + 1, y, z + ½] C = [x, y + 1, z + ½]

(14) P21/c {X, S, C}

(C−1

S)2 = XSXS

−1 = XCX

−1C

−1 = E

Y = S2 Z = C

2

S = [ x + 1, y + ½, z + ½] C = [x, y + ½, z + ½]

8

(15) C2/c {R, C, S, R2 = SRS−1

}

(RC)2 = (SC)

2 = (SC

−1)2 = RS

2RS

−2 = E

X = R2R Y = S2 Z = C

2 W = SR

R = [ x , y, z ] S = [ x + ½, y + ½, z ] C = [x, y , z + ½] R2 = [ x , y, z ]

Orthorhombic

(16) P222 {Z, R1, R2, R3, R4}

(R1R4)2 = (R4R2)

2 = (R2R3)

2 = (R3R1)

2 = (R1Z)

2 = (R2Z)

2 = (R3Z)

2 = (R4Z)

2

X = R2R1 Y = R4R3

R1 = [ x , y, z ] R2 = [ x + 1, y, z ] R3 = [x, y , z ] R4 = [x, y + 1, z ]

(17) P2221 {X, Y, R1, R2} S = R2R1

(R2X)2 = R1XR1X

−1 = (R1Y)

2 = R2YR2Y

−1 = E

Z = S2

R1 = [x, y , z ] R2 = [ x , y, z + ½]

(18) P21212 {Z, S, R, R2 = SRS } S2 = SR

R22 = RZRZ

−1 = SZS

−1Z = E

X = S22 = R2R Y = S

2

R = [ x , y , z] S = [ x + ½, y + ½, z ] R2 = [ x + 1, y , z]

9

(19) P212121 {S1, S3, S4 = S3S1−1

S3} S2 = S3S1

S12S4

−2 = S2

2S3 S2

−2S3

−1 = S3

2S1 S3

−2S1

−1 = S1

2S2 S1

−2S2

−1 = E

X = S12 Y = S2

2 Z = S3

2

S1 = [x + ½, y + ½, z ] S3 = [ x , y , z + ½] S4 = [x + ½, y − ½, z ]

(20) C2221 {R1, R2, W, W2 = R1WR1} S = R2R1

(S2R1)

2 = (S

2R2)

2 = WSWS

−1 = W2SW2S

−1 = W2

2WW2

−2W

−1 = W

2W2W

−2W2

−1 = E

X = WW2 Y = WW2−1

Z = S2

R1 = [x, y , z ] R2 = [ x , y, z + ½]

(21) C222 {Z, R1, R2, R3}

(R1R2)2 = (ZR1)

2 = (ZR2)

2 = ZR3Z

−1R3 = (R3R1R3R2)

2 = E

X = (R3R2)2 Y = (R3R1)

2 W3 = R3R2R1

R1 = [x, y , z ] R2 = [ x , y, z ] R3 = [ x + ½, y + ½, z]

(22) F222 {R1, R2, R3, R4, R5 = R3R1R3, R6 = R2R4R2}

(R1R2)2 = (R2R5)

2 = (R3R4)

2 = (R3R6)

2 = (R5R1R4)

2 = (R4R6R1)

2 = R1R4R5R6 = E

X = (R3R2)2 Y = R5R1 Z = R6R4

W1 = R4R1 W2 = R3 R4 R2 W3 = R3R2R1 R1 = [x, y , z + ½] R2 = [ x , y, z + ½]

R3 = [ x + ½, y + ½, z] R4 = [x, y + ½, z ]

R5 = [x, y + 1, z + ½] R6 = [x, y + ½, z + 1]

10

(23) I222 {R1, R2, S, R3 = SR1S, R4 = SR2S } R5 = R1R2

R32 = R4

2 = R5

2 = (R1R2)

2 = (R1R4)

2 = (R2R3)

2 = (R3R4)

2 = E

X = R3R1 Y= R4R2 Z = S2 W = R3R4R5

R1 = [ x , y, z ] R2 = [x, y , z ] S = [ x + ½, y + ½, z + ½]

R3 = [ x + 1, y, z ] R4 = [x, y + 1, z ]

(24) I212121 {R1, R2, R3, R4 = R1R3R1, R5 = R2R4R2, R6 = R2R3R2}

S1 = R3R2 S2 = R3R1 S3 = R1R2

S12R1S1

−2R1 = S2

2R2S2

−2R2 = S3

2R3S3

−2R3 = E

X = S12 Y = S2

2 Z = S3

2 W = S3R4

R1 = [x, y , z + ½] R2 = [ x , y, z ] R3 = [ x + ½, y + ½, z]

R4 = [ x + ½, y − ½, z] R5 = [ x − ½, y − ½, z] [ x − ½, y + ½, z]

(25) Pmm2 {Z, M1, M2, M3, M4} R = M1M2

(M1M2)2 = (M2M3)

2 = (M3M4)

2 =

M1ZM1Z−1

= M2ZM2Z−1

= M3ZM3Z−1

= M4ZM4Z−1

= E

X = M4M2 Y = M3M1

M1 = [x, y , z] M2 = [ x , y, z] M3 = [x, y + 1, z] M4 = [ x + 1, y, z]

(26) Pmc21 {Y, M1, M2, C} S = M1C

M1CM1C−1

= M2CM2C−1

= M1YM1Y−1

= M2YM2Y−1

= CYC−1

Y = E

X =M2M1 Z = C2

M1 = [ x , y, z] M2 = [ x + 1, y, z] C = [x, y , z + ½]

11

(27) Pcc2 {X, C, R1, R2} C2 = R1C

R1CR1C−1

= R2CR2C−1

= (R1X)2 = (R2X)

2 = CXC

−1X = E

Y = R2R1 Z = C2

C = [ x , y, z + ½] R1 = [ x , y , z] R2 = [ x , y + 1, z]

(28) Pma2 {Z, M, R1, R2, M2 = R1MR1} A = R1M

MZMZ−1

= M2ZM2Z−1

= R1ZR1Z−1

= R2ZR2Z−1

= R2R1MR1R2M = E

X = M2M Y = R2R1

M = [ x , y, z] R1 = [ x + ½, y , z] R2 = [ x + ½, y + 1, z] M2 = [ x + 1, y, z]

(29) Pca21 {Y, C, A} S = A−1

C

ACAC−1

= CYC−1

Y−1

= AYA−1

Y = E

X = A2 Z = C

2

A = [x + ½, y , z] C = [ x + ½, y, z + ½]

(30) Pnc2 {X, C, R, R2 = CRC−1

} N = CR

(RX)2 = CXC

−1X

−1 = C

2RC

−2R = E

Y = R2R Z = C2

R = [ x , y , z] C = [x, y + ½, z + ½] R2 = [ x , y + 1, z]

12

(31) Pmn21 {Y, M, S, M2 = SMS−1

} N = SM

SYS−1

Y = S2MS

−2M = MYMY

−1 = E

X = M2M Z = S2

M = [ x , y, z] S = [ x + ½, y + ½, z + ½] M2 = [ 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 = A2 Y = B

2

B = [ x + ½, y + ½, z] A = [x + ½, y + ½, z]

(33) Pna21 {A, S, A2 = SA−1

S−1

} N = SA

A2SA

−2S

−1 = S

2AS

−2A

−1 = E

X = A2 Y = AA2

−1 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} N2 = NR

NZN−1

Z−1

= RZRZ−1

= E

X = Z−1

N22 Y = Z

−1N

2

R = [ x , y , z] N = [x + ½, y + ½, z + ½]

13

(35) Cmm2 {Z, M1, M2, R}

(M1M2)2 = RZRZ

−1 = M1ZM1Z

−1 = M2ZM2Z

−1 = (M2RM1R)

2 = E

X = (RM1)2 Y = (RM2)

2 W = RM1M2

M1 = [ x , y, z] M2 = [x, y , z] R = [ x + ½, y + ½, z]

(36) Cmc21 {M, C, B, M2 = BMB−1

}

MCMC−1

= BCBC−1

= B2MB

−2M = E

X = BMB−1

M Y = B2 Z = C

2 W = BM

M = [ x , y, z] C = [x, y , z + ½] B = [ x + ½, y + ½, z] M2 = [ x + 1, y, z]

(37) Ccc2 {R1, R2, C, R3 = CR2C−1

} C2 = CR1

CR1C−1

R1 = C2R2C

−2R2 = (R1R2R3)

2 = E

X = R2R3 Y = R3R1R2R1 Z = C2 W = R2R1

R1 = [ x , y , z] R2 = [ x + ½, y + ½, z] C = [x, y , z + ½] R3 = [ x − ½, y + ½, z]

(38) Amm2 {M1, M2, M3, C, M4 = CM2C−1

} R = M1M2

(M1M2)2 = (M2M3)

2 = (M3M4)

2 = (M4M1)

2 = CM1C

−1M1 = CM3C

−1M3 = E

X = M3M1 Y = M4M2 Z = C2 W = CM1R

M1 = [ x , y, z] M2 = [x, y , z] M3 = [ x + 1, y, z] C = [x, y + ½, z + ½]

M4 = [x, y + 1, z]

14

(39) Abm2 {M, R1, R2, C, M2 = R1MR1} B = R1M

M2CMC−1

= M2C−1

MC = M2R2MR2 = R1CR1C−1

= R2CR2C−1

= C2MC

−2M = E

X = R2R1 Y = B2 Z = C

2 W1 = MC

M = [x, y , z] R1 = [ x , y + ½, z] R2 = [ x + 1, y + ½, z] C = [x, y + ½, z + ½]

M2 = [x, y + 1, z]

(40) Ama2 {M, R, S, M2 = RMR, R2 = SRS−1

} A = MR

M2SMS−1

= (MSMRS−1

)2 = S

2RS

−2R = S

2MS

−2M = (MR)

2S(MR)

2S

−1 = E

X = M2M Y = R2R Z = S2 W1 = SR

R = [ x + ½, y , z] M = [ x , y, z] S = [ x + ½, y + ½, z + ½]

M2 = [ x + 1, y, z] R2 = [ 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 = A2 Y = B

2 Z = C

2 W1 = BC

A = [x + ½, y + ½, z] B = [ x + ½, y + ½, z] C = [ x + ½, y, z + ½]

(42) Fmm2 {M1, M2, R, C, M3 = RM1R}

M3CMC−1

= (M1M2)2 = M2CM2C

−1 = CRC

−1R = M1C

2M1C

−2 = RM1RCM1C

−1 = E

X = (RM1)2 Y = (RM2)

2 Z = C

2

W1 = RM2C W2 = CM1 W3 = RM1M2

M1 = [ x , y, z] M2 = [x, y , z] R = [ x + ½, y + ½, z] C = [ x + ½, y, z + ½]

M3 = [ x + 1, y, z]

15

(43) Fdd2 {D1, D2, R1 = D2−1

D1, S = D2D1, R2 = SR1S−1

, D′ = SD1S−1

}

(D1D2−1

)2 = (D2D1)

2(D1D2)

−2 = D1

2D2

2D1

−2D2

−2 = D1

2(D2D1D2)D1

−2(D2D1D2)

−1 =

D22(D1D2D1)D2

−2(D1D2D1)

−1 = E

X = D23(D1D2D1)

−1 Y = D1

3(D2D1D2)

−1 Z = S

2

W1 = D12 W2 = D2

2 W3 = (D1D2D1)

−1D2D1

2

D1 = [ x + ¼, y + ¼, z + ¼] D2 = [x + ¼, y + ¼, z + ¼]

R1 = [ x , y , z] S = [ x + ½, y , z + ½] R2 = [ x + 1, y , z] D′ = [ x + ¾, y − ¼, z]

(44) Imm2 {M1, M2, S, M3 = SM1S−1

, M4 = SM2S−1

} R = M1M2

(M1M2)2 = S

2M1S

−2M1 = S

2M2S

−2M2 = (M4M1)

2 = (M3M2)

2 = E

X = M3M1 Y = M4M2 Z = S2 W = SM1M2

M1 = [ x , y, z] M2 = [x, y , z] S = [ x + ½, y + ½, z + ½]

M3 = [ x + 1, y, z] M4 = [x, y + 1, z]

(45) Iba2 {A, B, S} R = A−1

B

(AB)2 = (AB

−1)2 = SAS

−1A = SBS

−1B = (S

−1BSA)

2 = (S

−1ASB)

2 = E

X = A2 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, M2 = RMR, R2 = CRC−1

} A = RM

CMC−1

M = C2RC

−2R = CM2C

−1 M2 = E

X = M2M Y = R2R Z = C2 W = CRM

M = [ x , y, z] R = [ x + ½, y , z] C = [x, y + ½, z + ½]

M2 = [ x + 1, y, z] R2 = [ x + ½, y + 1, z]

16

(47) Pmmm {M1, M2, M3, M4, M5, M6}

(M1M3)2 = (M2M3)

2 = (M3M4)

2 = (M4M1)

2 = (M1M5)

2 = (M2M5)

2 = (M3M5)

2 = (M3M5)

2 =

(M4M5)2 = (M1M6)

2 = (M2M6)

2 = (M3M6)

2 = (M4M6)

2 = (M5M6)

2 = E

X = M2M1 Y = M3M4 Z = M6M5

M1 = [ x , y, z] M2 = [ x + 1, y, z] M3 = [x, y + 1, z]

M4 = [x, y , z] M5 = [x, y, z ] M6 = [x, y, z + 1]

(48) Pnnn { R1, R2, I1, I2, R3 = I1R1I2, R4 = I1R2I2}

N1 = I2R1 N2 = I2R2 N3 = I2R1R2

R32 = R4

2 = (R1R2)

2 = I1I2R1R2I2I1R1R2 =

(N1N2N3)2 = (N3N2N1)

2 = E

X = R3R1 Y = R4R2 Z = I2I1

(N12 = YZ N2

2 = ZX N3

2 = XY)

R1 = [x, y , z ] R2 = [ x , y, z ]

I1 = [ x + ½, y + ½, z − ½] I2 = [ x + ½, y + ½, z + ½]

N1 = [ x + ½, y + ½, z + ½] N2 = [x + ½, y + ½, z + ½] N3 = [x + ½, y + ½, z + ½]

R3 = [x + 1, y , z ] R4 = [ x , y + 1, z ]

(49) Pccm { R1, R2, R3, R4, M, M2 = R1MR1} C1 = R1M C2 = R2M

M2R3MR3 = (R1R2)2 = (R2R3)

2 = (R3R4)

2 = (R4R1)

2 =

(R1R2M)2 = (R2R3M)

2 = (R3R4M)

2 = (R4R1M)

2 = E

X = R3R1 Y = R4R2 Z = C12 = C2

2 = M2M

R1 = [ x , y, z + ½] R2 = [x, y , z + ½]

R3 = [ x + 1, y, z + ½] R4 = [x, y + 1, z + ½] M = [x, y, z ]

M2 = [x, y, z + 1]

17

(50) Pban {Z, R1, R2, I} B = IR1 A = IR2 N = IR1R2

(R1R2)2 = (AB

2 = (ABN)

2 = (IZ)

2 = (R1Z)

2 = (R2Z)

2 = E

X = B2 Y = A

2

R1 = [ x , y, z ] R2 = [x, y , z ] I = [ x + ½, y + ½, z ]

(51) Pmma {M1, M2, M3, R1, R2, M4 = R1M2R1} A = R1M2

(M1M2)2 = (M2M3)

2 = (M3M4)

2 = (M4M1)

2 = (R1M1)

2 = (R1M3)

2 = (R2M1)

2 = (R2M3)

2 =

M4R2M2R2 = E

X = M4M2 Y = M3M1 Z = R2R1

M1 = [x, y , z] M2 = [ x , y, z] M3 = [x, y + 1, z]

R1 = [ x + ½, y, z ] R2 = [ x + ½, y, z + 1] M4 = [ x + 1, y, z]

(52) Pnna {R1, R2, I1, I2, R3 = R2R1R2, R4 = I1R2I1}

N1 = R3I1 N2 = I1 R2 R1 A = I1R2

R4I2R2I2 = R3R4R1R4 = R4R2R1R2R4R1 = (R1I1I2)2 = E

X = R4R2 Y = R3R1 Z = I2I1

R1 = [x, y , z + ½] R2 = [ x , y + ½, z]

I1 = [ x + ½, y + ½, z ] I2 = [ x + ½, y + ½, z + 1]

R3 = [x, y + 1, z + ½] R4 = [ x + 1, y + ½, z]

(53) Pmna {Y, M, R1, R2, M2 = R1MR1} N = R1R2YM A = R1M

(MR2)2 = YMY

−1M = YR1Y

−1R1 = (YR2)

2 = (M2R2)

2= E

X = M2M Z = (R1R2)2

M = [ x , y, z] R1 = [ x + ½, y, z + ½] R2 = [x, y + 1, z ] M2 = [ x + 1, y, z]

18

(54) Pcca {R1, R2, R3, C, R4 = R2R1R2] C2 = R3C A = C−1

R1

R4CR1C−1

= (R1R3R4)2 = C

2R2C

−2R2 = C

2R3C

−2R3 = E

X = R4R1 Y = R2R3 Z = C2

R1 = [ x , y, z + ½] R2 = [ x + ½, y , z] R3 = [ x + ½, y + 1, z]

C = [ x + ½, y, z + ½] R4 = [ x , y + 1, z + ½]

(55) Pbam {B, A, M1, M2}

M1AM1A−1

= M2AM2A−1

= M1BM1B−1

= M2BM2B−1

= (AB)2 = (AB

−1)2 = E

X = A2 Y = B

2 Z = M2M1

B = [ x + ½, y + ½, z] A = [x + ½, y + ½, z]

M1 = [x, y, z ] M2 = [x, y, z + 1]

(56) Pccn {C, R1, R2, I, I2 = CIC] C2 = CR N = C−1

R2CI

I22 = CR1C

−1R1 = CR2C

−1R2 = (IR1R2)

2 = (IC

2)2 = E

X = I2I Y = R2R1 Z = C2

C = [ x + ½, y, z + ½] R1 = [ x + ½, y − ½, z] R2 = [ x + ½, y + ½, z]

I2 = [ x + 1, y , z ]

19

(57) Pbcm {M, R, I1, I2, M2 = I1MI1, R2 = I1RI1] B = I1R C = I1M

M2RMR = M2I2MI2 = M2R2MR2 = R2I2RI2 = RR2M2M = E

X = I2I1 Y = B2 Z = C

2

M = [x, y, z ] R = [x, y , z + ½]

I1 = [ x , y + ½, z + ½] I2 = [ x + 1, y + ½, z + ½]

M2 = [x, y, z + 1] R2 = [x, y + 1, z + ½]

(58) Pnnm {M, S1, S2, M2 = S1MS1−1

} N1 = S2M N2 = S1M

M2S2MS2−1

= (S1−1

S2)2 = (S1S2)

2 = (S1S2M)

2 = (S1

−1S2M)

2 = E

X = S12 Y = S2

2 Z = M2M

M = [x, y, z ] S1 = [x + ½, y + ½, z + ½] S2 = [ x + ½, y + ½, z + ½]

M2 = [x, y, z + 1]

(59) Pmmn {Z, M1, M2, I} N = IM1M2

(M1M2)2 = (IM1IM2)

2 = (IM1M2IM1IM2)

2 = M1ZM1Z

−1 = M2ZM2Z

−1 = (IZ)

2 = E

X = (IM1)2 Y = (IM2)

2

M1 = [ x , y, z] M2 = [x, y , z] I = [ x + ½, y + ½, z ]

(60) Pbcn {B, S1, S2} C = S2B N = S2B−1

S1

(S2S1)2 = (S2S1

−1)2 = BS2BS2

−1 = (BS1)

2 = (BS1

−1)2 = E

X = S12 Y = B

2 Z = S2

2

S1 = [x + ½, y + ½, z ] S2 = [ x + ½, y + ½, z + ½] B = [ x + ½, y + ½, z]

20

(61) Pbca {B, C, A}

BCBC−1

= CACA−1

= ABAB−1

= (ABC)2 = (CBA)

2 = E

X = A2 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, M2 = IMI, I2 = SIS } N = MS A = SI

I22 = M2SMS

−1 = (I2M2M)

2 = MI2I1MI1I2) = (IS

2)2 = S

2MS

−2M = E

X = I2I Y = M2M Z = S2

M = [x, y , z] I = [ x , y + ½, z + ½] S = [ x + ½, y + ½, z + ½]

M2 = [x, y +1, z] I2 = [ x + 1, y + ½, z + ½]

(63) Cmcm {M1, M2, R, I, M3 = IM2I } C = M2R

M3RM2R = (M1M2)2 = (M1M3)

2 = (M1R)

2 = (IM1IR)

2 = E

X = (IM1)2 Y = (RI)

2 Z = M2M3 W = IM1R

M1 = [ x , y, z] M2 = [x, y, z ] R = [x, y , z + ½] I = [ x + ½, y + ½, z + ½]

M3 = [x, y, z + 1]

(64) Cmca {M, C, R, I, R2 = IRI} A = CIM

IRICRC = (RM)2 = (IRIM)

2 = CMC

−1M = CIC

−1I = (C

2RIR)

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] R2 = [x, y + 1, z + ½]

21

(65) Cmmm {M1, M2, M3, M4, R}

(M1M2)2 = (RM1RM2)

2 = (RM3)

2 = (RM4)

2 =

(M1M3)2 = (M2M3)

2 = (M1M4)

2 = (M2M4)

2 = E

X = (RM1)2 Y = (RM2)

2 Z = M4M3 W = RM1M2

M1 = [ x , y, z] M2 = [x, y , z] M3 = [x, y, z ] M4 = [ x, y, z + 1]

R = [ x + ½, y + ½, z]

(66) Cccm {M, R1, R2, R3, M4 = R1MR1} C1 = R2MR1R2 C2 = R1MR1R2

M4R2MR2 = (R1R2)2 = (MR1R2)

2 = (MR3)

2 = (MR4)

2 = (R3R1R3R2)

2 = E

X = (R3R1)2 Y = (R3R2)

2 Z = M4M W = R3R1R2

M = [x, y, z ] R1 = [ x , y, z + ½] R2 = [x, y , z + ½] R3 = [ x , y , z]

M4 = [x, y, z + 1]

(67) Cmma {Z, M1, M2, R1, R2} A = R1M1

(M1M2)2 = (R1R2)

2 = (M1R2)

2 = (M2R1)

2 = (ZR1)

2 = (ZR2)

2 = ZM1Z

−1M1 = ZM2Z

−1M2 = E

X = (R1M1)2 Y = (M2R2)

2 W = R1R2M1M2

M1 = [ x , y, z] M2 = [x, y , z] R1 = [ x + ½, y, z ] R2 = [x, y + ½, z ]

(68) Ccca {R1, R2, R3, C, R4 = R3R2R3} C2 = CR3 A = CR3R1

R4CR2C = (R1R2)2 = (R1R4)

2 = CR3C

−1R3 = (C

2R1)

2 = (C

2R2)

2 = E

X = (R3R1)2 Y = (R3R2)

2 Z = C

2 W = R3R2R1

R1 = [ x , y, z ] R2 = [x, y , z ] R3 = [ x + ½, y + ½, z] C = [x, y + ½, z + ½]

R4 = [x, y + 1, z ]

22

(69) Fmmm {M1, M2, M3, R1, R2, M4 = R1M3R1}

M4R2M3R2 = (M2M3)2 = (M3M1)

2 = (M1M2)

2 = (R1R2)

2 = (R1M2)

2 = (R2M1)

2 =

(R1R2M3)2 = E

X = (R1M1)2 Y = (R2M2)

2 Z = M4M3 W = R1R2M2M1

M1 = [ x , y, z] M2 = [x, y , z] M3 = [x, y, z ]

R1 = [ x + ½, y, z + ½] R2 = [x, y + ½, z + ½] M4 = [x, y, z + 1]

(70) Fddd {D1, D2, D3,

S = D2D1, R1 = D2−1

D1, R2 = SR1S−1

, R3 = D3−1

D2, R4 = D3D1−1

, D′ = SD1S−1

}

SR3R4 = (D2D3−1

)2 = (D3D1

−1)2 = (D1D2

−1)2 = (D1D2D3

−1)2 = (D1D2D3)

2 = (D2D1D3)

2 =

D22D3

2D2

−2D3

−2 = D3

2D1

2D3

−2D1

−2 = D1

2D2

2D1

−2D2

−2 = E

X = (D2D3)2 Y = (D3D1)

2 Z = (D1D2)

2

W1 = D12 W2 = D2

2 W3 = D3

2

D1 = [ x + ¼, y + ¼, z + ¼] D2 = [x + ¼, y + ¼, z + ¼] D3 = [x + ¼, y + ¼, z + ¼]

R1 = [ x , y , z] S = [ x + ½, y , z + ½] R2 = [ x + 1, y , z]

R3 = [x, y , z ] R4 = [ x + ½, y, z + ½] D′ = [ x + ¾, y − ¼, z]

(71) Immm {M1, M2, M3, I, M4 = IM3I }

(M1M2)2 = (M2M3)

2 = (M3M1)

2 = (IM1IM2)

2 = (IM2IM3)

2 = (IM3IM1)

2 = E

X = (IM1)2 Y = (IM2)

2 Z = M4M3 W = IM3M2M1

M1 = [ x , y, z] M2 = [x, y , z] M3 = [x, y, z ]

I = [ x + ½, y + ½, z + ½] M4 = [x, y, z + 1]

23

(72) Ibam {M, R1, R2, I, M2 = IMI } B = IR2 A = IR1

M2R1MR1 = M2R2MR2 = (R1R2)2 = (MR1R2)

2 = (R1R2IR1IR2I)

2 = (R1IR1(IM)

2)2 =

(R2IR2(IM)2)2 =

(IMR2IR1)2 =E

X = A2 Y = B

2 Z = M2M W = IMR2R1

M = [x, y, z ] R1 = [ x , y, z + ½] R2 = [x, y , z + ½] I = [ x + ½, y + ½, z + ½]

M2 = [x, y, z + 1]

(73) Ibca {R1, R2, C, I, R3 = IR1I } B = IR1 A = IR2

R3R2R1R2 = R3CR1C = CR2C−1

R2 = BCBC−1

= CACA−1

= ABAB−1

=

(ABC)2 = (CBA)

2 = (CI)

2 = E

X = A2 Y = B

2 Z = C

2 W = IC

−1BA

R1 = [x, y , z + ½] R2 = [ x , y + ½, z] C = [x, y + ½, z + ½]

I = [ x + ½, y + ½, z + ½] R3 = [x + 1, y , z + ½]

(74) Imma {M1, M2, R1, R2, M3 = R1M1R1, M4 = R2M2R2} A = R1M1

(M1M2)2 = (M2M3)

2 = (M3M4)

2 = (M4M1)

2 = (R1M2)

2 = (R1M4)

2 = (R2M1)

2 = (R2M3)

2 = E

X = M3M1 Y = M4M2 Z = (R1R2)2 W = R1R2M1M2

M1 = [ x , y, z] M2 = [x, y , z] R1 = [ x + ½, y, z + ½] R2 = [x, y + ½, z ]

M3 = [ x + 1, y, z] M4 = [x, y + 1, z]

24

Tetragonal

(75) P4 {Z, R, F}

ZFZ−1

F−1

= ZRZ−1

R = (RF−1

)4 = E

X = RF2 Y = FRF

R = [ x + 1, y , z]

(76) P41 {X, S, Y = SXS−1

)

XS2XS

−2 = E

Y = SXS−1

Z = S4

S = [ y , x, z + ¼]

(77) P42 {R1, R2, S, Y = SR2R1S−1

}

R1SR1S−1

= R2R1S2R2R1S

−2 = (R2R1SR2 S

−1)2 = E

X = R2R1 Y = SR2R1S−1

Z = R1S2

R1 = [ x , y , z] R2 = [ x + 1, y , z] S = [ y , x, z + ½]

(78) P43 {X, S, Y = SX−1

S−1

}

XS2XS

−2 = E

Y = SX−1

S−1

Z = S4

S = [y, x , z + ¼]

25

(79) I4 {F, S, F2 = SFS−1

}

(FF2)2 = S

2FS

−2F

−1 = E

X = F2F−1

Y = F2−1

F Z = S2 W = SF

2

S = [ x + ½, y + ½, z + ½] F2 = [ y + 1, x, z]

(80) I41 {R, S, R2 = SRS−1

, R3 = S−1

RS, R4 = S2RS

−2}

R1R2R3R4 = (RR3R2)2 = RS

4RS

−4 = E

X = R2R Y = R4R Z = S4 W = S

2R

R = [ x , y , z] S = [ y + ½, x, z + ¼]

R2 = [ x + 1, y , z] R3 = [ x + 1, y + 1, z] R4 = [ 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

−1Z = E

X = R F2 Y = R2 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)2R(W F

2)−2

= F (W F2)2F

−1(W F

2)−2

= (W F2)2( F

2W)

−2 = E

X = R F2 Y = F R F Z = (W F

2)2

R = [ x + 1, y , z] F 2 = [y, x + 1, z ]

26

(83) P4/m {F, M1, M2, R}

(RF) 4

= (M1F)4 = (M2F)

4 = (M1R)

2 = (M2R)

2 = FM1F

−1M1 = FM2 F

−1M2 = E

X = RF2 Y = R2F

2 Z = M2M1

M1 = [x, y, z ] M2 = [x, y, z + 1] R = [ x + 1, y , z]

(84) P42/m { F , R, M1, M2}

(RM1)2 = (RM2)

2 = ( F

2M1)

2 = ( F

2M2)

2 = M1M2 F M1M2 F

−1 = (R F )

4 = E

X = R F2 Y = F

−1R F

−1 Z = M2M1

F = [y, x , z + ½] R = [ x + 1, y , z] M1 = [x, y, z ] M2 = [x, y, z + 1]

(85) P4/n {Z, F, I} N = IF2

FZF−1

Z−1

= (IZ)2 = (FI)

4 = E

X = F−1

IFI Y = FIF−1

I

I = [ x + ½, y + ½, z ]

(86) P42/n { F , I1, I2, F 2 = I2 F I1} N = I1 F2 S = F I2 R = I1I2S

2

(I1I2S2) 2

= F I1I2 F−1

I1I2 = ( F2I1 F I2 F I1)

2 = ( F

2I2 F I1 F I2)

2 = E

X = I2 F−1

I1 F Y = I1 F I2 F−1

Z = I2I1

F = [y, x , z + ½] I1 = [ x + ½, y + ½, z ] I2 = [ x + ½, y + ½, z + 1]

F 2 = [y, x + 1, z + ½]

27

(87) I4/m {F, M, I, M2 = IMI}

(IF)4 = MFMF

−1 = M2FM2F

−1 = (FIF

−1M2M)

2 = E

X = IFIF−1

Y = IF−1

IF Z = M2M W = IMF2

M = [x, y, z ] I = [ x + ½, y + ½, z + ½]

M2 = [x, y, z + 1]

(88) I41/a {S, F , F 2 = S F S } A = F S

X = A2 Y = F 2 F Z = S

4 W = F S

−2F

S = [ y , x + ½, z + ¼] F = [y, x , z ] F 2 = [y, x + 1, z ]

(89) P422 {Z, R1, R2, R3} F = R3R1

(R1R2)2 = (R2R3)

4 = (R3R1)

4 = (R2R3R1)

2 = (R1Z)

2 = (R2Z)

2 = (R3Z)

2 = E

X = R2R3R1R3 Y = R3R2R3R1

R1 = [x, y , z ] R2 = [ x , y, z ] R3 = [y, x , z ]

(90) P4212 {Z, F, R} S = RF

(RZ)2 = (RF)

4 = FZF

−1Z = (F

−1RFR)

2 = E

X = (RF−1

)2 Y = S

2

R = [ y + ½, x + ½, z ]

28

(91) P4122 {X, R1, R2, Y = R2X−1

R2} S = R2R1

(R1X)2 = (XR2)

2(X

−1R2)

2 = (X(R2R1)

2R2)

2 = E

Y = R2X−1

R2 Z = S2

R1 = [ x , y, z ] R2 = [ y , x , z + ¼]

(92) P41212 {S1, S2, S3 = S12S2

−1S1

2} R = S2S1

−1

S22S3

−2 = (S1S2

−1)2 = (S1S2)

2 = (S1S3

−1)2 = (S1S3)

2 = S1

4S2

2S1

−4S2

−2 = E

X = S1S2−2

S1−1

Y = S22 Z = S1

4

S1 = [ y , x, z + ¼] S2 = [ x + ½, y + ½, z + ½] S3 = [ x + ½, y − ½, z + ½]

(93) P4222 {R1, R2, R3, R4, R5, R6 = R2R5R1} S = R5R2

(R2R5R1)2 = (R1R2)

2 = (R2R3)

2 = (R3R4)

2 = (R4R1)

2 =

(R3R4R5)2 = (R3R4R6)

2 = (R3R6R5)

2 = R4R2R5R2R5R2R4R5R2R5 = E

X = R3R1 Y = R4R2 Z = R5R6

R1 = [ x , y, z ] R2 = [x, y , z ] R3 = [ x + 1, y, z ] R4 = [x, y + 1, z ]

R5 = [y, x, z + ½] R6 = [y, x, z − ½]

(94) P42212 {R1, R2, R3, S}

(R1R2)2 = (R3R2R3R1)

2 = SR3S

−1R3 = R1R3R2S

2R1R3R2S

−2 = R1R2R3S

2R1R2R3S

−2 = E

X = R3R2R1 Y = R1R3R2 Z = S2R3

R1 = [y, x, z ] R2 = [ y , x , z ] R3 = [ x + 1, y , z] S = [ y + ½, x − ½, z + ½]

29

(95) P4322 (X, R1, R2, Y = R2XR2} S = R2R1

(R1X)2 = (X

−1R2)

2(XR2)

2 = (X

−1(R2R1)

2R2)

2 = E

Y = R2XR2 Z = (R2R1)2

R1 = [ x , y, z ] R2 = [y, x, z + ¼]

(96) P43212 {S1, S2, S3 = S12S2

−1S1

2} R = S2S1

−1

S22S3

−2 = (S1S2

−1)2 = (S1S2)

2 = (S1S3

−1)2 = (S1S3)

2 = S1

4S2

2S1

−4S2

−2 = E

X = S1S2−2

S1−1

Y = S22 Z = S1

4

S1 = [y, x , z + ¼] S2 = [ x + ½, y + ½, z + ½] S3 = [ x + ½, y − ½, z + ½]

(97) I422 {F, R1, R2, F2 = R1F−1

R1}

(FF2)2 = (FR2)

2 = (FR1R2R1)

2 = E

X = (R1F−1

)2 Y = (R1F)

2 Z = (R2R1)

2 W = R2R1F

2

R1 = [ y + ½, x + ½, z + ½] R2 = [y, x, z ] F2 = [y, x + 1, z ]

(98) I4122 {R1, R2, R3, R4 = SR3S−1

, R5 = S2R3S

−2, R6 = S

−1R3S} S = R2R1

R3R4R5R6 = (R1R4)2 = (R1R6)

2 = (R4R3R6)

2 = E

X = R3R4 Y = R3R6 Z = S4 W = R3S

2

R1 = [ y , x , z ] R2 = [x, y , z + ¼] R3 = [ x + ½, y + ½, z]

R4 = [ x − ½, y + ½, z] R5 = [ x − ½, y − ½, z] R6 = [ x + ½, y − ½, z]

30

(99) P4mm {Z, M1, M2, M3} F = M1M3

(M1M2)2 = (M2M3)

4 = (M3M1)

4 = M1ZM1Z

−1 = M2ZM2Z

−1 = M3ZM3Z

−1 = E

X = M3M2M3M1 Y = M3M1M3M2

M1 = [ x , y, z] M2 = [x, y , z] M3 = [ x + ½, y + ½, z]

(100) P4bm {Z, M, F} B = MF

(MFMF−1

)2 = ZMZ

−1M = ZFZ

−1F

−1 = E

X = (MF−1

)2 Y = B

2

M = [ y + ½, x + ½, z] F = [ y , x, z]

(101) P42cm {M, R, C} S = CM

CRC−1

R = (MCMC−1

)2 = (RCMC

−1RM)

2 = E

X = RC−1

RMCM Y = MRC−1

RMC Z = C2

M = [y, x, z] R = [ x + 1, y , z] C = [x, y , z + ½]

(102) P42nm {M1, M2, R, S}

(M1M2)2 = SRS

−1R = (RM2RM1)

2 = RM1M2S

2RM1M2S

−2 = M2RM1S

2M2RM1S

−2 = E

X = RM1M2 Y = M1RM2 Z = RS2

M1 = [y, x, z] M2= [ x , y , z] R = [ x + 1, y , z] S = [ y + ½, x + ½, z + ½]

31

(103) P4cc {F1, F2, C}

CF1C−1

F1 = CF2C−1

F2 = F1F2F12F2F1F2

2 = E

X = F2F1−1

Y = F2−1

F1 Z = C2

F1 = [ y , x, z] F2 = [ y + 1, x, z] C = [y, x, z + ½]

(104) P4nc {F1, F2, C} N = CF12

CF1C−1

F2 = CF2C−1

F1 = F1F2F12F2F1F2

2 = E

X = F2F1 −1

Y = F2 −1

F1 Z = C2

F1 = [ y , x, z] F2 = [ y + 1, x, z] C = [ y + ½, x + ½, z + ½]

(105) P42mc {M1, M2, C, M3 = CM1C−1

, M4 = CM2C−1

} S = CM2

(M1M2)2 = (M1CM1C

−1)2 = (M2CM2C

−1)2 = M1C

2M1C

−2 = M2C

2M2C

−2 = E

X = M4M1 Y = M3M2 Z = C2

M1 = [ x + 1, y, z ] M2 = [x, y , z ] C = [y, z, z + ½]

M3 = [x, y + 1, z ] M4 = [ x , y, z ]

(106) P42bc {A, C, B = CA−1

C−1

}

C2AC

−2A

−1 = A

2CA

−2C

−2 = E

X = A2 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]

32

(107) I4mm {M1, M2, C, M3 = CM1C−1

} F = M2M1

(M2M1)4 = (CM1C

−1M1)

2 = CM2C

−1M2 = C

2M1C

−2M1 = E

X = CM1C−1

M2M1M2 Y = M2CM1C−1

M2M1 Z = C2 W = CM1M2M1

M1 = [x, y , z] M2 = [y, x, z] C = [ y + ½, x + ½, z + ½] M3 = [ 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

2CM

C = [y, x, z + ½] M = [ y + ½, x + ½, z]

(109) I41md {M, S, M2 = SMS−1

, M3 = S2MS

−2, M4 = S

−1MS } D = MS

(MSMS−1

)2 = S

4MS

−4M = E

X = S2MS

−2M Y = S

−1MS

2MS

−1 Z = S

4 W = D

2

M = [ x , y, z] S = [ y + ½, x, z + ¼]

M2 = [x, y , z] M3 = [ x + 1, y, z] M4 = [x, y + 1, z]

(110) I41cd {A, S, B = S−1

AS}

(AB)2 = (AB

−1)2 = S

4AS

−4A

−1 = E

X = A2 Y = B

2 Z = S

4 W = AB

−1S

2 = ASA

−1S

A = [x + ½, y + ½, z] S = [y + ½, x , z + ¼] B = [ x + ½, y + ½, z]

33

(111) P 4 2m {Z, R1, R2, M} F = MR1

(R1R2)2 = (R1M)

4 = (R2M)

4 = (R1Z)

2 = (R2Z)

2 = MZMZ

−1 = E

X = R1MR2M Y = R2MR1M

R1 = [ x + 1, y, z ] R2 = [x, y , z ] M = [y, x, z]

(112) P 4 2c {R1, R2, C, R3 = CR2C−1

, R4 = CR1C−1

} F = R2C

(C2R1)

2 = (C

2R2)

2 = (R1C)

4 = (R2C)

4 = E

X = R1R3 Y = R4R2 Z = C2

R1 = [ x + 1, y, z + ½] R2 = [x, y , z + ½] C = [y, x, z + ½]

R3 = [ x , y, z + ½] R4 = [ x + 1, y, z + ½]

(113) P 4 21m { F , Z, M} S = M F−1

(M F M F−1

)2 = MZMZ

−1 = Z F

−1Z F = E

X = (M F )2 Y = (M F

−1)2

M = [ y + ½, x + ½, z]

(114) P 4 21c { F , C, F 2 = C F C} S = C F

(C F C F−1

)2 = F C

2F

−1C

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]

34

(115) P 4 m2 {Z, M1, M2, R} F = M1R

(M1M2)2 = (M1R)

4 = (M2R)

4 = M1ZM1Z

−1 = M2ZM2Z

−1 = (RZ)

2 = E

X = M2RM1R Y = RM2RM1

M1 = [x, y , z] M2 = [ x + 1, y, z] R = [y, x, z ]

(116) P 4 c2 { F , R, C}

RCRC−1

= F2C F

2C

−1 = ( F R)

4 = ( F C)

2 = E

X = F2R 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

−1Z = (RZ)

2 = E

X = (R F )2 Y = B

2

R = [ y + ½, x + ½, z ]

(118) P 4 n2 { F , R1, R2, F 2 = R2 F−1

R1} N = F R2

( F R1R2)4 = ( F R1 F

−1R2)

2 = F R2R1 F

−1R2R1 = F R1R2 F

−1R1R2 = E

X = R2 F R1 F Y = F−1

R2 F R1 F2 Z = R2R1

R1 = [ y + ½, x + ½, z − ½] R2 = [ y + ½, x + ½, z + ½]

F 2 = [ y , x + 1, z]

35

(119) I 4 m2 {M, R1, R2, R3 = R2R1R2, M2 = R1MR1} F = R1M

M2R3MR3 = (MM2)2 = (R1M)

4 = (R3M)

4 = (R3M)

4 = (R2M)

4 = (R2M2)

4 =

M(R2R1)2M(R1R2)

2 = E

X = MR2M2R2 Y = M2R2MR2 Z = R3R1 W = R2MR1M

M = [ x , y, z] R1 = [y, x, z ] R2 = [ y + ½, x + ½, z + ½]

R3 = [y, x, z + 1] M2 = [x, y , z]

(120) I 4 c2 { F , R1, R2, R3 = R2R1R2} C = R1 F−1

R3 F R1 F = ( F R1)2 = ( F R3)

2 = ( F R2 F

−1R2)

2 = E

X = (R2 F )2 Y = (R2 F

−1)2 Z = R1R3 W = R1R2 F

2

F = [ y , x, z + ½] R1 = [y, x, z ] R2 = [ y + ½, x + ½, z + ½]

R3 = [y, x, z + 1]

(121) I 4 2m {M, R, C, R2 = CRC} F = MR

(C2R)

2 = (MR)

4 = (RC)

4 = (CMRM)

4 = MC

2MC

−2 = MRMC

2MRMC

−2 = E

X = R2MRM Y = R2MR2R Z = C2 W = CRMR

M = [y, x, z] R = [x, y , z ] C = [ y + ½, x + ½, z + ½] R2 = [ x + 1, y, z ]

(122) I 4 2d { F , R, R2 = F R F−1

, F 2 = RR2 F R2R} D = R F

F2R F R F

−1R F

2R F

−1R 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 ]

R2 = [x, y + ½, z + ½] F 2 = [y + 1, x , z + ¼]

36

(123) P4/mmm {M1, M2, M3, M4, M5} F = M1M2

(M2M3)4 = (M3M1)

2 = (M1M2)

4 = (M1M4)

2 = (M2M4)

2 = (M3M4)

2 =

(M1M5)2 = (M2M5)

2 = (M3M5)

2 = E

X = M2M3M2M1 Y = M3M2M1M2 Z = M5M4

M1 = [ x , y, z] M2 = [y, x, z] M3 = [x, y + 1, z]

M4 = [x, y, z ½] M5 = [x, y, z + 1]

(124) P4/mcc {M, R1, R2, R3, M2 = R1MR1}

F = R1R3 C1 = R1M C2 = R3M

M2R2MR2 = M2R3MR3 = (R1R2)2 = (R1MR1R3)

4 = (R3R1)

4 = (R3R2)

4 = (R1R2M)

2 =

(R1MR1MR1)2 = (R2MR1MR1)

2 = (R3M R1MR1)

2 = E

X = R3R2R3R1 Y = R2R3R1R3 Z = M2M

M = [x, y, z ] R1 = [ x , y, z + ½] R2 = [x, y + 1, z + ½] R3 = [y, x, z + ½]

M2 = [x, y, z + 1]

(125) P4/nbm {Z, M, R1, R2} F = R1R2 N = MFR1 B = MF

(R1R2)4 = (R1M)

4 = (R2M)

2 = MZMZ

−1 = (R1Z)

2 = (R2Z)

2 = E

X = (MR2R1)2 Y = B

2

M = [ y + ½, x + ½, z] R1 = [ x , y, z ] R2 = [y, x, z ]

(126) P4/nnc {R1, R2, C, R3 = CR1C} F = R1R2 N1 = CRF−1

N2 = CF

(R1R2)4 = (R1C)

4 = (R2C)

2 = (R1C

2)2 = E

X = (R2CR1)2 Y = R1CFC R2 Z = C

2

R1 = [ x , y, z ] R2 = [y, x, z ] C = [ y + ½, x + ½, z + ½]

R3 = [x, y + 1, z ]

37

(127) P4/mbm {F, M1, M2, M3} B = M2F

(M2M3)2 = (M1M2)

2 = M1FM1F

−1 = (M2FM2F

−1)2 = (FM3)

4 = FM3F

−1M3 = E

X = (M2F−1

)2 Y = (M2F)

2 Z = M3M1

M1 = [x, y, z ] M2 = [ y + ½, x + ½, z] M3 = [x, y, z + 1]

(128) P4/mnc {F, M, R, M2 = RMR } N = RMF C = RM

(FRF−1

R)2 = FMF

−1M = FRMRF

−1RMR = E

X = (RF−1

)2 Y = (RF)

2 Z = M2M

M = [x, y, z ] R = [ y + ½, x + ½, z + ½] M2 = [x, y, z + 1]

(129) P4/nmm {Z, M1, M2, R} F = M2M1 N = R1FM2

RM2RR2MR2 = (RM1)2 = (RM2)

4 = (R2M1)

2 = (M1 RM2R)

2 = (M1M2)

4 = (R RM2R)

4 =

(R2 RM2R)4 = (M2 RM2R)

4 = R2RM2RR2M2 = E

X = M1 RM2RM1M2 Y = M1M2M1 RM2R

M1 = [y, x, z] M2 = [ x , y, z]

R = [ y + ½, x + ½, z ] R2 = [ y + ½, x + ½, z + 1]

(130) P4/ncc {F, R, C} N = RF2C

(RC2)2 = FC

2F

−1C

−2 = F

2CF

−2C

−1 = F

2(RFRF

−1R)F

2(RF

−1RFR) = E

X = (RF)2 Y = (FR)

2 Z = C

2 = C2

2

R = [y + ½, x + ½, z + ½] C = [y, x, z + ½]

38

(131) P42/mmc {M1, M2, M3, R, M4 = RM3R} C = RM3

(M1R)4 = (M2R)

4 = (M1M2)

2 = (M1M3)

2 = (M2M3)

2 = (M1M4)

2 = (M2M4)

2 = (RM4)

2 =

(M1M2M3)2 = (M1M2M4)

2 = E

X = RM2RM1 Y = RM1RM2 Z = C2

M1 = [ x , y, z] M2 = [x, y , z] M3 = [x, y, z ] R = [y + ½, x + ½, z + ½]

M4 = [x, y, z + 1]

(132) P42/mcm {M1, M2, R1, R2, M3 = R1M2R1} C = R1M2 S = M1C

M3R2M2R2 = (R1R2)2 = (M1M2)

2 = (M1M3)

2 = (R1M1)

4 = (R2M1)

4 = (R1R2M2)

2 = E

X = R2M1R1M1 Y = R1M1R2M1 Z = C2

M1 = [y, x, z] M2 = [x, y, z ] R1 = [x, y , z + ½] R2 = [ x + 1, y, z + ½]

M3 = [x, y, z + 1]

(133) P42/nbc {R1, R2, C, R3 = CR2C} N = C−1

R2R1R2 B = C−1

R2R1

(CR2C)2 = (R1C)

2 = (R2C

2)2 = E

X = C2R1R2R1 Y = C

2R1R3R1 Z = C

2

R1 = [y, x, z ] R2 = [x, y , z + ½] C = [ y + ½, x + ½, z + ½]

R3 = [x, y + 1, z + ½]

(134) P42/nnm {M, R1, R2, R3, R4 = R3R1R2} N1 = R3R1MR1 N2 = R1R3M

(R1R4)2 = (R1M)

4 = (R4M)

4 = (R2M)

2 = (R3M)

2 = (R1R2R3)

2 = (MR1MR2R3)

2 = E

X = R4MR1M Y = MR4MR1 Z = R3R2

M = [y, x, z]

R1 = [x, y , z + ½] R2 = [ y + ½ , x + ½, z ] R3 = [ y + ½, x + ½, z + 1]

R4 = [ x + 1, y, z + ½]

39

(135) P42/mbc { F , M, R, M2 = RMR} S = F−1

M B = R F−1

C = RM

M2 F M F−1

= ( F2M)

2 = F C

2F

−1C

2 = F

2(R F R F

−1R) F

2(R F

−1R 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] M2 = [x, y, z + 1]

(136) P42/mnm { F , M1, M2, M3 = F M2 F−1

S = F−1

M2 N = M1 F−1

M2

(M1M2)2 = (M1M3)

2 = ( F M1 F )

2 = ( F

2M2)

2 = ( F

2M3)

2 =

F2(M1 F M1 F

−1M1) F

2(M1 F

−1M1 F M1) = (M1M2M3)

2 = (M1 F M2 F M2)

2 = E

X = (M1 F )2 Y = (M1 F

−1)2 Z = M3M2

F = [y, x , z + ½] M1 = [ y + ½, x + ½, z] M2 = [x, y, z ] M3 = [x, y, z + 1]

(137) P42/nmc {R, M, C, M2 = CMC−1

} S = MC−1

N = RMCM

(RM)4 = (RC)

2 = (MM2)

2 = (C

2MR)

4 = MC

2MC

−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] M2 = [x, y , z]

(138) P42/ncm {R, C, M, R2 = CRC, M2 = CMC−1

} S = CM N = MR1R2

(R1R2)2 = (RM)

2 = (R2M2)

2 = (MM2)

2 = (RC

2)2 = MC

2MC

−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]

R2 = [ y , x , z + ½] M2 = [ y + ½, x − ½, z]

40

(139) I4/mmm {M1, M2, M3, R} F = M2M1

(RM1)4 = (RM2)

2 = (M1M2)

4 = (M1M3)

2 = (M2M3)

2 = (M1RM2M1M3RM2)

2 = E

X = (RM2M1)2 Y = (M2M1R)

2 Z = (RM3)

2 W = RM1M2M1M3

M1 = [x, y , z] M2 = [y, x, z] M3 = [x, y, z ] R = [ y + ½, x + ½, z + ½]

(140) I4/mcm {M1, M2, R1, R2, M3 = R1M1R1} F = R1R2 C = R1R2M

M3R2M1R2 = (R2M2)2 = (R1R2)

4 = (M1M2)

2 = E

X = M2R2R1R2M2R1 Y = M2R1M2R2R1R2 Z = M3M1 W = R2R1R2R1M1

M1 = [x, y, z ] M2 = [ y + ½, x + ½, z] R1 = [ x , y, z + ½] R2 = [y, x, z + ½]

M3 = [x, y, z + 1]

(141) I41/amd { R1, R2, M, M2 = SMS−1

, M3 = S2MS

−2, M4 = S

−1MS}

S = R2R1 A = R2SM D = AR2

(R1M)2 = (R1M3)

2 = (R2M)

4 = (R2M2)

4 = (R2M3)

4 = (R2M4)

4 = (MM2)

2 = E

X = A2 Y = R2A

2R2 Z = S

4 W = D

2

R1 = [x, y + ½, z ] R2 = [y, x, z + ¼] M = [ x , y, z]

M2 = [x, y , z] M3 = [ x + 1, y, z] M4 = [x, y + 1, z]

(142) I41/acd {R1, R2, B, A1 = R2BR2} S = R2R1

A = R2BS C = B−1

S2 D = BS

S4C

−2 = (BR1)

2 = (BR2)

4 = B(R2R1R2)B

−1(R2R1R2) = AC

2A

−1C

2 = CA

2C

−1A

−2 =

BA2B

−1A

2 = (R2C)

4 = E

X = A2 Y = B

2 Z = C

2 W = D

2

R1 = [x, y + ½, z ] R2 = [y, x, z + ¼] B = [ x + ½, y + ½, z]

A1 = [x + ½, y + ½, z]

41

Trigonal

(143) P3 {Z, Q1, Q2}

(Q1Q2)3 = ZQ1Z

−1Q1

−1 = ZQ2Z

−1Q2

−1 = E

X = Q1−1

Q2 Y = Q2Q1−1

Q1 = [−y + 1, x − y, z] Q2 = [−y + 1, x − y + 1, z]

(144) P31 {X, S, Y = SXS−1

, XY }

XYX−1

Y−1

= S3XS

−3X

−1 = E

Z = S3

S = [−y, x − y, z + 1/3]

(145) P32 {X, S, Y = S−1

XS XY }

XYX−1

Y−1

= S3XS

−3X

−1 = E

Z = S3

S = [y − x, −x, z + 1/3]

(146) R3 {S, Q, Q2 = SQS−1

, Q3 = S−1

QS}

QS3Q

−1S

−3 = Q2QQ2

−1Q

−1 = (QQ2)

3 = (QS)

2QS

−2 = E

X = Q2Q−1

Y = QQ2Q Z = S3 T = SQ

−1

S = [2/3 − y, 1/3 + x − y, 1/3 + z] Q2 = [−y + 1, x − y, z] Q3 = [−y + 1, x − y + 1, z]

42

(147) P 3 {Z, Q , I}

( Q I)3 = (ZI)

2 = Q Z Q

−1Z = E

X = Q I Q2 Y = Q

−1IQ

−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

2Q )

2 = E

X = I Q−1

IQ Y = I Q I Q−1

Z = S3 T = IQ

−1S

S = [2/3 − y, 1/3 + x − y, 1/3 + z] I = [1 − x, 1 − y, −z]

(149) P312 {Z, R1, R2, R3} Q = R2R1

(R2R3)3 = (R3R1)

3 = (R1R2)

3 = (R1Z)

2 = (R2Z)

2 = (R3Z)

2 = E

X = R3R1R3R2 Y = R3R2R3R1

R1 = [x, x − y, −z] R2 = [y − x, y, −z] R3 = [1 − y, 1 − x, −z]

(150) P321 {Q, R1, R2, Q2 = R1Q−1

R1}

Q2R2QR2 = (QQ2)3 = E

X = Q−1

Q2 Y = Q2Q−1

Z = R2R1

Q = [1 − y, x − y, z] R1 = [y, x, −z] R2 = [y, x, 1 − z]

Q2 = [1 − y, 1 + x − y, z]

43

(151) P3112 {X, R1, R2, Y = R2R1XR1R2, XY}

(R2X)2 = (X

−1R1)

2(XR1)

2 = (XR1R2R1R2R1)

2 = E

Y = R2R1XR1R2 Z = (R2R1)3

R1 = [x, x − y, −z] R2 = [−x + y, y, 1/3 − z]

(152) P3121 {X, R1, R2, Y = R1XR1, XY}

(R1X)2(R1X

−1)2 = X(R1R2)

2X

−1(R1R2)

2 = E

Y = R1XR1 Z = (R2R1)3

R1 = [y, x, −z] R2 = [−x, −x + y, 1/3 − z]

(153) P3212 {X, R1, R2, Y = R1R2XR2R1, XY }

(R1X)2 = (R2X)

2(R2X

−1)2 = (XR2R1R2R1R2)

2 = E

Y = R1R2XR2R1 Z = (R2R1)3

R1 = [−x + y, y, −z] R2 = [x, x − y, 1/3 − z]

(154) P3221 {X, R1, R2, Y = R2XR2, XY }

(R2X)2(R2X

−1)2 = X(R2R1)

2X

−1 (R2R1)

2 = E

Y = R2XR2 Z = (R2R1)3

R1 = [−x, −x + y, −z] R2 = [y, x, 1/3 − z]

44

(155) R32 {Q R1, R2, Q2 = R2Q−1

R2, Q3 = R1Q2−1

R1 }

QR1Q−1

R1 = (QQ2)3 = (QQ2Q)R1R2R1R2R1(QQ2Q)

−1R1R2R1R2R1 = E

X = Q2Q−1

Y = QQ2Q Z = (R2R1)3

R1 = [y, x, −z] R2 = [2/3 − x, 1/3 − x + y, 1/3 − z]

Q2 = [1 − y, x − y, z] Q3 = [1 − y, 1 + x − y, z]

(156) P3m1 {Z, M1, M2, M3} Q = M2M1

(M2M3)3 = (M3M1)

2 = (M1M2)

2 = MiZMiZ

−1 = E

X = M1M3M1M2 Y = M2M3M2M1

M1 = [x, x − y, z] M2 = [−x + y, y, z] M3 = [1 − y, 1 − x, z]

(157) P31m {Z, M, Q}

(QMQ−1

M)3 = QZQ

−1Z

−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, Q1, Q2} B = (Q1Q2)−1

(Q1Q2)3 = Q1CQ1C

−1 = Q2CQ2C

−1 = E

X = Q1Q2−1

Y = Q2Q1−1

Z = C2

C = [1 − y, 1 − x, ½ + z] Q1 = [1 − y, x − y, z] Q2 = [1 − y, 1 + x − y, z]

45

(159) P31c {C, Q, Q2 = CQ−1

C−1

}

(QQ2)3 = C

2QC

−2Q

−1 = E

X = Q−1

Q2 Y = Q2Q−1

Z = C2

C = [y, x, ½ + z] Q = [1 − y, x − y, z] Q2 = [1 − y, 1 + x − y, z]

(160) R3m {M, S, M2 = SMS−1

, M3 = S−1

MS} Q = S2MS

−2M

MS3MS

−3 = (M2M)

3 = E

X = MSMS−1

MS−1

MS Y = SXS−1

Z = S3 T = SMS

−1MS

M = [x, x − y, z] S = [2/3 − y, 1/3 + x − y, 1/3 + z]

M2 = [−y, −x, z] M3 = [−x + y, y, z]

(161) R3c {S, C, S2 = CSC−1

, Q = S−1

S2, Q2 = S2S−1

}

(Q1Q2)3 = S

3C

−2 = E

X = Q−1

Q2 Y = Q2Q−1

Z = C2 T = S2

2S

−1

S = [2/3 − y, 1/3 + x − y,

½ + z] C = [x, x − y, ½ + z]

S2 = [2/3 − x + y, 1/3 − x, ½ + z] Q = [1 − y, x − y, z] Q2 = [1 − y, 1 + x − y, z]

(162) P 3 1m {Z, M, R1, R2} Q = MR2

(R1R2)3 = (MR1)

2 = (MR2)

6 = (ZR1)

2 = (ZR2)

2 = MZMZ

−1 = E

X = R1R2(MR2)2 Y = R2MR2R1R2M

M = [x − y, −y, z] R1 = [1 − x + y, y, −z] R2 = [x, x − y, −z]

46

(163) P 3 1c {C, R1, R2, R3 = CR1C} Q = C−1

R1

R3C−1

R2C = (R2R3)3 = (R3R1)

3 = (R1R2)

3 = (CR2)

2 = (R1C)

6 = E

X = R2R3R2R1 Y = R2R1R2R3 Z = C2

C = [y, x, ½ + z] R1 = [−x + y, y, ½ − z] R2 = [−y, −x, ½ − z]

R3 = [x, x − y, ½ − z]

(164) P 3 m1 { M1, M2, R1, R2, M3 = R1M1R1} Q = M1R

M3R2M1R2 = (M2M3)3 = (M3M1)

3 = (M1M2)

3 = (M1R1)

6 = (M2R1)

2 = (M2R2)

2 = E

X = M1M2M1M3 Y = M2M3M2M1 Z = R2R1

M1 = [x, x − y, z] M2 = [1 − y, 1 − x, z] R = [y, x, −z] R2 = [y, x, 1 − z]

M3 = [−x + y, y, z]

(165) P 3 c1 {R, C, Q} Q = RQ−1

CQ

(QRQ−1

R)3 = (RQ

−1CQ)

2Q = RCRC

−1 = CQC

−1RQR = C

2QRQC

−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, R1, R2, M2 = R1MR1, M3 = R2MR2} Q = MR1

(MR2R1R2)2 = (MR1R2R1)

2 = (MR2)

6 = (MR1)

6 = E

X = MM3MM2 Y = M3M2M3M Z = (R2R1)3 T = R2MR1M

M = [x, x − y, z] R1 = [y, x, −z] R2 = [1/3 + x − y, 2/3 − y, 1/3 − z]

M2 = [−x + y, y, z] M3 = [1 − y, 1 − x, z]

47

(167) R 3 c { Q , I, R1, R2, }

( Q I)6 = ( Q

−2IQ I Q I)

2 = Q R2 Q

−1R1 = (I(R2R1)

3)2 = Q

2IQIR2R1Q

−2IQ

−1I R1R2 = E

X = I Q−1

IQ Y = I Q I Q

−1 Z = (R2R1)

3 T = R2R1 Q

2

I = [1 − x, 1 − y, −z]

R1 = [2/3 − x, 1/3 + y −x, −1/6 − z] R2 = [1/3 + x − y, 2/3 − y, 1/6 − z]

48

Hexagonal

(168) P6 {Z, H, Q}

QZQ−1

Z−1

= HZH−1

Z−1

= (Q−1

H)2 = E

X = QH−2

Y = (QH2Q)

−1

Q = [1 − y, x − y, z]

(169) P61 {X, S, Y = S2XS

−2 XY = SXS

−1}

S3XS

−3X = X(SXS

−1)X

−1(SXS

−1)−1

= E

Y = S2XS

−2 Z = S

6

S = [x − y, x, z + 1/6]

(170) P65 {X, S, Y = S2XS

−2 XY = SXS

−1}

S3XS

−3X = X(SXS

−1)X

−1(SXS

−1)−1

= E

Y = S2XS

−2 Z = S

−6

S = [x − y, x, z − 1/6]

(171) P62 {R, S, R2 = SRS−1

, R3 = S−1

RS}

(RR3S)2 = (RR2R3)

2 = S

3RSRS

−3RS

−1R = E

X = R2R Y = RR3 Z = S3

R = [1 − x, 1 − y, z] S = [−x + y + 1, −x + 1, z + 1/3]

R2 = [2 − x, 1 − y, z] R3 = [1 − x, −y, z]

49

(172) P64 {R, S, R2 = SRS−1

, R3 = S−1

RS }

(RR3S)2 = (RR2R3)

2 = S

3RSRS

−3RS

−1R = E

X = R3R Y = RR2 Z = S3

R = [1 − x, 1 − y, z] S = [−y + 1, x − y, z + 1/3]

R2 = [1 − x, −y, z] R3 = [2 − x, 1 − y, z]

(173) P63 {Q, S, Q2 = SQS−1

}

(QQ2)3 = S

2QS

−2Q

−1 = E

X = Q−1

Q2 Y = Q2Q−1

Z = S2

Q = [1 − y, x − y, z] S = [1 − x, 1 − y, ½ + z] Q2 = [1 − y, 1 + x − y, z]

(174) P 6 {Z, H 1, H 2}

( H 1 H 2)3 = ( H 1

4H 2)

6 = ( H 2

4H 1)

6 = H 1

3H 2

−3 = H 1Z H 1

−1Z = H 2Z H 2

−1Z

= H 12( H 1 H 2)

−1H 2( H 2 H 1)

−1 = E

X = H 1 H 2−1

Y = H 2−1

H 1

H 1 = [1 + y − x, 1 − x, −z] H2 = [y − x, 1 − x, −z]

(175) P6/m {Z, H , R}

(R H )6 = (R H

2)6 = (R H

3)2 = ( H

4R H R)

6 = H Z H

−1Z = H

3R H

−3R = = ZRZ

−1R =

( H2R( H

−1R)

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]

50

(176) P63/m { H 1, H 2, I, I2 = H 13I H 1

3}

H 13H 2

3 = ( H 1 H 2)

3 = ( H 1 H 2

−2)6 = H 1

2I2I H 1

−1 I2I = H 2

2I2I H 2

−1 I2I = E

X = H 1 H 2−1

Y = H 2−1

H 1 Z = I2I

H 1 = [1 + y − x, 1 − x, ½ − z] H2 = [y − x, 1 − x, ½ − z]

I = [1 − x, 1 − y, −z] I2 = [1 − x, 1 − y, 1 − z]

(177) P622 {Z, R1, R2, R3} H = R1R2

(R2R3)3 = (R3R1)

2 = (R1R2)

6 = (R1Z)

2 = (R2Z)

2 = (R3Z)

2 = E

X = R2R3H−2

Y = R1XR1

R1 = [y, x, −z] R2 = [x, x − y, −z] R3 = [1 − x, 1 − y, −z]

(178) P6122 {X, R1, R2, Y = S−1

XS, XY} S = R2R1

(R1X)2 = (XR2)

2(X

−1R2)

2 = SR2XSR2SX

−1 = E

Y = S−1

XS Z = S6

R1 = [x − y, −y, −z] R2 = [x, x − y, 1/6 − z]

(179) P6522 {X, R1, R2, Y = S−1

XS, XY} S = R1R2

(R1X)2 = (XR2)

2(X

−1R2)

2 = SR2XSR2SX

−1 = E

Y = S−1

XS Z = S6

R1 = [x − y, −y, 1/6 − z] R2 = [x, x − y, −z]

51

(180) P6222 {R1, R2, R3, R4 = R1R3R1, R5 = R2R4R2}

(R2R3)2 = (R1R5)

2 = ((R2R1)

3R3)

2 = (R3R4R5)

2 = E

X = R5R3 Y = R3R4 Z = (R2R1)3

R1 = [x, x − y, −z] R2 = [1 − y, 1 − x, 1/3 − z] R3 = [1 − x, 1 − y, z]

R4 = [1 − x, −y, z] R5 = [2 − x, 1 − y, z]

(181) P6422 {R1, R2, R3, R4 = R1R3R1, R5 = R2R4R2}

(R2R3)2 = (R1R5)

2 = ((R2R1)

3R3)

2 = (R3R4R5)

2 = E

X = R5R3 Y = R3R4 Z = (R1R2)3

R1 = [x, x − y, 1/3 − z] R2 = [1 − y, 1 − x, −z] R3 = [1 − x, 1 − y, z]

R4 = [1 − x, −y, z] R5 = [2 − x, 1 − y, z]

(182) P6322 { R1, R2, R3, R4, R5 = R1R4R1}

(R2R3)3 = (R3R1)

3 = (R1R2)

3 = R5R4R2R1R3R1R4R5R1R3 R1R2 = E

X = R3R1R3R2 Y = R1R2R3R2 Z = R5R4

R1 = [1 − y, 1 − x, ½ − z] R2 = [y − x, y, ½ − z] R3 = [x, x − y, ½ − z]

R4 = [y, x, −z] R5 = [y, x, ½ − z]

(183) P6mm {Z, M1, M2, M3} H = M1M2

(M2M3)3 = (M3M1)

2 = (M1M2)

6 = M1ZM1Z

−1 = M2ZM2Z

−1 = M3ZM3Z

−1 = E

X = M2M3(M2M1)2 Y = (M3M1M2)

2

M1 = [y, x, z] M2 = [x, x − y, z] M3 = [1 − y, 1 − x, z]

52

(184) P6cc {H, C, R} C2 = HC

RCRC−1

= (RH)3 = (H

−2RHRHR)

2 = R(C2H)R(C2H)

−1 = E

X = RC2−1

RC2 Y = RC−1

HRH−1

C Z = C2

C = [y, x, ½ + z] R = [1 − x, 1 − y, z]

(185) P63cm {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) P63mc {M1, M2, C, M3 = CM2C−1

}

(M1M2)3 = M1CM1C

−1 = (M2CM2C

−1)3 = M2C

2M2C

−1 = E

X = M1M2M3 Y = M3M2M3M1 Z = C2

M1 = [1 − y, 1 − x, z] M2 = [x, x − y, z] C = [y, x, ½ + z]

M3 = [1 − x, 1 − y, z]

(187) P 6 m2 {M1, M2, M3, M4, M5} Q = M3M2M4

(M1M2)3 = (M2M3)

3 = (M3M1)

3 = (M1M4)

2 = (M2M4)

2 = (M3M4)

2 = (M1M5)

2 = (M2M5)

2 =

(M3M5)2 = E

X = M1M2M1M3 Y = M3M2M3M1 Z = M5M4

M1 = [1 − y, 1 − x, z] M2 = [x, x − y, z] M3 = [−x + y, y, z]

M4 = [x, y, −z] M5 = [x, y, 1 − z]

53

(188) P 6 c2 {M, R1, R2, R3, M2 = R1MR1} H = MR1R3 C = MR1

M2R2MR2 = M2R3MR3 = (R1R2)3 = (R2R3)

3 = (R3R1)

3 = R1MR1 = R2MR2 = R3MR3 = E

R1MR1 = R2MR2 = R3MR3

X = R1R2R1R3 Y = R3R2R3R1 Z = M2M1

M = [x, y, −z] M2 = [x, y, 1 − z]

R1 = [x, x − y, ½ − z] R2 = [1 − y, 1 − x, ½ − z] R3 = [−x + y, y, ½ − z]

(189) P 6 2m {M, M1, M2, Q} H = QMQ−1

M1

(QMQ−1

M1)6 = (QMQ

−1M2)

6 = (MM1)

2 = (MM2)

2 = QM1Q

−1M1 = QM2Q

−1M2 =

(MQMQ−1

)3 = E

X = (Q−1

M)2 Y = (MQ

−1)2 Z = M2M1

Q = [1 − y, x − y, z] M = [y, x, z] M1 = [x, y, −z] M2 = [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 {M1, M2, M3, M4, M5} H = M1M2

(M2M3)3 = (M3M1)

2 = (M1M2)

6 = (M4M1)

2 = (M4M2)

2 = (M4M3)

2 = (M5M1)

2 = (M5M2)

2 =

(M5M3)2 = E

X = M2M3H−2

Y = M1XM1 Z = M5M4

M1 = [y, x, z] M2 = [x, x − y, z] M3 = [1 − y, 1 − x, z]

M4 = [x, y, −z] M5 = [x, y, 1 − z]

54

(192) P6/mcc {M, R1, R2, R3, M2 = R1MR1}

H = R1R2 C1 = M1R2R1R2 C2 = M1R1

(R2R3)3 = (R3R1)

2 = (R1R2)

6 = MR2R3MR3R2 = MR3R1MR1R3 = MR1R2MR2R1 = E

X = R2R3 H−2

Y = R1XR1 Z = M2M

M = [x, y, −z] M2 = [x, y, 1 − z]

R1 = [y, x, ½ − z] R2 = [x, x − y, ½ − z] R3 = [1 − x, 1 − y, ½ − z]

(193) P63/mcm {M1, M2, R1, R2, M3 = R1M2R1} C = R1M2

M3R2M1R2 = (R1R2)3 = (M1R1)

6 = (M1R2)

2 = (M1M2)

2 = (M1M3)

2 = M1C

2M1C

−2 =

(R2C2)2 = E

X = (R2R1M1)2 Y = (M1R2R1)

2 Z = C

2

M1 = [y, x, z] M2 = [x, y, −z] M2 = [x, y, 1 − z]

R1 = [x, x − y, ½ − z] R2 = [1 − y, 1 − x, ½ − z]

(194) P63/mmc {M1, M2, M3, R, M4 = RM3R} C = RM3

(M1M2)3 = (M1R)

6 = (M2R)

2 = (M3M1)

2 = (M3M2)

2 = E

X = (M2M1R)2 Y = (RM2M1)

2 Z = C

2

M1 = [x, x − y, z] M2 = [1 − y, 1 − x, z] M3 = [x, y, −z]

R = [y, x, ½ − z] M4 = [x, y, 1 − z]

55

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.

56

(195) P23 {Q, R1, R2, R3 = R1R2}

K1 = QR1 L1 = Q−1

R1 K2 = QR2 L2 = Q−1

R2

(R1R2)2 = (QR3)

3 = (L1K2)

2 = (K1K2)

3 = (L1L2)

3 = (K1L2

−1)3 = K1

3L2

−3 = QR3K2

2L1K1 =

K12L1

2K1L1

2K1

2L1 = K2

2L2

2K2L2

2K2

2L2 = E

Z = QR3L1

R1 = [x, 1 − y, −z] R2 = [1 − x, y, −z] R3 = [1 − x, 1 − y, z]

(196) F23 {Q, Q2, R}

(Q−1

Q2)2 = (QQ2)

3 = (QR)

3 = (Q2R)

3 = E

Z = (Q2RQ−1

)2 W = RQQ2Q

Q2 = [−z, x, −y] R = [½ − x, ½ − y, z]

(197) I23 {S1, S2, R = S1S2} Q = S2−1

S1

K = QR = S2−1

S12S2 L = Q

−1R = S1

−1S2

2S1

(S1S2)2 = (S2

−1S1)

3 = (S1

2S2

2)2 = (S1

2S2

−2)3 = S1

3S2

3S1

−3S2

−3 =

K2L

2KL

2K

2L = E

Z = (S1−1

S22)2 W = S1

3Z

S1 = [½ + y, ½ − z, ½ − x] S2 = [½ + z, ½ − x, ½ − y] R = [1 − x, y, −z]

Axes of S1 = 32(λ,λ − 1/3, 2/3 − λ)[1/6,

1/6, −1/6] and

S2 = 32(λ, 2/3 − λ, λ − 1/3)[1/6, −

1/6, 1/6] marked in turquoise.

57

(198) P213 {Q, S1, S2, Q2 = S1Q, Q3 = S2Q, Q4 = QS1}

S1S22 S1

−1S2

2 = S2S1

2 S2

−1S1

2 = QS2

2Q

−1S1

−2 = E

Z = S2

2

S1 = [½ + x, ½ − y, −z] S2 = [½ − x, 1 − y, ½ + z]

Q2 = [½ + z, ½ − x, −y] Q3 = [½ − z, 1 − x, −½ + y] Q4 = [−z, ½ + x, ½ − y]

(199) I213 {Q, R1, R2, S, R3 = QR2Q−1

} S2 = R3R1

K1 = QR1 L1 = Q−1

R1 K2 = QR2 L2 = Q−1

R2

L1K12K2L2

2 = K2

3S

−3 = K1L1R2R1 = (K2L1)

2S

−2 = K2

2L2L1 = (L1K2)

2(K2L2)

2 =

L1K1L12K2

2 = (K1K2

2)2 = S2

2R2S2

−2R2 = K1

2L1

2K1L1

2K1

2L1 = K2

2L2

2K2L2

2K2

2L2 = E

Z = S22 W = K1

−3S2

2

R1 = [½ − x, y, −z] R2 = [1 − x, ½ − y, z] S = [z, 1 − x, ½ − y]

R3 = [x, 1 − y, ½ − z]

(200) Pm 3 {Q, M1, M2, M3 = Q−1

M1Q} Q = (M2Q)3Q

(M1M2)2= (M2M3)

2 = (M1M2M3)

2 = (M1QM1Q

−1)2 = (M2QM2Q

−1)2 = (M2Q)

6 = E

Z = QM1Q−1

M2

M1 = [x, 1 − y, z] M2 = [x, y, −z] M3 = [1 − x, y, z]

The two shaded triangular facets are related through

S = 31(λ, 5/6 − λ, λ − 1/6)[ −1/6,

1/6, −1/6]

S1 and S2 are the the 2-fold screw transformations indicated in

red. The two blue shaded facets are related through the threefold

rotation Q3 = 3(λ, 1 − λ, ½ − λ). The two yellow shaded facets

are similarly related through Q4 = 3(λ, λ + ½, −λ). (Axes of Q3

and Q4 have been omitted in the figure.)

58

(201) Pn 3 { Q 1, Q 2, R = Q 2−1

Q 1} N = R Q 13

Q = Q 1−2

Q2 = Q 2−2

K = QR L = Q−1

R K2 = Q2R L2 = Q2−1

R

( Q 1 RQ 1−1

R)2 = RQ 1

−1Q 2RQ 2

−1Q 1

= ( Q 1

3RQ 1R)

3 =

K2L

2KL

2K

2L = K2

2L2

2K2L2

2K2

2L2 = E

Z = ( Q 12RQ 1

−1R)

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)3Q

(RQ)3 = (RM)

2 = (MQ)

6 = (MQMQ

−1)2 = E

Z = (Q−1

RQM)2 W = R(QM)

2Q

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 2−1

)2 = (Q 1

2Q 2

−2)2 = ( Q 1

2Q 2

2)3 = E

Z = Q 13Q 2

3 W = D

2

Q 1 = [¼ − z, ¼ − x, ¼ − y] Q 2 = [¼ + z, ¼ − x, −¼ + y]

59

(204) Im 3 { Q , M, M2 = Q M Q−1

}

I = Q3 Q = Q

4

(QM)6 = (MQMQ

−1)2 = (IMIQMQ

−1)2 = E

Z = (M Q3)2 W = QZQ

−1(MQ)

3

Q = [½ − z, ½ − x, ½ − y] M = [x, y, −z] M2 = [1 + x, y, −z]

(205) Pa 3 { Q 1, Q 2, Q = ( Q 1 Q 2)−1

}

A = ( Q 22Q 1)

−1 Q = Q 2

3 ( Q 1 Q 2

−1)2Q

Q 16 = Q 2

6 = (Q 1 Q 2)

3 = (Q 1

2Q 2)

2 = ( Q 1

2Q 2

−2)2 = ( Q 1 Q 2

−1)2Q 1

3Q 2

3 =

Q12(Q2

−1Q1Q2

−1)(Q1

−1Q2Q1

−1)Q2

2(Q1

−1Q2Q1

−1)(Q2

−1Q1Q2

−1) = E

Z = ( Q 2−1

Q 1)2

Q 1 = [½ + z, x, ½ − y] Q 2 = [z, ½ − x, −½ + y]

(206) Ia 3 { Q , R, I2 = ( Q2R)

2Q

−1R } A = Q

2RQ

Q = Q−2

I = Q3 K = Q R L = Q

−1R

K6 = L

6 = (KL)

2K

3L

3 = QI2Q

−1I2 = Q RILKL = Q(RI)

2Q

−1(RI2)

2 = K

2L

2KL

2K

2L = E

Z = (KL)−2

W = I2I

R = [x, ½ − y, −z] I2 = [½ − x, ½ − y, ½ − z]

The asymmetric unit has six facets, related in pairs through Q,

Q1 = 3 (½ − λ, ½ + λ, λ; ½, ½, 0) and Q2 = 3 (−λ, ½ − λ, λ; 0, ½, 0)

60

(207) P432 {F, R1, R2} Q = F−1

R2

K = QR1 L = Q−1

R1

(R1R2)4 = (R2F)

3 = (QR2)

4 = (F

2R1)

2 = (F

2R2)

4 = (QFR1)

2 = (FQR1)

4 = (FR1F

−1R1)

2 =

(FR1FR2R1R2)2 = K

2L

2KL

2K

2L = (KL

2KL

−3)2 = E

Y = F2R2R1R2

F = [1 + y , x, z] R1 = [1 − x, y, −z] R2 = [1 − x, z, y]

(208) P4232 {R1, R2, R3, R4} Q = R2R1 Q2 = R4R3

R5 = R1R3 R6 = R2R4

(R1R2)3 = (R3R4)

3 = (R1R3)

2 = (R2R4)

2 = R1R3R2R4 = E

X = R2(R1R4)3R1

R1 = [½ − z, ½ − y, ½ − x] R2 = [½ − x, ½ − z, ½ − y]

R3 = [½ + z, ½ − y, −½ + x] R4 = [½ − x, ½ + z, −½ + y]

(209) F432 {F, R1, R2} Q = R1F−1

(R1R2)2 = (FR1)

3 = (FR2)

3 = (R1F

2)4 = (R2F

2)4 = (R1R2F)

4 = E

Y = (R1R2F2)2 W = R2F

2R1F

2

F = [x, −z, y] R1 = [y, x, −z] R2 = [½ − y, ½ − x, −z]

61

(210) F4132 {Q, R1, R2, R3 = Q−1

R2} S = R3R1

(QR1)3 = (QR2)

2 = (QR2R1Q(R2R1)

3)2 = ((R2R1)

3Q)

2 = E

Z = S4 W = R2R1R2QR1Q

−1

R1 = [x, −y, −z] R2 = [¼ − z, ¼ − y, ¼ − x] R3 = [¼ − y, ¼ − x, ¼ − z]

(211) I432 {R1, R2, R3} Q = R2R1 F = QR3Q

(R2R1)3 = (QR3)

4 = (FR1F

−1R1)

2 = (Q(FR1)

2)2 = (R1FQF)

4 = E

Z = (R1R2R1R3)2 W = Q

−1R1R3F

2

R1 = [½ − z, ½ − y, ½ − x] R2 = [½ − x, ½ − z, ½ − y] R3 = [y, x, −z]

(212) P4332 {R1, R2, R3, S, Q2 = R3R1} Q = R2R1

G = QR3 J = Q−1

R3

(R1R2)3 = (R1R3)

3 = S

4Q2S

−2Q

−1 = S

4(Q2

−1Q)

2 = S

2G

−2 = S

4Q

−1 (R2R3)

2Q =

(QS4Q

−1)S

2(QS

4Q

−1)S

−2 = (Q

−1S

4Q)S

2(Q

−1S

4Q)S

−2 = (G

2J

3)2 = E

Z = S4

R1 = [¼ − z, ¼ − y, ¼ − x] R2 = [¼ − x, ¼ − z, ¼ − y]

R3 = [¾ − x, ¼ + z, −¼ + y] S = [−¼ + y, ¾ − x, ¼ + z]

Q2 = [½ + z, ½ − x, −y]

The two shaded triangular facets are related

through the fourfold screw transformation

S = 41−(¼, ½, z)[0, 0, ¼], indicated in blue

62

(213) P4132 {R1, R2, R3, S, Q2 = R1R3} Q = R1R2

G = QR3 J = Q−1

R3

(R1R2)3 = (R1R3)

3 = S

4Q2

−1S

−2Q = S

4(QQ2

−1)2 = S

2J

−2 = S

4Q(R2R3)

2Q

−1 =

(QS4Q

−1)S

2(QS

4Q

−1)S

−2 = (Q

−1S

4Q)S

2(Q

−1S

4Q)S

−2 = (G

2J

3)2 = E

Z = S4

R1 = [¼ − x, ¼ − z, ¼ − y] R2 = [¼ − z, ¼ − y, ¼ − x]

R3 = [¼ + z, ¾ − y, −¼ + x] S = [¾ − y, −¼ + x, ¼ + z]

Q2 = [−z, ½ − x, −½ + y]

(214) I4132 {R1, R2, R3, R4, R5 = R2R3R2, Q = R1R5, S = R2R3}

(R1R2)2 = (R1R3)

3 = (R1R4)

3(R4R3)

3 = S

4Q

−1(R4R2)

4Q = (S

3R1S

2R1)

2 = E

Z = (R2R4)4 W = S

4Q(R1R4)

3Q

−1

R1 = [¼ − x, ¼ − z, ¼ − y] R2 = [x, −y, ½ − z] R3 = [−¼ + z, −¼ − y, ¼ + x]

R4 = [¼ + y, −¼ + x, ¼ − z] R5 = [¼ − z, ¼ − y, ¼ − x] S = [−¼ + z, ¼ + y, ¼ − x]

Shaded facets related through

S = 41(0, y, ½)[0, ¼, 0]

The two shaded triangular facets are related

through the fourfold screw transformation

S = 41(½, ¼, z)[0, 0, ¼], indicated in blue

63

(215) P 4 3m {M1, M2, R, M3 = RM1R} F = M1R Q = M1M2

K = QR L = Q−1

R

(M1M2)3 = (M3M2)

3 = (M1R)

4 = (M2R)

4 = (M3R)

4 = K

2L

2KL

2K

2L = E

Y = F

2M2RM2

M1 = [y, x, z] M2 = [x, z, y] R = [1 − x, y, −z] M3 = [1 − y, 1 − x, z]

F = [y, 1 − x, −z]

(216) F 4 3m {M1, M2, M3, M4} F = M1M2M3 Q = M3M1

(M1M2)2 =(M2M3)

3 = (M3M1)

3 = (M1M4)

3 = (M2M4)

3 = (M3M4)

2 = (M1M2M3)

4 =

(M2M3M4)4 = (M3M1M4)

4 = (M1M2M4)

4 = E

X = M4M1M2M4M3M1M2M3 W = M4M1M2M3M2M1

M1 = [x, z, y] M2 = [x, −z, −y] M3 = [y, x, z] M4 = [½ − y, ½ − x, z]

(217) I 4 3m { F , M, M2 = F−1

M F } Q = MM2

G = F M J = F−1

M K = Q F2 L = Q

−1F

2

(MM2)3 = ( F

2M)

4 = QG

3Q

−1J

3 = J

3G

3J

−3G

−3 = M( F Q)

2M( F Q)

−2 = K

2L

2KL

2K

2L = E

Z = ( F Q)−2

W = ZJ3

F = [½ − z, ½ − y, −½ + x] M = [y, x, z] M2 = [x, z, y]

64

Shaded triangular facets related through

S = 31(λ, 5/6 − λ, λ − 1/6)[−1/6,

1/6, −1/6]???

(218) P 4 3n {Q, F 1, F 2 } N = F 1−1

F 22

K1 = Q F 12 L1 = Q

−1F 1

2 K2 = Q F 2

2 L2 = Q

−1F 1

2

( F 12F 2

2)2 = (Q F 1

2F 2

2)3 = (Q F 1)

4 = (Q

−1F 2)

4 = ( F 1Q

−1)2( F 2Q)

−2 = (L1K2)

2 = (K1K2)

3

= (L1L2)3 = (K1L2

−1)3 = K1

3L2

−3 = K1

2L1

2K1L1

2K1

2L1 = K2

2L2

2K2L2

2K2

2L2 = E

Z = (Q−1

F 1)2

F 1 = [½ + z, ½ − y, ½ − x] F 2 = [½ + y, ½ − x, ½ − z]

(219) F 4 3c { F 1, F 2} Q = F 2 F 1−1

C = F 2 F 1−1

F 22

F 1

( F 2 F 1−1

)3 = ( F 1 F 2)

3 = ( F 1

2F 2

−1)4 = ( F 2

2F 1

−1)4 = E

Z = C2 W = (F2

−1F1

2)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 22Q

−1} D = R F 2

K1 = Q F 12 L1 = Q

−1F 1

2 K2 = Q F 2

2 L2 = Q

−1F 2

2

( F 1 F 2−1

)3 = (Q F 1)

4 = (Q

−1F 2)

4 = ( F 1S)

4 = ( F 2S)

4 = K1

3L2

−3 = K2

3Q

−1L1

−3Q =

K12L1

2K1L1

2K1

2L1 = K2

2L2

2K2L2

2K2

2L2 = E

Z = (L1K1)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

S = 31(λ, 5/6 − λ, λ − 1/6)[−1/6,

1/6, −1/6]

65

(221) Pm 3 m {M1, M2 M3, M4} Q = M3M1 Q = M2(M3M2M1)2

(M2M3)4 = (M3M1)

3 = (M1M2)

2 = (M1M4)

4 = (M2M4)

2 = (M3M4)

2 = (M1M2M3)

6 =

(M2M3M4)4 = (M3M1M4)

6 = (M1M2M4)

4 = E

Z = QM4 Q−1

M2

M1 = [y, x, z] M2 = [x, y, −z] M3 = [x, z, y] M4 = [1 − x, y, z]

(222) Pn 3 n { Q , F } N1 = Q2

F Q3

F Q−2

N2 = Q2

F Q2( F

2Q )

2

( Q F )2 = ( Q F

2)6 = (Q F

−1)2 = ( Q F Q F

−1)2 = ( Q F Q

−1F )

3 = ( Q

3F

2)4 = E

Z = ( Q2

F )2

F = [½ + z, ½ − y, ½ − x] Q = [½ − z, ½ − x, ½ − y]

(223) Pm 3 n { R1, R2, M, M1, M2 = R1M1R1} Q = R2R1

Q = (MQ)3Q N = R1(MQ)

3

(R1R2)3

= M1M2R1R2 = (MR1)4 = (MR2)

4 = (MM1)

4 = (MM2)

4 = (M1R1)

6 = (M1R2)

2 =

(MQ)6 = E

Z = (R1QM)2

R1 = [½ − z, ½ − y, ½ − x] R2 = [½ − x, ½ − z, ½ − y]

M = [x, y, −z] M1 = [x, z, y] M2 = [z, y, x]

66

(224) Pn 3 m {R1, R2, M1, M2 = R2M1R2, M3 = R1M2R1}

Q = M1M2 Q = R2M1 N = QR2M1

(R1R2)2 = (R1M1)

4 = (R1M2)

2 = (R1M3)

2 = (R2M1)

6 = (R2M2)

6 = (M1M2)

3 = (R1R2M1)

4 =

(R1R2M2)4 = E

Z = Q−1

R1R2M1M3

R1 = [1 − x, y, −z] R2 = [½ − z, ½ − y, ½ − x]

M1 = [y, x, z] M2 = [x, z, y] M3 = [x, −z, −y]

Q = [½ − z, ½ − x, ½ − y]

(225) Fm 3 m {M1, M2, M3, M4} Q = M3M2 Q = M1(M2M3M1)2

(M3M1)2 = (M1M2)

4 = (M2M3)

3 = (M1M4)

2 = (M2M4)

3 = (M3M4)

2 = (M1M2M3)

6 =

(M2M3M4)4 = (M3M1M4)

2 = (M1M2M4)

6 = E

Z = (M2M4M3M2M1)2 W = M4(M1M2)

2M3(M1M2)

2

M1 = [x, y, −z] M2 = [x, z, y] M3 = [y, x, z] M4 = [½ − y, ½ − x, z]

(226) Fm 3 c {F, M, R} Q = (MRF)2M C = Q

2RQ

3

(RF)3 = (RF

2)4 = FMF

−1M = (MR)

4 = (FRM)

6 = ((RF)

2M)

6 = E

Z = C2 W = (FRMR)

2

F = [½ − y, x, z] M = [x, y, −z] R = [½ − x, z, y]

67

(227) Fd 3 m {M1, M2, R, M3 = RM1R, M4 = RM2R}

Q = RM1 D = M2R(M1R)2

(M1M2)2 = (M1R)

6 = (M2R)

6 = (M1M4)

3 = (M1M3M4)

4 = E

Y = (M1M2R)4 W = D

2

M1 = [y, x, z] M2 = [½ − y, ½ − x, z] R = [¼ − z, ¼ − y, ¼ − x]

M3 = [x, z, y] M4 = [x, −z, −y] Q = [¼ − x, ¼ − y, ¼ − z]

(228) Fd 3 c { F , R, F 2 = R F R} Q = R F = [¼ − z, ¼ − x, ¼ − y]

D = F RQ−2

C = Q2

F F 22Q

2

( F R)6 = ( F F 2)

3 = ( F F 2

−1)3 = ( F F 2

2F

−1F 2

2)2 = E

Z = C2 W = D

2

F = [½ − y, x, −z] R = [¼ + z, ¼ − y, −¼ + x] F 2 = [½ − x, −z, y]

(229) Im 3 m {M1, M2, R, M3 = RM1R} Q = RM1

(RM1)6

= (RM2)4 = (M1M2)

2 = (M2M3)

4 = (M1M2M3)

6 = E

Z = (QRM2)2 W = (RM1)

3(M1M2M3)

3

M1 = [y, x, z] M2 = [x, y, −z] R = [½ − z, ½ − y, ½ − x] M3 = [x, z, y]

Q = [½ − x, ½ − y, ½ − z]

68

(230) Ia 3 d {Q , F , R, R2 = RQ−1

, R3 = Q F2Q

−1R} A = F

2I D = RI

Q = Q−2

I = Q3

(RQ)2 = (Q

−1F

2QR)

2 = ( F R)

6 = ( Q RQ

−1R)

2 = (( F

2I F

2Q )

2Q )

2 = ( F

2R F R F

−1R)

2 = E

X = A2 W = D

2

F = [¼ + y, ¼ − x, −¼ − z] R = [¼ − x, ¼ − z, ¼ −y]

R2 = [¼ − z, ¼ − y, ¼ − x] R3 = [¼ − x, ¼ + z, −¼ + y]

69

Appendix

Matrix methods

The realization of each abstract group as a group of Euclidean transformations in three-

dimensional Euclidean space corresponds to a representation of that abstract group in

terms of 4×4 matrices. For example (see group 195), R3 = [1 − x, 1 − y, z] denotes the

effect of multiplying x

y

z

1

x

y

z

1

on the left by matrix

1 1

1 1

1 0

0 0 0 1

−

−

, which can for convenience be abridged to the 3×4 matrix 1 1

1 1

1 0

−

−

because

the final row is always 0 0 0 1. If the final column of such a 3×4 matrix (which corresponds

to the shift of the origin brought about by the transformation) is null, this also can be

omitted. Thus, for example, (QR3)3 = E is represented by

QR3 = 1 1 1 1 0

1 1 1 1 1

1 1 0 1 1

−

− −

−

=

;

21 0 1 1

1 1 1 0

1 1 1 1

−

− −

− −

=

;

31 0 1 0 1 1 1 0

1 1 1 1 1 0 1 0

1 1 1 1 1 1 1 0

E−

− − −

− − −

= = =

This matrix method has been used extensively in the process of finding and checking the

generating relations.

An interesting exercise in the application of this method is to check the relations satisfied

by a twofold rotation R and a threefold rotation Q. For the cubic space groups there are

just six possible ways in which the two axes can be related, as indicated below. K = QR

and L = Q−1

R.

K3 = E K

4 = E

(KL)3 = E K

2 = E

70

K2L

2KL

2K

2L = E (K

2L

3)2 = E