Hall symbols

Hall, S. R.; Grosse-Kunstleve, R. W.

doi:10.1107/97809553602060000552

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 112-114

Section A1.4.2.3. Hall symbols

S. R. Hall^b ^* and R. W. Grosse-Kunstleve^c

A1.4.2.3. Hall symbols

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The explicit-origin space-group notation proposed by Hall (1981a) is based on a subset of the symmetry operations, in the form of Seitz matrices, sufficient to uniquely define a space group. The concise unambiguous nature of this notation makes it well suited to handling symmetry in computing and database applications.

Table A1.4.2.7 lists space-group notation in several formats. The first column of Table A1.4.2.7 lists the space-group numbers with axis codes appended to identify the non-standard settings. The second column lists the Hermann–Mauguin symbols in computer-entry format with appended codes to identify the origin and cell choice when there are alternatives. The general forms of the Hall notation are listed in the fourth column and the computer-entry representations of these symbols are listed in the third column. The computer-entry format is the general notation expressed as case-insensitive ASCII characters with the overline (bar) symbol replaced by a minus sign.

The Hall notation has the general form: $[{\bf L}[{\bf N}_{\bf T}^{\bf A}]_{\bf 1}\ldots{}[{\bf N}_{\bf T}^{\bf A}]_{\bf p}{\bf V}.\eqno({\rm A}1.4.2.4)]$ L is the symbol specifying the lattice translational symmetry (see Table A1.4.2.2). The integral translations are implicitly included in the set of generators. If L has a leading minus sign, it also specifies an inversion centre at the origin. $[[{\bf N}_{\bf T}^{\bf A}]_{\bf n}]$ specifies the 4 × 4 Seitz matrix S _n of a symmetry element in the minimum set which defines the space-group symmetry (see Tables A1.4.2.3 to A1.4.2.6 ), and p is the number of elements in the set. V is a change-of-basis operator needed for less common descriptions of the space-group symmetry.

Table A1.4.2.3| top | pdf |
Translation symbol T

The symbol T specifies the translation elements of a Seitz matrix. Alphabetical symbols (given in the first column) specify translations along a fixed direction. Numerical symbols (given in the third column) specify translations as a fraction of the rotation order |N| and in the direction of the implied or explicitly defined axis.

Translation symbol	Translation vector	Subscript symbol	Fractional translation
a	$[{1 \over 2},0,0]$	1 in 3₁	$[{1 \over 3}]$
b	$[0,{1 \over 2},0]$	2 in 3₂	$[{2 \over 3}]$
c	$[0,0,{1 \over 2}]$	1 in 4₁	$[{1 \over 4}]$
n	$[{1 \over 2}, {1 \over 2}, {1\over 2}]$	3 in 4₃	$[{3 \over 4}]$
u	$[{1 \over 4},0,0]$	1 in 6₁	$[{1 \over 6}]$
v	$[0,{1 \over 4},0]$	2 in 6₂	$[{1 \over 3}]$
w	$[0,0,{1 \over 4}]$	4 in 6₄	$[{2 \over 3}]$
d	$[{1\over 4},{1\over 4},{1\over 4}]$	5 in 6₅	$[{5 \over 6}]$

Table A1.4.2.4| top | pdf |
Rotation matrices for principal axes

The 3 × 3 matrices for proper rotations along the three principal unit-cell directions are given below. The matrices for improper rotations (−1, −2, −3, −4 and −6) are identical except that the signs of the elements are reversed.

Axis	Symbol A	Rotation order
Axis	Symbol A	1	2	3	4	6
a	x	$[\pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}]$	$[\pmatrix{1&0&0\cr0&\bar{1}&0\cr0&0&\bar{1}\cr}]$	$[\pmatrix{1&0&0\cr0&0&\bar{1}\cr0&1&\bar{1}\cr}]$	$[\pmatrix{1&0&0\cr0&0&\bar{1}\cr0&1&0\cr}]$	$[\pmatrix{1&0&0\cr0&1&\bar{1}\cr0&1&0\cr}]$
b	y	$[\pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}]$	$[\pmatrix{\bar{1}&0&0\cr0&1&0\cr0&0&\bar{1}\cr}]$	$[\pmatrix{\bar{1}&0&1\cr0&1&0\cr\bar{1}&0&0\cr}]$	$[\pmatrix{0&0&1\cr0&1&0\cr\bar{1}&0&0\cr}]$	$[\pmatrix{0&0&1\cr0&1&0\cr\bar{1}&0&1\cr}]$
c	z	$[\pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}]$	$[\pmatrix{\bar{1}&0&0\cr0&\bar{1}&0\cr0&0&1\cr}]$	$[\pmatrix{0&\bar{1}&0\cr1&\bar{1}&0\cr0&0&1\cr}]$	$[\pmatrix{0&\bar{1}&0\cr1&0&0\cr0&0&1\cr}]$	$[\pmatrix{1&\bar{1}&0\cr1&0&0\cr0&0&1\cr}]$

Table A1.4.2.5| top | pdf |
Rotation matrices for face-diagonal axes

The symbols for face-diagonal twofold rotations are 2′ and 2′′. The face-diagonal axis direction is determined by the axis of the preceding rotation N^x, N^y or N^z. Note that the single prime ′ is the default and may be omitted.

Preceding rotation	Rotation	Axis	Matrix
N^x	2′	b − c	$[\pmatrix{\bar{1}&0&0\cr0&0&\bar{1}\cr0&\bar{1}&0\cr}]$
N^x	2′′	b + c	$[\pmatrix{\bar{1}&0&0\cr0&0&1\cr0&1&0\cr}]$
N^y	2′	a − c	$[\pmatrix{0&0&\bar{1}\cr0&\bar{1}&0\cr\bar{1}&0&0\cr}]$
N^y	2′′	a + c	$[\pmatrix{0&0&1\cr0&\bar{1}&0\cr1&0&0\cr}]$
N^z	2′	a − b	$[\pmatrix{0&\bar{1}&0\cr\bar{1}&0&0\cr0&0&\bar{1}\cr}]$
N^z	2′′	a + b	$[\pmatrix{0&1&0\cr1&0&0\cr0&0&\bar{1}\cr}]$

Table A1.4.2.6| top | pdf |
Rotation matrix for the body-diagonal axis

The symbol for the threefold rotation in the a + b + c direction is 3*. Note that for cubic space groups the body-diagonal axis is implied and the asterisk * may be omitted.

Axis	Rotation	Matrix
a + b + c	3*	$[\pmatrix{0&0&1\cr1&0&0\cr0&1&0\cr}]$

The matrix symbol $[{\bf N}_{\bf T}^{\bf A}]$ is composed of three parts: N is the symbol denoting the |N|-fold order of the rotation matrix (see Tables A1.4.2.4, A1.4.2.5 and A1.4.2.6), T is a subscript symbol denoting the translation vector (see Table A1.4.2.3) and A is a superscript symbol denoting the axis of rotation.

The computer-entry format of the Hall notation contains the rotation-order symbol N as positive integers 1, 2, 3, 4, or 6 for proper rotations and as negative integers −1, −2, −3, −4 or −6 for improper rotations . The T translation symbols 1, 2, 3, 4, 5, 6, a, b, c, n, u, v, w, d are described in Table A1.4.2.3. These translations apply additively [e.g. ad signifies a ( $[{3 \over 4}, {1\over 4}, {1 \over 4}]$ ) translation]. The A axis symbols x, y, z denote rotations about the axes a, b and c, respectively (see Table A1.4.2.4). The axis symbols ′′ and ′ signal rotations about the body-diagonal vectors a + b (or alternatively b + c or c + a) and a − b (or alternatively b − c or c − a) (see Table A1.4.2.5). The axis symbol * always refers to a threefold rotation along a + b + c (see Table A1.4.2.6).

The change-of-basis operator V has the general form (v _x, v _y, v _z). The vectors v _x, v _y and v _z are specified by $[\eqalign{{v}_x &={r}_{1, \, 1}X+{r}_{1, \, 2}Y+{r}_{1, \, 3}Z+{\bf t}_1\cr {v}_y &={r}_{2, \, 1}X+{r}_{2, \, 2}Y+{r}_{2, \, 3}Z+{\bf t}_2\cr{v}_z &={r}_{3, \, 1}X+{r}_{3, \, 2}Y+{r}_{3, \, 3}Z+{\bf t}_3\cr},]$ where $[{r}_{i, \, j}]$ and $[{\bf t}_i]$ are fractions or real numbers. Terms in which $[{r}_{i, \, j}]$ or $[{\bf t}_i]$ are zero need not be specified. The 4 × 4 change-of-basis matrix operator V is defined as $[{\bf V} = \pmatrix{{r}_{1, \, 1}&{r}_{1, \, 2}&{r}_{1, \, 3}&{\bf t}_1\cr {r}_{2, \, 1}&{r}_{2, \, 2}&{r}_{2, \, 3}&{\bf t}_2\cr {r}_{3, \, 1}&{r}_{3, \, 2}& {r}_{3, \, 3}& {\bf t}_3\cr0&0&0&1\cr}.]$ The transformed symmetry operations are derived from the specified Seitz matrices S _n as $[{\bf S}_{\bf n}^{\prime}={\bf V}\cdot{\bf S}_{\bf n}\cdot{\bf V}^{-1}]$ and from the integral translations t(1, 0, 0), t(0, 1, 0) and t(0, 0, 1) as $[({\bf t}_{\bf n}^{\prime}, {\bf 1})^{T}={\bf V}\cdot({\bf t}_{\bf n}, {\bf 1})^{T}.]$

A shorthand form of V may be used when the change-of-basis operator only translates the origin of the basis system. In this form v _x, v _y and v _z are specified simply as shifts in twelfths, implying the matrix operator $[{\bf V}=\pmatrix{1&0&0&{{{v}_x}/12}\cr 0&1&0&{{{v}_y}/12}\cr 0 & 0 & 1 & {{{v}_z}/12}\cr 0 & 0 & 0 & 1\cr}.]$ In the shorthand form of V, the commas separating the vectors may be omitted.

A1.4.2.3.1. Default axes

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For most symbols the rotation axes applicable to each N are implied and an explicit axis symbol A is not needed. The rules for default axis directions are:

(i) the first rotation or roto-inversion has an axis direction of c;
(ii) the second rotation (if |N| is 2) has an axis direction of a if preceded by an |N| of 2 or 4, a−b if preceded by an |N| of 3 or 6;
(iii) the third rotation (if |N| is 3) has an axis direction of a + b + c.

A1.4.2.3.2. Example matrices

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The following examples show how the notation expands to Seitz matrices.

The notation $[\bar{\it 2}_c^{x}]$ represents an improper twofold rotation along a and a c/2 translation: $[-2{\rm xc}=\pmatrix{-1&0&0&0\cr0&1&0&0\cr0&0&1&{1 \over 2}\cr0&0&0&1}.]$

The notation $[{\it 3}^*]$ represents a threefold rotation along a + b + c: $[3^*=\pmatrix{0&0&1&0\cr1&0&0&0\cr0&1&0&0\cr0&0&0&1}.]$

The notation $[{\it 4}_{vw}]$ represents a fourfold rotation along c (implied) and translation of b/4 and c/4: $[4{\rm vw}=\pmatrix{0&-1&0&0\cr1&0&0&{1 \over 4}\cr0&0&1&{1 \over 4}\cr0&0&0&1}.]$

The notation 6₁ 2 (0 0 −1) represents a 6₁ screw along c, a twofold rotation along a − b and an origin shift of −c/12. Note that the 6₁ matrix is unchanged by the shifted origin whereas the 2 matrix is changed by −c/6. $[\eqalign{&61\;2\;(0\;0\;{-1})\cr&\quad{}=\pmatrix{1&-1&0&0\cr1&0&0&0\cr0&0&1&{1 \over 6}\cr 0&0&0&1\cr},\pmatrix{0&-1&0&0\cr-1&0&0&0\cr 0&0&-1&#38>;{5 \over 6}\cr0&0&0&1\cr}.\cr}]$ The change-of-basis vector (0 0 −1) could also be entered as (x, y, z − 1/12).

The reverse setting of the R-centred lattice (hexagonal axes) is specified using a change-of-basis transformation applied to the standard obverse setting (see Table A1.4.2.2). The obverse Seitz matrices are $[\openup3pt{\rm R}\;3=\pmatrix{1&0&0&{1 \over 3}\cr0&1&0&{2 \over 3}\cr0&0&1&{2 \over 3}\cr0&0&0&1\cr},\pmatrix{1&0&0&{2 \over 3}\cr0&1&0&{1 \over 3}\cr0&0&1&{1 \over 3}\cr0&0&0&1\cr},\pmatrix{0&-1&0&0\cr1&-1&0&0\cr0&0&1&0\cr0&0&0&1}.]$ The reverse-setting Seitz matrices are $[\displaylines{{\rm R}\;3\;({\rm -x,-y,z})\hfill\cr\quad= \pmatrix{1&0&0&{1 \over 3}\cr0&1&0&{2 \over 3}\cr0&0&1&{1 \over 3}\cr0&0&0&1\cr},\pmatrix{1&0&0&{2 \over 3}\cr0&1&0&{1 \over 3}\cr0&0&1&{2 \over 3}\cr0&0&0&1\cr},\pmatrix{0&-1&0&0\cr1&-1&0&0\cr0&0&1&0\cr0&0&0&1}.\cr}]$

Table A1.4.2.7| top | pdf |
Hall symbols

The first column, n:c, lists the space-group numbers and axis codes^† separated by a colon. The second column lists the Hermann–Mauguin symbols in computer-entry format. The third column lists the Hall symbols in computer-entry format and the fourth column lists the Hall symbols as described in Tables A1.4.2.2–A1.4.2.6 .

n:c	H–M entry	Hall entry	Hall symbol
1	P 1	p 1	P 1
2	P -1	-p 1	$[\overline{\rm P}]$ 1
3:b	P 1 2 1	p 2y	P $[2^{\rm y}]$
3:c	P 1 1 2	p 2	P 2
3:a	P 2 1 1	p 2x	P $[2^{\rm x}]$
4:b	P 1 21 1	p 2yb	P $[2_{\rm b}^{\rm y}]$
4:c	P 1 1 21	p 2c	P $[2_{\rm c}]$
4:a	P 21 1 1	p 2xa	P $[2_{\rm a}^{\rm x}]$
5:b1	C 1 2 1	c 2y	C $[2^{\rm y}]$
5:b2	A 1 2 1	a 2y	A $[2^{\rm y}]$
5:b3	I 1 2 1	i 2y	I $[2^{\rm y}]$
5:c1	A 1 1 2	a 2	A 2
5:c2	B 1 1 2	b 2	B 2
5:c3	I 1 1 2	i 2	I 2
5:a1	B 2 1 1	b 2x	B $[2^{\rm x}]$
5:a2	C 2 1 1	c 2x	C $[2^{\rm x}]$
5:a3	I 2 1 1	i 2x	I $[2^{\rm x}]$
6:b	P 1 m 1	p -2y	P $[\overline{2}^{\rm y}]$
6:c	P 1 1 m	p -2	P $[\overline{2}]$
6:a	P m 1 1	p -2x	P $[\overline{2}^{\rm x}]$
7:b1	P 1 c 1	p -2yc	P $[\overline{2}_{\rm c}^{\rm y}]$
7:b2	P 1 n 1	p -2yac	P $[\overline{2}^{\rm y}_{\rm ac}]$
7:b3	P 1 a 1	p -2ya	P $[\overline{2}_{\rm a}^{\rm y}]$
7:c1	P 1 1 a	p -2a	P $[\overline{2}_{\rm a}]$
7:c2	P 1 1 n	p -2ab	P $[\overline{2}_{\rm ab}]$
7:c3	P 1 1 b	p -2b	P $[\overline{2}_{\rm b}]$
7:a1	P b 1 1	p -2xb	P $[\overline{2}_{\rm b}^{\rm x}]$
7:a2	P n 1 1	p -2xbc	P $[\overline{2}^{\rm x}_{\rm bc}]$
7:a3	P c 1 1	p -2xc	P $[\overline{2}_{\rm c}^{\rm x}]$
8:b1	C 1 m 1	c -2y	C $[\overline{2}^{\rm y}]$
8:b2	A 1 m 1	a -2y	A $[\overline{2}^{\rm y}]$
8:b3	I 1 m 1	i -2y	I $[\overline{2}^{\rm y}]$
8:c1	A 1 1 m	a -2	A $[\overline{2}]$
8:c2	B 1 1 m	b -2	B $[\overline{2}]$
8:c3	I 1 1 m	i -2	I $[\overline{2}]$
8:a1	B m 1 1	b -2x	B $[\overline{2}^{\rm x}]$
8:a2	C m 1 1	c -2x	C $[\overline{2}^{\rm x}]$
8:a3	I m 1 1	i -2x	I $[\overline{2}^{\rm x}]$
9:b1	C 1 c 1	c -2yc	C $[\overline{2}_{\rm c}^{\rm y}]$
9:b2	A 1 n 1	a -2yab	A $[\overline{2}^{\rm y}_{\rm ab}]$
9:b3	I 1 a 1	i -2ya	I $[\overline{2}_{\rm a}^{\rm y}]$
9:-b1	A 1 a 1	a -2ya	A $[\overline{2}_{\rm a}^{\rm y}]$
9:-b2	C 1 n 1	c -2yac	C $[\overline{2}^{\rm y}_{\rm ac}]$
9:-b3	I 1 c 1	i -2yc	I $[\overline{2}_{\rm c}^{\rm y}]$
9:c1	A 1 1 a	a -2a	A $[\overline{2}_{\rm a}]$
9:c2	B 1 1 n	b -2ab	B $[\overline{2}_{\rm ab}]$
9:c3	I 1 1 b	i -2b	I $[\overline{2}_{\rm b}]$
9:-c1	B 1 1 b	b -2b	B $[\overline{2}_{\rm b}]$
9:-c2	A 1 1 n	a -2ab	A $[\overline{2}_{\rm ab}]$
9:-c3	I 1 1 a	i -2a	I $[\overline{2}_{\rm a}]$
9:a1	B b 1 1	b -2xb	B $[\overline{2}_{\rm b}^{\rm x}]$
9:a2	C n 1 1	c -2xac	C $[\overline{2}^{\rm x}_{\rm ac}]$
9:a3	I c 1 1	i -2xc	I $[\overline{2}_{\rm c}^{\rm x}]$
9:-a1	C c 1 1	c -2xc	C $[\overline{2}_{\rm c}^{\rm x}]$
9:-a2	B n 1 1	b -2xab	B $[\overline{2}^{\rm x}_{\rm ab}]$
9:-a3	I b 1 1	i -2xb	I $[\overline{2}_{\rm b}^{\rm x}]$
10:b	P 1 2/m 1	-p 2y	$[\overline{\rm P}]$ $[2^{\rm y}]$
10:c	P 1 1 2/m	-p 2	$[\overline{\rm P}]$ 2
10:a	P 2/m 1 1	-p 2x	$[\overline{\rm P}]$ $[2^{\rm x}]$
11:b	P 1 21/m 1	-p 2yb	$[\overline{\rm P}]$ $[2_{\rm b}^{\rm y}]$
11:c	P 1 1 21/m	-p 2c	$[\overline{\rm P}]$ $[2_{\rm c}]$
11:a	P 21/m 1 1	-p 2xa	$[\overline{\rm P}]$ $[2_{\rm a}^{\rm x}]$
12:b1	C 1 2/m 1	-c 2y	$[\overline{\rm C}]$ $[2^{\rm y}]$
12:b2	A 1 2/m 1	-a 2y	$[\overline{\rm A}]$ $[2^{\rm y}]$
12:b3	I 1 2/m 1	-i 2y	$[\overline{\rm I}]$ $[2^{\rm y}]$
12:c1	A 1 1 2/m	-a 2	$[\overline{\rm A}]$ 2
12:c2	B 1 1 2/m	-b 2	$[\overline{\rm B}]$ 2
12:c3	I 1 1 2/m	-i 2	$[\overline{\rm I}]$ 2
12:a1	B 2/m 1 1	-b 2x	$[\overline{\rm B}]$ $[2^{\rm x}]$
12:a2	C 2/m 1 1	-c 2x	$[\overline{\rm C}]$ $[2^{\rm x}]$
12:a3	I 2/m 1 1	-i 2x	$[\overline{\rm I}]$ $[2^{\rm x}]$
13:b1	P 1 2/c 1	-p 2yc	$[\overline{\rm P}]$ $[2_{\rm c}^{\rm y}]$
13:b2	P 1 2/n 1	-p 2yac	$[\overline{\rm P}]$ $[2^{\rm y}_{\rm ac}]$
13:b3	P 1 2/a 1	-p 2ya	$[\overline{\rm P}]$ $[2^{\rm y}_{\rm a}]$
13:c1	P 1 1 2/a	-p 2a	$[\overline{\rm P}]$ $[2_{\rm a}]$
13:c2	P 1 1 2/n	-p 2ab	$[\overline{\rm P}]$ $[2_{\rm ab}]$
13:c3	P 1 1 2/b	-p 2b	$[\overline{\rm P}]$ $[2_{\rm b}]$
13:a1	P 2/b 1 1	-p 2xb	$[\overline{\rm P}]$ $[2_{\rm b}^{\rm x}]$
13:a2	P 2/n 1 1	-p 2xbc	$[\overline{\rm P}]$ $[2^{\rm x}_{\rm bc}]$
13:a3	P 2/c 1 1	-p 2xc	$[\overline{\rm P}]$ $[2^{\rm x}_{\rm c}]$
14:b1	P 1 21/c 1	-p 2ybc	$[\overline{\rm P}]$ $[2^{\rm y}_{\rm bc}]$
14:b2	P 1 21/n 1	-p 2yn	$[\overline{\rm P}]$ $[2^{\rm y}_{\rm n}]$
14:b3	P 1 21/a 1	-p 2yab	$[\overline{\rm P}]$ $[2^{\rm y}_{\rm ab}]$
14:c1	P 1 1 21/a	-p 2ac	$[\overline{\rm P}]$ $[2_{\rm ac}]$
14:c2	P 1 1 21/n	-p 2n	$[\overline{\rm P}]$ $[2_{\rm n}]$
14:c3	P 1 1 21/b	-p 2bc	$[\overline{\rm P}]$ $[2_{\rm bc}]$
14:a1	P 21/b 1 1	-p 2xab	$[\overline{\rm P}]$ $[2^{\rm x}_{\rm ab}]$
14:a2	P 21/n 1 1	-p 2xn	$[\overline{\rm P}]$ $[2^{\rm x}_{\rm n}]$
14:a3	P 21/c 1 1	-p 2xac	$[\overline{\rm P}]$ $[2^{\rm x}_{\rm ac}]$
15:b1	C 1 2/c 1	-c 2yc	$[\overline{\rm C}]$ $[2^{\rm y}_{\rm c}]$
15:b2	A 1 2/n 1	-a 2yab	$[\overline{\rm A}]$ $[2^{\rm y}_{\rm ab}]$
15:b3	I 1 2/a 1	-i 2ya	$[\overline{\rm I}]$ $[2^{\rm y}_{\rm a}]$
15:-b1	A 1 2/a 1	-a 2ya	$[\overline{\rm A}]$ $[2^{\rm y}_{\rm a}]$
15:-b2	C 1 2/n 1	-c 2yac	$[\overline{\rm C}]$ $[2^{\rm y}_{\rm ac}]$
15:-b3	I 1 2/c 1	-i 2yc	$[\overline{\rm I}]$ $[2^{\rm y}_{\rm c}]$
15:c1	A 1 1 2/a	-a 2a	$[\overline{\rm A}]$ $[2_{\rm a}]$
15:c2	B 1 1 2/n	-b 2ab	$[\overline{\rm B}]$ $[2_{\rm ab}]$
15:c3	I 1 1 2/b	-i 2b	$[\overline{\rm I}]$ $[2_{\rm b}]$
15:-c1	B 1 1 2/b	-b 2b	$[\overline{\rm B}]$ $[2_{\rm b}]$
15:-c2	A 1 1 2/n	-a 2ab	$[\overline{\rm A}]$ $[2_{\rm ab}]$
15:-c3	I 1 1 2/a	-i 2a	$[\overline{\rm I}]$ $[2_{\rm a}]$
15:a1	B 2/b 1 1	-b 2xb	$[\overline{\rm B}]$ $[2^{\rm x}_{\rm b}]$
15:a2	C 2/n 1 1	-c 2xac	$[\overline{\rm C}]$ $[2^{\rm x}_{\rm ac}]$
15:a3	I 2/c 1 1	-i 2xc	$[\overline{\rm I}]$ $[2^{\rm x}_{\rm c}]$
15:-a1	C 2/c 1 1	-c 2xc	$[\overline{\rm C}]$ $[2^{\rm x}_{\rm c}]$
15:-a2	B 2/n 1 1	-b 2xab	$[\overline{\rm B}]$ $[2^{\rm x}_{\rm ab}]$
15:-a3	I 2/b 1 1	-i 2xb	$[\overline{\rm I}]$ $[2^{\rm x}_{\rm b}]$
16	P 2 2 2	p 2 2	P 2 2
17	P 2 2 21	p 2c 2	P $[2_{\rm c}]$ 2
17:cab	P 21 2 2	p 2a 2a	P $[2_{\rm a}]$ $[2_{\rm a}]$
17:bca	P 2 21 2	p 2 2b	P 2 $[2_{\rm b}]$
18	P 21 21 2	p 2 2ab	P 2 $[2_{\rm ab}]$
18:cab	P 2 21 21	p 2bc 2	P $[2_{\rm bc}]$ 2
18:bca	P 21 2 21	p 2ac 2ac	P $[2_{\rm ac}]$ $[2_{\rm ac}]$
19	P 21 21 21	p 2ac 2ab	P $[2_{\rm ac}]$ $[2_{\rm ab}]$
20	C 2 2 21	c 2c 2	C $[2_{\rm c}]$ 2
20:cab	A 21 2 2	a 2a 2a	A $[2_{\rm a}]$ $[2_{\rm a}]$
20:bca	B 2 21 2	b 2 2b	B 2 $[2_{\rm b}]$
21	C 2 2 2	c 2 2	C 2 2
21:cab	A 2 2 2	a 2 2	A 2 2
21:bca	B 2 2 2	b 2 2	B 2 2
22	F 2 2 2	f 2 2	F 2 2
23	I 2 2 2	i 2 2	I 2 2
24	I 21 21 21	i 2b 2c	I $[2_{\rm b}]$ $[2_{\rm c}]$
25	P m m 2	p 2 -2	P 2 $[\overline{2}]$
25:cab	P 2 m m	p -2 2	P $[\overline{2}]$ 2
25:bca	P m 2 m	p -2 -2	P $[\overline{2}\ \overline{2}]$
26	P m c 21	p 2c -2	P $[2_{\rm c}]$ $[\overline{2}]$
26:ba-c	P c m 21	p 2c -2c	P $[2_{\rm c}]$ $[\overline{2}_{\rm c}]$
26:cab	P 21 m a	p -2a 2a	P $[\overline{2}_{\rm a}]$ $[2_{\rm a}]$
26:-cba	P 21 a m	p -2 2a	P $[\overline{2}\ 2_{\rm a}]$
26:bca	P b 21 m	p -2 -2b	P $[\overline{2}\ \overline{2}_{\rm b}]$
26:a-cb	P m 21 b	p -2b -2	P $[\overline{2}_{\rm b}]$ $[\overline{2}]$
27	P c c 2	p 2 -2c	P 2 $[\overline{2}_{\rm c}]$
27:cab	P 2 a a	p -2a 2	P $[\overline{2}_{\rm a}]$ 2
27:bca	P b 2 b	p -2b -2b	P $[\overline{2}_{\rm b}]$ $[\overline{2}_{\rm b}]$
28	P m a 2	p 2 -2a	P 2 $[\overline{2}_{\rm a}]$
28:ba-c	P b m 2	p 2 -2b	P 2 $[\overline{2}_{\rm b}]$
28:cab	P 2 m b	p -2b 2	P $[\overline{2}_{\rm b}]$ 2
28:-cba	P 2 c m	p -2c 2	P $[\overline{2}_{\rm c}]$ 2
28:bca	P c 2 m	p -2c -2c	P $[\overline{2}_{\rm c}]$ $[\overline{2}_{\rm c}]$
28:a-cb	P m 2 a	p -2a -2a	P $[\overline{2}_{\rm a}]$ $[\overline{2}_{\rm a}]$
29	P c a 21	p 2c -2ac	P $[2_{\rm c}]$ $[\overline{2}_{\rm ac}]$
29:ba-c	P b c 21	p 2c -2b	P $[2_{\rm c}]$ $[\overline{2}_{\rm b}]$
29:cab	P 21 a b	p -2b 2a	P $[\overline{2}_{\rm b}]$ $[2_{\rm a}]$
29:-cba	P 21 c a	p -2ac 2a	P $[\overline{2}_{\rm ac}]$ $[2_{\rm a}]$
29:bca	P c 21 b	p -2bc -2c	P $[\overline{2}_{\rm bc}]$ $[\overline{2}_{\rm c}]$
29:a-cb	P b 21 a	p -2a -2ab	P $[\overline{2}_{\rm a}]$ $[\overline{2}_{\rm ab}]$
30	P n c 2	p 2 -2bc	P 2 $[\overline{2}_{\rm bc}]$
30:ba-c	P c n 2	p 2 -2ac	P 2 $[\overline{2}_{\rm ac}]$
30:cab	P 2 n a	p -2ac 2	P $[\overline{2}_{\rm ac}]$ 2
30:-cba	P 2 a n	p -2ab 2	P $[\overline{2}_{\rm ab}]$ 2
30:bca	P b 2 n	p -2ab -2ab	P $[\overline{2}_{\rm ab}]$ $[\overline{2}_{\rm ab}]$
30:a-cb	P n 2 b	p -2bc -2bc	P $[\overline{2}_{\rm bc}]$ $[\overline{2}_{\rm bc}]$
31	P m n 21	p 2ac -2	P $[2_{\rm ac}]$ $[\overline{2}]$
31:ba-c	P n m 21	p 2bc -2bc	P $[2_{\rm bc}]$ $[\overline{2}_{\rm bc}]$
31:cab	P 21 m n	p -2ab 2ab	P $[\overline{2}_{\rm ab}]$ $[2_{\rm ab}]$
31:-cba	P 21 n m	p -2 2ac	P $[\overline{2}\ 2_{\rm ac}]$
31:bca	P n 21 m	p -2 -2bc	P $[\overline{2}\ \overline{2}_{\rm bc}]$
31:a-cb	P m 21 n	p -2ab -2	P $[\overline{2}_{\rm ab}]$ $[\overline{2}]$
32	P b a 2	p 2 -2ab	P 2 $[\overline{2}_{\rm ab}]$
32:cab	P 2 c b	p -2bc 2	P $[\overline{2}_{\rm bc}]$ 2
32:bca	P c 2 a	p -2ac -2ac	P $[\overline{2}_{\rm ac}]$ $[\overline{2}_{\rm ac}]$
33	P n a 21	p 2c -2n	P $[2_{\rm c}]$ $[\overline{2}_{\rm n}]$
33:ba-c	P b n 21	p 2c -2ab	P $[2_{\rm c}]$ $[\overline{2}_{\rm ab}]$
33:cab	P 21 n b	p -2bc 2a	P $[\overline{2}_{\rm bc}]$ $[2_{\rm a}]$
33:-cba	P 21 c n	p -2n 2a	P $[\overline{2}_{\rm n}]$ $[2_{\rm a}]$
33:bca	P c 21 n	p -2n -2ac	P $[\overline{2}_{\rm n}]$ $[\overline{2}_{\rm ac}]$
33:a-cb	P n 21 a	p -2ac -2n	P $[\overline{2}_{\rm ac}]$ $[\overline{2}_{\rm n}]$
34	P n n 2	p 2 -2n	P 2 $[\overline{2}_{\rm n}]$
34:cab	P 2 n n	p -2n 2	P $[\overline{2}_{\rm n}]$ 2
34:bca	P n 2 n	p -2n -2n	P $[\overline{2}_{\rm n}]$ $[\overline{2}_{\rm n}]$
35	C m m 2	c 2 -2	C 2 $[\overline{2}]$
35:cab	A 2 m m	a -2 2	A $[\overline{2}]$ 2
35:bca	B m 2 m	b -2 -2	B $[\overline{2}\ \overline{2}]$
36	C m c 21	c 2c -2	C $[2_{\rm c}]$ $[\overline{2}]$
36:ba-c	C c m 21	c 2c -2c	C $[2_{\rm c}]$ $[\overline{2}_{\rm c}]$
36:cab	A 21 m a	a -2a 2a	A $[\overline{2}_{\rm a}]$ $[2_{\rm a}]$
36:-cba	A 21 a m	a -2 2a	A $[\overline{2}\ 2_{\rm a}]$
36:bca	B b 21 m	b -2 -2b	B $[\overline{2}\ \overline{2}_{\rm b}]$
36:a-cb	B m 21 b	b -2b -2	B $[\overline{2}_{\rm b}]$ $[\overline{2}]$
37	C c c 2	c 2 -2c	C 2 $[\overline{2}_{\rm c}]$
37:cab	A 2 a a	a -2a 2	A $[\overline{2}_{\rm a}]$ 2
37:bca	B b 2 b	b -2b -2b	B $[\overline{2}_{\rm b}]$ $[\overline{2}_{\rm b}]$
38	A m m 2	a 2 -2	A 2 $[\overline{2}]$
38:ba-c	B m m 2	b 2 -2	B 2 $[\overline{2}]$
38:cab	B 2 m m	b -2 2	B $[\overline{2}]$ 2
38:-cba	C 2 m m	c -2 2	C $[\overline{2}]$ 2
38:bca	C m 2 m	c -2 -2	C $[\overline{2}\ \overline{2}]$
38:a-cb	A m 2 m	a -2 -2	A $[\overline{2}\ \overline{2}]$
39	A b m 2	a 2 -2b	A 2 $[\overline{2}_{\rm b}]$
39:ba-c	B m a 2	b 2 -2a	B 2 $[\overline{2}_{\rm a}]$
39:cab	B 2 c m	b -2a 2	B $[\overline{2}_{\rm a}]$ 2
39:-cba	C 2 m b	c -2a 2	C $[\overline{2}_{\rm a}]$ 2
39:bca	C m 2 a	c -2a -2a	C $[\overline{2}_{\rm a}]$ $[\overline{2}_{\rm a}]$
39:a-cb	A c 2 m	a -2b -2b	A $[\overline{2}_{\rm b}]$ $[\overline{2}_{\rm b}]$
40	A m a 2	a 2 -2a	A 2 $[\overline{2}_{\rm a}]$
40:ba-c	B b m 2	b 2 -2b	B 2 $[\overline{2}_{\rm b}]$
40:cab	B 2 m b	b -2b 2	B $[\overline{2}_{\rm b}]$ 2
40:-cba	C 2 c m	c -2c 2	C $[\overline{2}_{\rm c}]$ 2
40:bca	C c 2 m	c -2c -2c	C $[\overline{2}_{\rm c}]$ $[\overline{2}_{\rm c}]$
40:a-cb	A m 2 a	a -2a -2a	A $[\overline{2}_{\rm a}]$ $[\overline{2}_{\rm a}]$
41	A b a 2	a 2 -2ab	A 2 $[\overline{2}_{\rm ab}]$
41:ba-c	B b a 2	b 2 -2ab	B 2 $[\overline{2}_{\rm ab}]$
41:cab	B 2 c b	b -2ab 2	B $[\overline{2}_{\rm ab}]$ 2
41:-cba	C 2 c b	c -2ac 2	C $[\overline{2}_{\rm ac}]$ 2
41:bca	C c 2 a	c -2ac -2ac	C $[\overline{2}_{\rm ac}]$ $[\overline{2}_{\rm ac}]$
41:a-cb	A c 2 a	a -2ab -2ab	A $[\overline{2}_{\rm ab}]$ $[\overline{2}_{\rm ab}]$
42	F m m 2	f 2 -2	F 2 $[\overline{2}]$
42:cab	F 2 m m	f -2 2	F $[\overline{2}]$ 2
42:bca	F m 2 m	f -2 -2	F $[\overline{2}\ \overline{2}]$
43	F d d 2	f 2 -2d	F 2 $[\overline{2}_{\rm d}]$
43:cab	F 2 d d	f -2d 2	F $[\overline{2}_{\rm d}]$ 2
43:bca	F d 2 d	f -2d -2d	F $[\overline{2}_{\rm d}]$ $[\overline{2}_{\rm d}]$
44	I m m 2	i 2 -2	I 2 $[\overline{2}]$
44:cab	I 2 m m	i -2 2	I $[\overline{2}]$ 2
44:bca	I m 2 m	i -2 -2	I $[\overline{2}\ \overline{2}]$
45	I b a 2	i 2 -2c	I 2 $[\overline{2}_{\rm c}]$
45:cab	I 2 c b	i -2a 2	I $[\overline{2}_{\rm a}]$ 2
45:bca	I c 2 a	i -2b -2b	I $[\overline{2}_{\rm b}]$ $[\overline{2}_{\rm b}]$
46	I m a 2	i 2 -2a	I 2 $[\overline{2}_{\rm a}]$
46:ba-c	I b m 2	i 2 -2b	I 2 $[\overline{2}_{\rm b}]$
46:cab	I 2 m b	i -2b 2	I $[\overline{2}_{\rm b}]$ 2
46:-cba	I 2 c m	i -2c 2	I $[\overline{2}_{\rm c}]$ 2
46:bca	I c 2 m	i -2c -2c	I $[\overline{2}_{\rm c}]$ $[\overline{2}_{\rm c}]$
46:a-cb	I m 2 a	i -2a -2a	I $[\overline{2}_{\rm a}]$ $[\overline{2}_{\rm a}]$
47	P m m m	-p 2 2	$[\overline{\rm P}]$ 2 2
48:1	P n n n:1	p 2 2 -1n	P 2 2 $[\overline{1}_{\rm n}]$
48:2	P n n n:2	-p 2ab 2bc	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm bc}]$
49	P c c m	-p 2 2c	$[\overline{\rm P}]$ 2 $[2_{\rm c}]$
49:cab	P m a a	-p 2a 2	$[\overline{\rm P}]$ $[2_{\rm a}]$ 2
49:bca	P b m b	-p 2b 2b	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm b}]$
50:1	P b a n:1	p 2 2 -1ab	P 2 2 $[\overline{1}_{\rm ab}]$
50:2	P b a n:2	-p 2ab 2b	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm b}]$
50:1cab	P n c b:1	p 2 2 -1bc	P 2 2 $[\overline{1}_{\rm bc}]$
50:2cab	P n c b:2	-p 2b 2bc	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm bc}]$
50:1bca	P c n a:1	p 2 2 -1ac	P 2 2 $[\overline{1}_{\rm ac}]$
50:2bca	P c n a:2	-p 2a 2c	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm c}]$
51	P m m a	-p 2a 2a	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm a}]$
51:ba-c	P m m b	-p 2b 2	$[\overline{\rm P}]$ $[2_{\rm b}]$ 2
51:cab	P b m m	-p 2 2b	$[\overline{\rm P}]$ 2 $[2_{\rm b}]$
51:-cba	P c m m	-p 2c 2c	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm c}]$
51:bca	P m c m	-p 2c 2	$[\overline{\rm P}]$ $[2_{\rm c}]$ 2
51:a-cb	P m a m	-p 2 2a	$[\overline{\rm P}]$ 2 $[2_{\rm a}]$
52	P n n a	-p 2a 2bc	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm bc}]$
52:ba-c	P n n b	-p 2b 2n	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm n}]$
52:cab	P b n n	-p 2n 2b	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm b}]$
52:-cba	P c n n	-p 2ab 2c	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm c}]$
52:bca	P n c n	-p 2ab 2n	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm n}]$
52:a-cb	P n a n	-p 2n 2bc	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm bc}]$
53	P m n a	-p 2ac 2	$[\overline{\rm P}]$ $[2_{\rm ac}]$ 2
53:ba-c	P n m b	-p 2bc 2bc	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm bc}]$
53:cab	P b m n	-p 2ab 2ab	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm ab}]$
53:-cba	P c n m	-p 2 2ac	$[\overline{\rm P}]$ 2 $[2_{\rm ac}]$
53:bca	P n c m	-p 2 2bc	$[\overline{\rm P}]$ 2 $[2_{\rm bc}]$
53:a-cb	P m a n	-p 2ab 2	$[\overline{\rm P}]$ $[2_{\rm ab}]$ 2
54	P c c a	-p 2a 2ac	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm ac}]$
54:ba-c	P c c b	-p 2b 2c	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm c}]$
54:cab	P b a a	-p 2a 2b	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm b}]$
54:-cba	P c a a	-p 2ac 2c	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm c}]$
54:bca	P b c b	-p 2bc 2b	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm b}]$
54:a-cb	P b a b	-p 2b 2ab	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm ab}]$
55	P b a m	-p 2 2ab	$[\overline{\rm P}]$ 2 $[2_{\rm ab}]$
55:cab	P m c b	-p 2bc 2	$[\overline{\rm P}]$ $[2_{\rm bc}]$ 2
55:bca	P c m a	-p 2ac 2ac	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm ac}]$
56	P c c n	-p 2ab 2ac	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm ac}]$
56:cab	P n a a	-p 2ac 2bc	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm bc}]$
56:bca	P b n b	-p 2bc 2ab	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm ab}]$
57	P b c m	-p 2c 2b	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm b}]$
57:ba-c	P c a m	-p 2c 2ac	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm ac}]$
57:cab	P m c a	-p 2ac 2a	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm a}]$
57:-cba	P m a b	-p 2b 2a	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm a}]$
57:bca	P b m a	-p 2a 2ab	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm ab}]$
57:a-cb	P c m b	-p 2bc 2c	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm c}]$
58	P n n m	-p 2 2n	$[\overline{\rm P}]$ 2 $[2_{\rm n}]$
58:cab	P m n n	-p 2n 2	$[\overline{\rm P}]$ $[2_{\rm n}]$ 2
58:bca	P n m n	-p 2n 2n	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm n}]$
59:1	P m m n:1	p 2 2ab -1ab	P 2 $[2_{\rm ab}]$ $[\overline{1}_{\rm ab}]$
59:2	P m m n:2	-p 2ab 2a	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm a}]$
59:1cab	P n m m:1	p 2bc 2 -1bc	P $[2_{\rm bc}]$ 2 $[\overline{1}_{\rm bc}]$
59:2cab	P n m m:2	-p 2c 2bc	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm bc}]$
59:1bca	P m n m:1	p 2ac 2ac -1ac	P $[2_{\rm ac}]$ $[2_{\rm ac}]$ $[\overline{1}_{\rm ac}]$
59:2bca	P m n m:2	-p 2c 2a	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm a}]$
60	P b c n	-p 2n 2ab	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm ab}]$
60:ba-c	P c a n	-p 2n 2c	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm c}]$
60:cab	P n c a	-p 2a 2n	$[\overline{\rm P}]$ $[2_{\rm a}]$ $[2_{\rm n}]$
60:-cba	P n a b	-p 2bc 2n	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm n}]$
60:bca	P b n a	-p 2ac 2b	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm b}]$
60:a-cb	P c n b	-p 2b 2ac	$[\overline{\rm P}]$ $[2_{\rm b}]$ $[2_{\rm ac}]$
61	P b c a	-p 2ac 2ab	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm ab}]$
61:ba-c	P c a b	-p 2bc 2ac	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm ac}]$
62	P n m a	-p 2ac 2n	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm n}]$
62:ba-c	P m n b	-p 2bc 2a	$[\overline{\rm P}]$ $[2_{\rm bc}]$ $[2_{\rm a}]$
62:cab	P b n m	-p 2c 2ab	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm ab}]$
62:-cba	P c m n	-p 2n 2ac	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm ac}]$
62:bca	P m c n	-p 2n 2a	$[\overline{\rm P}]$ $[2_{\rm n}]$ $[2_{\rm a}]$
62:a-cb	P n a m	-p 2c 2n	$[\overline{\rm P}]$ $[2_{\rm c}]$ $[2_{\rm n}]$
63	C m c m	-c 2c 2	$[\overline{\rm C}]$ $[2_{\rm c}]$ 2
63:ba-c	C c m m	-c 2c 2c	$[\overline{\rm C}]$ $[2_{\rm c}]$ $[2_{\rm c}]$
63:cab	A m m a	-a 2a 2a	$[\overline{\rm A}]$ $[2_{\rm a}]$ $[2_{\rm a}]$
63:-cba	A m a m	-a 2 2a	$[\overline{\rm A}]$ 2 $[2_{\rm a}]$
63:bca	B b m m	-b 2 2b	$[\overline{\rm B}]$ 2 $[2_{\rm b}]$
63:a-cb	B m m b	-b 2b 2	$[\overline{\rm B}]$ $[2_{\rm b}]$ 2
64	C m c a	-c 2ac 2	$[\overline{\rm C}]$ $[2_{\rm ac}]$ 2
64:ba-c	C c m b	-c 2ac 2ac	$[\overline{\rm C}]$ $[2_{\rm ac}]$ $[2_{\rm ac}]$
64:cab	A b m a	-a 2ab 2ab	$[\overline{\rm A}]$ $[2_{\rm ab}]$ $[2_{\rm ab}]$
64:-cba	A c a m	-a 2 2ab	$[\overline{\rm A}]$ 2 $[2_{\rm ab}]$
64:bca	B b c m	-b 2 2ab	$[\overline{\rm B}]$ 2 $[2_{\rm ab}]$
64:a-cb	B m a b	-b 2ab 2	$[\overline{\rm B}]$ $[2_{\rm ab}]$ 2
65	C m m m	-c 2 2	$[\overline{\rm C}]$ 2 2
65:cab	A m m m	-a 2 2	$[\overline{\rm A}]$ 2 2
65:bca	B m m m	-b 2 2	$[\overline{\rm B}]$ 2 2
66	C c c m	-c 2 2c	$[\overline{\rm C}]$ 2 $[2_{\rm c}]$
66:cab	A m a a	-a 2a 2	$[\overline{\rm A}]$ $[2_{\rm a}]$ 2
66:bca	B b m b	-b 2b 2b	$[\overline{\rm B}]$ $[2_{\rm b}]$ $[2_{\rm b}]$
67	C m m a	-c 2a 2	$[\overline{\rm C}]$ $[2_{\rm a}]$ 2
67:ba-c	C m m b	-c 2a 2a	$[\overline{\rm C}]$ $[2_{\rm a}]$ $[2_{\rm a}]$
67:cab	A b m m	-a 2b 2b	$[\overline{\rm A}]$ $[2_{\rm b}]$ $[2_{\rm b}]$
67:-cba	A c m m	-a 2 2b	$[\overline{\rm A}]$ 2 $[2_{\rm b}]$
67:bca	B m c m	-b 2 2a	$[\overline{\rm B}]$ 2 $[2_{\rm a}]$
67:a-cb	B m a m	-b 2a 2	$[\overline{\rm B}]$ $[2_{\rm a}]$ 2
68:1	C c c a:1	c 2 2 -1ac	C 2 2 $[\overline{1}_{\rm ac}]$
68:2	C c c a:2	-c 2a 2ac	$[\overline{\rm C}]$ $[2_{\rm a}]$ $[2_{\rm ac}]$
68:1ba-c	C c c b:1	c 2 2 -1ac	C 2 2 $[\overline{1}_{\rm ac}]$
68:2ba-c	C c c b:2	-c 2a 2c	$[\overline{\rm C}]$ $[2_{\rm a}]$ $[2_{\rm c}]$
68:1cab	A b a a:1	a 2 2 -1ab	A 2 2 $[\overline{1}_{\rm ab}]$
68:2cab	A b a a:2	-a 2a 2b	$[\overline{\rm A}]$ $[2_{\rm a}]$ $[2_{\rm b}]$
68:1-cba	A c a a:1	a 2 2 -1ab	A 2 2 $[\overline{1}_{\rm ab}]$
68:2-cba	A c a a:2	-a 2ab 2b	$[\overline{\rm A}]$ $[2_{\rm ab}]$ $[2_{\rm b}]$
68:1bca	B b c b:1	b 2 2 -1ab	B 2 2 $[\overline{1}_{\rm ab}]$
68:2bca	B b c b:2	-b 2ab 2b	$[\overline{\rm B}]$ $[2_{\rm ab}]$ $[2_{\rm b}]$
68:1a-cb	B b a b:1	b 2 2 -1ab	B 2 2 $[\overline{1}_{\rm ab}]$
68:2a-cb	B b a b:2	-b 2b 2ab	$[\overline{\rm B}]$ $[2_{\rm b}]$ $[2_{\rm ab}]$
69	F m m m	-f 2 2	$[\overline{\rm F}]$ 2 2
70:1	F d d d:1	f 2 2 -1d	F 2 2 $[\overline{1}_{\rm d}]$
70:2	F d d d:2	-f 2uv 2vw	$[\overline{\rm F}]$ $[2_{\rm uv}]$ $[2_{\rm vw}]$
71	I m m m	-i 2 2	$[\overline{\rm I}]$ 2 2
72	I b a m	-i 2 2c	$[\overline{\rm I}]$ 2 $[2_{\rm c}]$
72:cab	I m c b	-i 2a 2	$[\overline{\rm I}]$ $[2_{\rm a}]$ 2
72:bca	I c m a	-i 2b 2b	$[\overline{\rm I}]$ $[2_{\rm b}]$ $[2_{\rm b}]$
73	I b c a	-i 2b 2c	$[\overline{\rm I}]$ $[2_{\rm b}]$ $[2_{\rm c}]$
73:ba-c	I c a b	-i 2a 2b	$[\overline{\rm I}]$ $[2_{\rm a}]$ $[2_{\rm b}]$
74	I m m a	-i 2b 2	$[\overline{\rm I}]$ $[2_{\rm b}]$ 2
74:ba-c	I m m b	-i 2a 2a	$[\overline{\rm I}]$ $[2_{\rm a}]$ $[2_{\rm a}]$
74:cab	I b m m	-i 2c 2c	$[\overline{\rm I}]$ $[2_{\rm c}]$ $[2_{\rm c}]$
74:-cba	I c m m	-i 2 2b	$[\overline{\rm I}]$ 2 $[2_{\rm b}]$
74:bca	I m c m	-i 2 2a	$[\overline{\rm I}]$ 2 $[2_{\rm a}]$
74:a-cb	I m a m	-i 2c 2	$[\overline{\rm I}]$ $[2_{\rm c}]$ 2
75	P 4	p 4	P 4
76	P 41	p 4w	P $[4_{\rm w}]$
77	P 42	p 4c	P $[4_{\rm c}]$
78	P 43	p 4cw	P $[4_{\rm cw}]$
79	I 4	i 4	I 4
80	I 41	i 4bw	I $[4_{\rm bw}]$
81	P -4	p -4	P $[\overline{4}]$
82	I -4	i -4	I $[\overline{4}]$
83	P 4/m	-p 4	$[\overline{\rm P}]$ 4
84	P 42/m	-p 4c	$[\overline{\rm P}]$ $[4_{\rm c}]$
85:1	P 4/n:1	p 4ab -1ab	P $[4_{\rm ab}]$ $[\overline{1}_{\rm ab}]$
85:2	P 4/n:2	-p 4a	$[\overline{\rm P}]$ $[4_{\rm a}]$
86:1	P 42/n:1	p 4n -1n	P $[4_{\rm n}]$ $[\overline{1}_{\rm n}]$
86:2	P 42/n:2	-p 4bc	$[\overline{\rm P}]$ $[4_{\rm bc}]$
87	I 4/m	-i 4	$[\overline{\rm I}]$ 4
88:1	I 41/a:1	i 4bw -1bw	I $[4_{\rm bw}]$ $[\overline{1}_{\rm bw}]$
88:2	I 41/a:2	-i 4ad	$[\overline{\rm I}]$ $[4_{\rm ad}]$
89	P 4 2 2	p 4 2	P 4 2
90	P 4 21 2	p 4ab 2ab	P $[4_{\rm ab}]$ $[2_{\rm ab}]$
91	P 41 2 2	p 4w 2c	P $[4_{\rm w}]$ $[2_{\rm c}]$
92	P 41 21 2	p 4abw 2nw	$[\rm{{P} 4_{abw} 2_{\hskip-4ptnw}}]$
93	P 42 2 2	p 4c 2	P $[4_{\rm c}]$ 2
94	P 42 21 2	p 4n 2n	P $[4_{\rm n}]$ $[2_{\rm n}]$
95	P 43 2 2	p 4cw 2c	P $[4_{\rm cw}]$ $[2_{\rm c}]$
96	P 43 21 2	p 4nw 2abw	P $[4_{\rm {\hskip-4pt\phantom b}nw}\ 2_{\rm abw}]$
97	I 4 2 2	i 4 2	I 4 2
98	I 41 2 2	i 4bw 2bw	I $[4_{\rm bw}]$ $[2_{\rm bw}]$
99	P 4 m m	p 4 -2	P 4 $[\overline{2}]$
100	P 4 b m	p 4 -2ab	P 4 $[\overline{2}_{\rm ab}]$
101	P 42 c m	p 4c -2c	P $[4_{\rm c}]$ $[\overline{2}_{\rm c}]$
102	P 42 n m	p 4n -2n	P $[4_{\rm n}]$ $[\overline{2}_{\rm n}]$
103	P 4 c c	p 4 -2c	P 4 $[\overline{2}_{\rm c}]$
104	P 4 n c	p 4 -2n	P 4 $[\overline{2}_{\rm n}]$
105	P 42 m c	p 4c -2	P $[4_{\rm c}]$ $[\overline{2}]$
106	P 42 b c	p 4c -2ab	P $[4_{\rm c}]$ $[\overline{2}_{\rm ab}]$
107	I 4 m m	i 4 -2	I 4 $[\overline{2}]$
108	I 4 c m	i 4 -2c	I 4 $[\overline{2}_{\rm c}]$
109	I 41 m d	i 4bw -2	I $[4_{\rm bw}]$ $[\overline{2}]$
110	I 41 c d	i 4bw -2c	I $[4_{\rm bw}]$ $[\overline{2}_{\rm c}]$
111	P -4 2 m	p -4 2	P $[\overline{4}]$ 2
112	P -4 2 c	p -4 2c	P $[\overline{4}\ 2_{\rm c}]$
113	P -4 21 m	p -4 2ab	P $[\overline{4}\ 2_{\rm ab}]$
114	P -4 21 c	p -4 2n	P $[\overline{4}\ 2_{\rm n}]$
115	P -4 m 2	p -4 -2	P $[\overline{4}\ \overline{2}]$
116	P -4 c 2	p -4 -2c	P $[\overline{4}\ \overline{2}_{\rm c}]$
117	P -4 b 2	p -4 -2ab	P $[\overline{4}\ \overline{2}_{\rm ab}]$
118	P -4 n 2	p -4 -2n	P $[\overline{4}\ \overline{2}_{\rm n}]$
119	I -4 m 2	i -4 -2	I $[\overline{4}\ \overline{2}]$
120	I -4 c 2	i -4 -2c	I $[\overline{4}\ \overline{2}_{\rm c}]$
121	I -4 2 m	i -4 2	I $[\overline{4}]$ 2
122	I -4 2 d	i -4 2bw	I $[\overline{4}\ 2_{\rm bw}]$
123	P 4/m m m	-p 4 2	$[\overline{\rm P}]$ 4 2
124	P 4/m c c	-p 4 2c	$[\overline{\rm P}]$ 4 $[2_{\rm c}]$
125:1	P 4/n b m:1	p 4 2 -1ab	P 4 2 $[\overline{1}_{\rm ab}]$
125:2	P 4/n b m:2	-p 4a 2b	$[\overline{\rm P}]$ $[4_{\rm a}]$ $[2_{\rm b}]$
126:1	P 4/n n c:1	p 4 2 -1n	P 4 2 $[\overline{1}_{\rm n}]$
126:2	P 4/n n c:2	-p 4a 2bc	$[\overline{\rm P}]$ $[4_{\rm a}]$ $[2_{\rm bc}]$
127	P 4/m b m	-p 4 2ab	$[\overline{\rm P}]$ 4 $[2_{\rm ab}]$
128	P 4/m n c	-p 4 2n	$[\overline{\rm P}]$ 4 $[2_{\rm n}]$
129:1	P 4/n m m:1	p 4ab 2ab -1ab	P $[4_{\rm ab}]$ $[2_{\rm ab}]$ $[\overline{1}_{\rm ab}]$
129:2	P 4/n m m:2	-p 4a 2a	$[\overline{\rm P}]$ $[4_{\rm a}]$ $[2_{\rm a}]$
130:1	P 4/n c c:1	p 4ab 2n -1ab	P $[4_{\rm ab}]$ $[2_{\rm n}]$ $[\overline{1}_{\rm ab}]$
130:2	P 4/n c c:2	-p 4a 2ac	$[\overline{\rm P}]$ $[4_{\rm a}]$ $[2_{\rm ac}]$
131	P 42/m m c	-p 4c 2	$[\overline{\rm P}]$ $[4_{\rm c}]$ 2
132	P 42/m c m	-p 4c 2c	$[\overline{\rm P}]$ $[4_{\rm c}]$ $[2_{\rm c}]$
133:1	P 42/n b c:1	p 4n 2c -1n	P $[4_{\rm n}]$ $[2_{\rm c}]$ $[\overline{1}_{\rm n}]$
133:2	P 42/n b c:2	-p 4ac 2b	$[\overline{\rm P}]$ $[4_{\rm ac}]$ $[2_{\rm b}]$
134:1	P 42/n n m:1	p 4n 2 -1n	P $[4_{\rm n}]$ 2 $[\overline{1}_{\rm n}]$
134:2	P 42/n n m:2	-p 4ac 2bc	$[\overline{\rm P}]$ $[4_{\rm ac}]$ $[2_{\rm bc}]$
135	P 42/m b c	-p 4c 2ab	$[\overline{\rm P}]$ $[4_{\rm c}]$ $[2_{\rm ab}]$
136	P 42/m n m	-p 4n 2n	$[\overline{\rm P}]$ $[4_{\rm n}]$ $[2_{\rm n}]$
137:1	P 42/n m c:1	p 4n 2n -1n	P $[4_{\rm n}]$ $[2_{\rm n}]$ $[\overline{1}_{\rm n}]$
137:2	P 42/n m c:2	-p 4ac 2a	$[\overline{\rm P}]$ $[4_{\rm ac}]$ $[2_{\rm a}]$
138:1	P 42/n c m:1	p 4n 2ab -1n	P $[4_{\rm n}]$ $[2_{\rm ab}]$ $[\overline{1}_{\rm n}]$
138:2	P 42/n c m:2	-p 4ac 2ac	$[\overline{\rm P}]$ $[4_{\rm ac}]$ $[2_{\rm ac}]$
139	I 4/m m m	-i 4 2	$[\overline{\rm I}]$ 4 2
140	I 4/m c m	-i 4 2c	$[\overline{\rm I}]$ 4 $[2_{\rm c}]$
141:1	I 41/a m d:1	i 4bw 2bw -1bw	I $[4_{\rm bw}]$ $[2_{\rm bw}]$ $[\overline{1}_{\rm bw}]$
141:2	I 41/a m d:2	-i 4bd 2	$[\overline{\rm I}]$ $[4_{\rm bd}]$ 2
142:1	I 41/a c d:1	i 4bw 2aw -1bw	I $[4_{\rm bw}]$ $[2_{\rm aw}]$ $[\overline{1}_{\rm bw}]$
142:2	I 41/a c d:2	-i 4bd 2c	$[\overline{\rm I}]$ $[4_{\rm bd}]$ $[2_{\rm c}]$
143	P 3	p 3	P 3
144	P 31	p 31	P $[3_{\rm 1}]$
145	P 32	p 32	P $[3_{\rm 2}]$
146:h	R 3:h	r 3	R 3
146:r	R 3:r	p 3*	P 3*
147	P -3	-p 3	$[\overline{\rm P}]$ 3
148:h	R -3:h	-r 3	$[\overline{\rm R}]$ 3
148:r	R -3:r	-p 3*	$[\overline{\rm P}]$ 3*
149	P 3 1 2	p 3 2	P 3 2
150	P 3 2 1	p 3 2"	P 3 2"
151	P 31 1 2	p 31 2 (0 0 4)	P $[3_{\rm 1}]$ 2 (0 0 4)
152	P 31 2 1	p 31 2"	P $[3_{\rm 1}]$ 2"
153	P 32 1 2	p 32 2 (0 0 2)	P $[3_{\rm 2}]$ 2 (0 0 2)
154	P 32 2 1	p 32 2"	P $[3_{\rm 2}]$ 2"
155:h	R 3 2:h	r 3 2"	R 3 2"
155:r	R 3 2:r	p 3* 2	P 3* 2
156	P 3 m 1	p 3 -2"	P 3 $[\overline{2}]$ "
157	P 3 1 m	p 3 -2	P 3 $[\overline{2}]$
158	P 3 c 1	p 3 -2"c	P 3 $[\overline{2}"_{\rm c}]$
159	P 3 1 c	p 3 -2c	P 3 $[\overline{2}_{\rm c}]$
160:h	R 3 m:h	r 3 -2"	R 3 $[\overline{2}]$ "
160:r	R 3 m:r	p 3* -2	P 3* $[\overline{2}]$
161:h	R 3 c:h	r 3 -2"c	R 3 $[\overline{2}"_{\rm c}]$
161:r	R 3 c:r	p 3* -2n	P 3* $[\overline{2}_{\rm n}]$
162	P -3 1 m	-p 3 2	$[\overline{\rm P}]$ 3 2
163	P -3 1 c	-p 3 2c	$[\overline{\rm P}]$ 3 $[2_{\rm c}]$
164	P -3 m 1	-p 3 2"	$[\overline{\rm P}]$ 3 2"
165	P -3 c 1	-p 3 2"c	$[\overline{\rm P}]$ 3 $[2^{"}_{\rm c}]$
166:h	R -3 m:h	-r 3 2"	$[\overline{\rm R}]$ 3 2"
166:r	R -3 m:r	-p 3* 2	$[\overline{\rm P}]$ 3* 2
167:h	R -3 c:h	-r 3 2"c	$[\overline{\rm R}]$ 3 $[2^{"}_{\rm c}]$
167:r	R -3 c:r	-p 3* 2n	$[\overline{\rm P}]$ 3* $[2_{\rm n}]$
168	P 6	p 6	P 6
169	P 61	p 61	P $[6_{\rm 1}]$
170	P 65	p 65	P $[6_{\rm 5}]$
171	P 62	p 62	P $[6_{\rm 2}]$
172	P 64	p 64	P $[6_{\rm 4}]$
173	P 63	p 6c	P $[6_{\rm c}]$
174	P -6	p -6	P $[\overline{6}]$
175	P 6/m	-p 6	$[\overline{\rm P}]$ 6
176	P 63/m	-p 6c	$[\overline{\rm P}]$ $[6_{\rm c}]$
177	P 6 2 2	p 6 2	P 6 2
178	P 61 2 2	p 61 2 (0 0 5)	P $[6_{\rm 1}]$ 2 (0 0 5)
179	P 65 2 2	p 65 2 (0 0 1)	P $[6_{\rm 5}]$ 2 (0 0 1)
180	P 62 2 2	p 62 2 (0 0 4)	P $[6_{\rm 2}]$ 2 (0 0 4)
181	P 64 2 2	p 64 2 (0 0 2)	P $[6_{\rm 4}]$ 2 (0 0 2)
182	P 63 2 2	p 6c 2c	P $[6_{\rm c}]$ $[2_{\rm c}]$
183	P 6 m m	p 6 -2	P 6 $[\overline{2}]$
184	P 6 c c	p 6 -2c	P 6 $[\overline{2}_{\rm c}]$
185	P 63 c m	p 6c -2	P $[6_{\rm c}]$ $[\overline{2}]$
186	P 63 m c	p 6c -2c	P $[6_{\rm c}]$ $[\overline{2}_{\rm c}]$
187	P -6 m 2	p -6 2	P $[\overline{6}]$ 2
188	P -6 c 2	p -6c 2	P $[\overline{6}_{\rm c}]$ 2
189	P -6 2 m	p -6 -2	P $[\overline{6}\;\overline{2}]$
190	P -6 2 c	p -6c -2c	P $[\overline{6}_{\rm c}]$ $[\overline{2}_{\rm c}]$
191	P 6/m m m	-p 6 2	$[\overline{\rm P}]$ 6 2
192	P 6/m c c	-p 6 2c	$[\overline{\rm P}]$ 6 $[2_{\rm c}]$
193	P 63/m c m	-p 6c 2	$[\overline{\rm P}]$ $[6_{\rm c}]$ 2
194	P 63/m m c	-p 6c 2c	$[\overline{\rm P}]$ $[6_{\rm c}]$ $[2_{\rm c}]$
195	P 2 3	p 2 2 3	P 2 2 3
196	F 2 3	f 2 2 3	F 2 2 3
197	I 2 3	i 2 2 3	I 2 2 3
198	P 21 3	p 2ac 2ab 3	P $[2_{\rm ac}]$ $[2_{\rm ab}]$ 3
199	I 21 3	i 2b 2c 3	I $[2_{\rm b}]$ $[2_{\rm c}]$ 3
200	P m -3	-p 2 2 3	$[\overline{\rm P}]$ 2 2 3
201:1	P n -3:1	p 2 2 3 -1n	P 2 2 3 $[\overline{1}_{\rm n}]$
201:2	P n -3:2	-p 2ab 2bc 3	$[\overline{\rm P}]$ $[2_{\rm ab}]$ $[2_{\rm bc}]$ 3
202	F m -3	-f 2 2 3	$[\overline{\rm F}]$ 2 2 3
203:1	F d -3:1	f 2 2 3 -1d	F 2 2 3 $[\overline{1}_{\rm d}]$
203:2	F d -3:2	-f 2uv 2vw 3	$[\overline{\rm F}]$ $[2_{\rm uv}]$ $[2_{\rm vw}]$ 3
204	I m -3	-i 2 2 3	$[\overline{\rm I}]$ 2 2 3
205	P a -3	-p 2ac 2ab 3	$[\overline{\rm P}]$ $[2_{\rm ac}]$ $[2_{\rm ab}]$ 3
206	I a -3	-i 2b 2c 3	$[\overline{\rm I}]$ $[2_{\rm b}]$ $[2_{\rm c}]$ 3
207	P 4 3 2	p 4 2 3	P 4 2 3
208	P 42 3 2	p 4n 2 3	P $[4_{\rm n}]$ 2 3
209	F 4 3 2	f 4 2 3	F 4 2 3
210	F 41 3 2	f 4d 2 3	F $[4_{\rm d}]$ 2 3
211	I 4 3 2	i 4 2 3	I 4 2 3
212	P 43 3 2	p 4acd 2ab 3	P $[4_{\rm acd}]$ $[2_{\rm ab}]$ 3
213	P 41 3 2	p 4bd 2ab 3	P $[4_{\rm bd}]$ $[2_{\rm ab}]$ 3
214	I 41 3 2	i 4bd 2c 3	I $[4_{\rm bd}]$ $[2_{\rm c}]$ 3
215	P -4 3 m	p -4 2 3	P $[\overline{4}]$ 2 3
216	F -4 3 m	f -4 2 3	F $[\overline{4}]$ 2 3
217	I -4 3 m	i -4 2 3	I $[\overline{4}]$ 2 3
218	P -4 3 n	p -4n 2 3	P $[\overline{4}_{\rm n}]$ 2 3
219	F -4 3 c	f -4a 2 3	F $[\overline{4}_{\rm a}]$ 2 3
220	I -4 3 d	i -4bd 2c 3	I $[\overline{4}_{\rm bd}]$ $[2_{\rm c}]$ 3
221	P m -3 m	-p 4 2 3	$[\overline{\rm P}]$ 4 2 3
222:1	P n -3 n:1	p 4 2 3 -1n	P 4 2 3 $[\overline{1}_{\rm n}]$
222:2	P n -3 n:2	-p 4a 2bc 3	$[\overline{\rm P}]$ $[4_{\rm a}]$ $[2_{\rm bc}]$ 3
223	P m -3 n	-p 4n 2 3	$[\overline{\rm P}]$ $[4_{\rm n}]$ 2 3
224:1	P n -3 m:1	p 4n 2 3 -1n	P $[4_{\rm n}]$ 2 3 $[\overline{1}_{\rm n}]$
224:2	P n -3 m:2	-p 4bc 2bc 3	$[\overline{\rm P}]$ $[4_{\rm bc}]$ $[2_{\rm bc}]$ 3
225	F m -3 m	-f 4 2 3	$[\overline{\rm F}]$ 4 2 3
226	F m -3 c	-f 4a 2 3	$[\overline{\rm F}]$ $[4_{\rm a}]$ 2 3
227:1	F d -3 m:1	f 4d 2 3 -1d	F $[4_{\rm d}]$ 2 3 $[\overline{1}_{\rm d}]$
227:2	F d -3 m:2	-f 4vw 2vw 3	$[\overline{\rm F}]$ $[4_{\rm vw}]$ $[2_{\rm vw}]$ 3
228:1	F d -3 c:1	f 4d 2 3 -1ad	F $[4_{\rm d}]$ 2 3 $[\overline{1}_{\rm ad}]$
228:2	F d -3 c:2	-f 4ud 2vw 3	$[\overline{\rm F}]$ $[4_{\rm ud}]$ $[2_{\rm vw}]$ 3
229	I m -3 m	-i 4 2 3	$[\overline{\rm I}]$ 4 2 3
230	I a -3 d	-i 4bd 2c 3	$[\overline{\rm I}]$ $[4_{\rm bd}]$ $[2_{\rm c}]$ 3

The codes appended to the space-group numbers listed in the first column identify the relationship between the symmetry elements and the crystal cell. Where no code is given the first choice listed below applies.

Monoclinic. Code = <unique axis><cell choice>: unique axis choices [cf. IT A (2005) Table 4.3.2.1 ] b, -b, c, -c, a, -a; cell choices [cf. IT A (2005) Table 4.3.2.1 ] 1, 2, 3.
Orthorhombic. Code = <origin choice><setting>: origin choices 1, 2; setting choices [cf. IT A (2005) Table4.3.2.1 ] abc, ba-c, cab, -cba, bca, a-cb.
Tetragonal, cubic. Code = <origin choice>: origin choices 1, 2.
Trigonal. Code = <cell choice>: cell choices h (hexagonal), r (rhombohedral).

The conventional primitive hexagonal lattice may be transformed to a C-centred orthohexagonal setting using the change-of-basis operator $[\openup3pt{\rm P}\;6\;({\rm x-1/2y, 1/2y, z})=\pmatrix{{1 \over 2}&-{3 \over 2}&0&0\cr{1 \over 2}&{1 \over 2}&0&0\cr0&0&1&0\cr0&0&0&1\cr}.]$ In this case the lattice translation for the C centring is obtained by transforming the integral translation t(0, 1, 0): $[\eqalign{{\bi V}\cdot\pmatrix{0&1&0&1\cr}^T&=\pmatrix{1&-{1\over 2}&0&0\cr0&{1 \over 2}&0&0\cr0&0&1&0\cr0&0&0&1\cr}\pmatrix{0\cr1\cr0\cr1\cr}\cr&=\pmatrix{-{1\over 2}&{1\over 2}&0&1\cr}^T.\cr}]$

The standard setting of an I-centred tetragonal space group may be transformed to a primitive setting using the change-of-basis operator $[{\rm I }\;4\;({\rm y+z,x+z,x+y})=\pmatrix{0&1&0&0\cr0&1&-1&0\cr-1&1&0&0\cr0&0&0&1\cr}.]$ Note that in the primitive setting, the fourfold axis is along a + b.

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn, 5th ed., corrected reprint. Heidelberg: Springer.Google Scholar

Hall, S. R. (1981a). Space-group notation with an explicit origin. Acta Cryst. A37, 517–525.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 112-114