Sydney R. Hall
Ralf W. Grosse-Kunstleve
The explicit-origin space group notation proposed by Hall (1981) [1], [2] is based on the minimum number of symmetry operations, in the form of Seitz matrices, needed 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.
The notation has the general form:
L [NAT]1 ...
[NAT]p
V
where L is the symbol specifying the lattice translational symmetry (see Table 1), NAT identifies the 4x4 Seitz matrix of a symmetry element in the minimum set which defines the space-group symmetry (see Tables 2 3, 4, and 5), and p is the number of elements in the set. V is a translation vector which shifts the origin of the generator matrices by fractions of the unit cell lengths a, b and c.
The matrix symbol NAT is composed of three parts:
Table 6 lists space group notation in several formats. The computer-entry representation of the Hall symbols is listed in column 3. The computer-entry format is the general notation expressed as case insensitive ASCII characters, with the overline (bar) symbol replaced by a minus sign. Column 1 of Table 6 contains the space-group number with an appended code which identifies the non-standard settings. Column 2 contains the full Hermann-Mauguin symbols in computer-entry format with appended codes which identify the origin and cell choice when there are alternatives.
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 a negative integers
-1, -2, -3, -4 or -6
for improper rotations.
The T translation symbols
1, 2, 3, 4, 5, a, b, c, n, u, v, w, d
are described in Table 2.
These translations apply additively
(e.g. ad signifies a (3/4,1/4,1/4)) translation).
The A axis symbols
x, y, z denote rotations
about the axes a, b, c,
respectively (see Table 3).
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 4).
The axis symbol * always refers to a 3-fold rotation along
a+b+c (see Table 5).
The
origin-shift translation vector V
has the construction (va vb
vc), where
va, vb and vc
denote the shifts in 12ths parallel to the cell edges
a, b and c, respectively.
va/12, vb/12
and vc/12
are the coordinates of the unshifted origin in the
shifted basis system. The shifted Seitz matrices Sn'
are derived from the unshifted matrices Sn with the
transformation
(1 0 0 va/12) (1 0 0 -va/12)
Sn' = (0 1 0 vb/12) * Sn * (0 1 0 -vb/12)
(0 0 1 vc/12) (0 0 1 -vc/12)
(0 0 0 1 ) (0 0 0 1 )
For most Hall 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:
Here are several simple examples of how NAT symbols expand to Seitz matrices.
(-1 0 0 0 )
-2xc = ( 0 1 0 0 )
( 0 0 1 1/2)
( 0 0 0 1 )
( 0 0 1 0 )
3* = ( 1 0 0 0 )
( 0 1 0 0 )
( 0 0 0 1 )
( 0 -1 0 0 )
4vw = ( 1 0 0 1/4)
( 0 0 1 1/4)
( 0 0 0 1 )
( 1 -1 0 0 ) ( 0 -1 0 0 )
61 2 (0 0 -1) = ( 1 0 0 0 ) (-1 0 0 0 )
( 0 0 1 1/6) ( 0 0 -1 5/6)
( 0 0 0 1 ) ( 0 0 0 1 )
The lattice symbol L specifies one or more Seitz matrices which are needed to generate the space-group symmetry elements. For noncentrosymmetric lattices the rotation matrices are for 1 (see Table 3). For centrosymmetric lattices the lattice symbols are preceded by a minus sign `-', rotations are 1 and -1, and the total number of generator matrices implied by each symbol is twice the number of implied lattice translations.
| Non-centrosymmetric symbol |
Number of lattice translations |
Implied lattice translation(s) |
|---|---|---|
| P | 1 | (0,0,0) |
| A | 2 | (0,0,0), (0,1/2,1/2) |
| B | 2 | (0,0,0), (1/2,0,1/2) |
| C | 2 | (0,0,0), (1/2,1/2,0) |
| I | 2 | (0,0,0), (1/2,1/2,1/2) |
| R | 3 | (0,0,0), (2/3,1/3,1/3), (1/3,2/3,2/3) |
| S | 3 | (0,0,0), (1/3,1/3,2/3), (2/3,2/3,1/3) |
| T | 3 | (0,0,0), (1/3,2/3,1/3), (2/3,1/3,2/3) |
| F | 4 | (0,0,0), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0) |
The unusual lattice symbols S and T are necessary to allow for obverse and reverse settings for all of 3x, 3y, and 3z, respectively. Table 1.1. summarizes the relationsships.
| Table 1.1. | Lattice symbol | ||
|---|---|---|---|
| Unique axis | R | S | T |
| 3z | obverse | - | reverse |
| 3y | reverse | obverse | - |
| 3x | - | reverse | obverse |
The symbol T specifies the translation elements of a Seitz matrix. Alphabetical symbols (column 1 below) specify translations along a fixed direction. Numerical symbols (column 3 below) 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 | 1/2,0,0 | 1 in 31 | 1/3 |
| b | 0,1/2,0 | 2 in 32 | 2/3 |
| c | 0,0,1/2 | 1 in 41 | 1/4 |
| n | 1/2,1/2,1/2 | 3 in 43 | 3/4 |
| u | 1/4,0,0 | 1 in 61 | 1/6 |
| v | 0,1/4,0 | 2 in 62 | 1/3 |
| w | 0,0,1/4 | 4 in 64 | 2/3 |
| d | 1/4,1/4,1/4 | 5 in 65 | 5/6 |
The 3x3 matrices for proper rotations along the three principal unit-cell directions. The matrices for improper rotations (-1, -2, -3, -4 and -6) are identical except that the signs are reversed.
Rotation Order: 1 2 3 4 6
Symbol
Axis A
( 1 0 0) ( 1 0 0) ( 1 0 0) ( 1 0 0) ( 1 0 0)
a x ( 0 1 0) ( 0 -1 0) ( 0 0 -1) ( 0 0 -1) ( 0 1 -1)
( 0 0 1) ( 0 0 -1) ( 0 1 -1) ( 0 1 0) ( 0 1 0)
( 1 0 0) ( -1 0 0) ( -1 0 1) ( 0 0 1) ( 0 0 1)
b y ( 0 1 0) ( 0 1 0) ( 0 1 0) ( 0 1 0) ( 0 1 0)
( 0 0 1) ( 0 0 -1) ( -1 0 0) ( -1 0 0) ( -1 0 1)
( 1 0 0) ( -1 0 0) ( 0 -1 0) ( 0 -1 0) ( 1 -1 0)
c z ( 0 1 0) ( 0 -1 0) ( 1 -1 0) ( 1 0 0) ( 1 0 0)
( 0 0 1) ( 0 0 1) ( 0 0 1) ( 0 0 1) ( 0 0 1)
The symbols for face-diagonal 2-fold rotations are 2' and 2". The face-diagonal axis direction is determined by the axis of the preceding rotation Nx, Ny or Nz. Note that the single quote symbol ' is the default and may be omitted.
Preceding
rotation: Nx Ny Nz
Notation: 2' 2" 2' 2" 2' 2"
Axis: b-c b+c a-c a+c a-b a+b
(-1 0 0) (-1 0 0) ( 0 0 -1) ( 0 0 1) ( 0 -1 0) ( 0 1 0)
( 0 0 -1) ( 0 0 1) ( 0 -1 0) ( 0 -1 0) (-1 0 0) ( 1 0 0)
( 0 -1 0) ( 0 1 0) (-1 0 0) ( 1 0 0) ( 0 0 -1) ( 0 0 -1)
The symbol for the 3-fold 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 Notation
( 0 0 1)
a+b+c 3* ( 1 0 0)
( 0 1 0)
The codes appended to space-group numbers listed in column 1 of Table 6 identify the relationship of the symmetry elements to the crystal cell. The appended codes are separated from the space-group number by a colon. When a code is omitted the first listed choice applies.
Monoclinic code = <unique axis><cell choice>
Unique axis choices(+ b -b c -c a -a
Cell choices(+ 1 2 3
Orthorhombic code = <origin choice><setting>
Origin choices 1 2
Setting choices(+ abc ba-c cab -cba bca a-cb
Tetragonal, Cubic code = <origin choice>
Origin choices 1 2
Trigonal code = <cell choice>
Cell choices H (hex) R (rhomb)
(+ cf. IT Vol. A 1983 Table 4.3.1
Number Hermann-Mauguin Hall
------ --------------- ----
1 P 1 P 1
2 P -1 -P 1
3:b P 1 2 1 P 2y
3:c P 1 1 2 P 2
3:a P 2 1 1 P 2x
4:b P 1 21 1 P 2yb
4:c P 1 1 21 P 2c
4:a P 21 1 1 P 2xa
5:b1 C 1 2 1 C 2y
5:b2 A 1 2 1 A 2y
5:b3 I 1 2 1 I 2y
5:c1 A 1 1 2 A 2
5:c2 B 1 1 2 B 2
5:c3 I 1 1 2 I 2
5:a1 B 2 1 1 B 2x
5:a2 C 2 1 1 C 2x
5:a3 I 2 1 1 I 2x
6:b P 1 m 1 P -2y
6:c P 1 1 m P -2
6:a P m 1 1 P -2x
7:b1 P 1 c 1 P -2yc
7:b2 P 1 n 1 P -2yac
7:b3 P 1 a 1 P -2ya
7:c1 P 1 1 a P -2a
7:c2 P 1 1 n P -2ab
7:c3 P 1 1 b P -2b
7:a1 P b 1 1 P -2xb
7:a2 P n 1 1 P -2xbc
7:a3 P c 1 1 P -2xc
8:b1 C 1 m 1 C -2y
8:b2 A 1 m 1 A -2y
8:b3 I 1 m 1 I -2y
8:c1 A 1 1 m A -2
8:c2 B 1 1 m B -2
8:c3 I 1 1 m I -2
8:a1 B m 1 1 B -2x
8:a2 C m 1 1 C -2x
8:a3 I m 1 1 I -2x
9:b1 C 1 c 1 C -2yc
9:b2 A 1 n 1 A -2yac
9:b3 I 1 a 1 I -2ya
9:-b1 A 1 a 1 A -2ya
9:-b2 C 1 n 1 C -2ybc
9:-b3 I 1 c 1 I -2yc
9:c1 A 1 1 a A -2a
9:c2 B 1 1 n B -2bc
9:c3 I 1 1 b I -2b
9:-c1 B 1 1 b B -2b
9:-c2 A 1 1 n A -2ac
9:-c3 I 1 1 a I -2a
9:a1 B b 1 1 B -2xb
9:a2 C n 1 1 C -2xbc
9:a3 I c 1 1 I -2xc
9:-a1 C c 1 1 C -2xc
9:-a2 B n 1 1 B -2xbc
9:-a3 I b 1 1 I -2xb
10:b P 1 2/m 1 -P 2y
10:c P 1 1 2/m -P 2
10:a P 2/m 1 1 -P 2x
11:b P 1 21/m 1 -P 2yb
11:c P 1 1 21/m -P 2c
11:a P 21/m 1 1 -P 2xa
12:b1 C 1 2/m 1 -C 2y
12:b2 A 1 2/m 1 -A 2y
12:b3 I 1 2/m 1 -I 2y
12:c1 A 1 1 2/m -A 2
12:c2 B 1 1 2/m -B 2
12:c3 I 1 1 2/m -I 2
12:a1 B 2/m 1 1 -B 2x
12:a2 C 2/m 1 1 -C 2x
12:a3 I 2/m 1 1 -I 2x
13:b1 P 1 2/c 1 -P 2yc
13:b2 P 1 2/n 1 -P 2yac
13:b3 P 1 2/a 1 -P 2ya
13:c1 P 1 1 2/a -P 2a
13:c2 P 1 1 2/n -P 2ab
13:c3 P 1 1 2/b -P 2b
13:a1 P 2/b 1 1 -P 2xb
13:a2 P 2/n 1 1 -P 2xbc
13:a3 P 2/c 1 1 -P 2xc
14:b1 P 1 21/c 1 -P 2ybc
14:b2 P 1 21/n 1 -P 2yn
14:b3 P 1 21/a 1 -P 2yab
14:c1 P 1 1 21/a -P 2ac
14:c2 P 1 1 21/n -P 2n
14:c3 P 1 1 21/b -P 2bc
14:a1 P 21/b 1 1 -P 2xab
14:a2 P 21/n 1 1 -P 2xn
14:a3 P 21/c 1 1 -P 2xac
15:b1 C 1 2/c 1 -C 2yc
15:b2 A 1 2/n 1 -A 2yac
15:b3 I 1 2/a 1 -I 2ya
15:-b1 A 1 2/a 1 -A 2ya
15:-b2 C 1 2/n 1 -C 2ybc
15:-b3 I 1 2/c 1 -I 2yc
15:c1 A 1 1 2/a -A 2a
15:c2 B 1 1 2/n -B 2bc
15:c3 I 1 1 2/b -I 2b
15:-c1 B 1 1 2/b -B 2b
15:-c2 A 1 1 2/n -A 2ac
15:-c3 I 1 1 2/a -I 2a
15:a1 B 2/b 1 1 -B 2xb
15:a2 C 2/n 1 1 -C 2xbc
15:a3 I 2/c 1 1 -I 2xc
15:-a1 C 2/c 1 1 -C 2xc
15:-a2 B 2/n 1 1 -B 2xbc
15:-a3 I 2/b 1 1 -I 2xb
16 P 2 2 2 P 2 2
17 P 2 2 21 P 2c 2
17:cab P 21 2 2 P 2a 2a
17:bca P 2 21 2 P 2 2b
18 P 21 21 2 P 2 2ab
18:cab P 2 21 21 P 2bc 2
18:bca P 21 2 21 P 2ac 2ac
19 P 21 21 21 P 2ac 2ab
20 C 2 2 21 C 2c 2
20:cab A 21 2 2 A 2a 2a
20:bca B 2 21 2 B 2 2b
21 C 2 2 2 C 2 2
21:cab A 2 2 2 A 2 2
21:bca B 2 2 2 B 2 2
22 F 2 2 2 F 2 2
23 I 2 2 2 I 2 2
24 I 21 21 21 I 2b 2c
25 P m m 2 P 2 -2
25:cab P 2 m m P -2 2
25:bca P m 2 m P -2 -2
26 P m c 21 P 2c -2
26:ba-c P c m 21 P 2c -2c
26:cab P 21 m a P -2a 2a
26:-cba P 21 a m P -2 2a
26:bca P b 21 m P -2 -2b
26:a-cb P m 21 b P -2b -2
27 P c c 2 P 2 -2c
27:cab P 2 a a P -2a 2
27:bca P b 2 b P -2b -2b
28 P m a 2 P 2 -2a
28:ba-c P b m 2 P 2 -2b
28:cab P 2 m b P -2b 2
28:-cba P 2 c m P -2c 2
28:bca P c 2 m P -2c -2c
28:a-cb P m 2 a P -2a -2a
29 P c a 21 P 2c -2ac
29:ba-c P b c 21 P 2c -2b
29:cab P 21 a b P -2b 2a
29:-cba P 21 c a P -2ac 2a
29:bca P c 21 b P -2bc -2c
29:a-cb P b 21 a P -2a -2ab
30 P n c 2 P 2 -2bc
30:ba-c P c n 2 P 2 -2ac
30:cab P 2 n a P -2ac 2
30:-cba P 2 a n P -2ab 2
30:bca P b 2 n P -2ab -2ab
30:a-cb P n 2 b P -2bc -2bc
31 P m n 21 P 2ac -2
31:ba-c P n m 21 P 2bc -2bc
31:cab P 21 m n P -2ab 2ab
31:-cba P 21 n m P -2 2ac
31:bca P n 21 m P -2 -2bc
31:a-cb P m 21 n P -2ab -2
32 P b a 2 P 2 -2ab
32:cab P 2 c b P -2bc 2
32:bca P c 2 a P -2ac -2ac
33 P n a 21 P 2c -2n
33:ba-c P b n 21 P 2c -2ab
33:cab P 21 n b P -2bc 2a
33:-cba P 21 c n P -2n 2a
33:bca P c 21 n P -2n -2ac
33:a-cb P n 21 a P -2ac -2n
34 P n n 2 P 2 -2n
34:cab P 2 n n P -2n 2
34:bca P n 2 n P -2n -2n
35 C m m 2 C 2 -2
35:cab A 2 m m A -2 2
35:bca B m 2 m B -2 -2
36 C m c 21 C 2c -2
36:ba-c C c m 21 C 2c -2c
36:cab A 21 m a A -2a 2a
36:-cba A 21 a m A -2 2a
36:bca B b 21 m B -2 -2b
36:a-cb B m 21 b B -2b -2
37 C c c 2 C 2 -2c
37:cab A 2 a a A -2a 2
37:bca B b 2 b B -2b -2b
38 A m m 2 A 2 -2
38:ba-c B m m 2 B 2 -2
38:cab B 2 m m B -2 2
38:-cba C 2 m m C -2 2
38:bca C m 2 m C -2 -2
38:a-cb A m 2 m A -2 -2
39 A b m 2 A 2 -2c
39:ba-c B m a 2 B 2 -2c
39:cab B 2 c m B -2c 2
39:-cba C 2 m b C -2b 2
39:bca C m 2 a C -2b -2b
39:a-cb A c 2 m A -2c -2c
40 A m a 2 A 2 -2a
40:ba-c B b m 2 B 2 -2b
40:cab B 2 m b B -2b 2
40:-cba C 2 c m C -2c 2
40:bca C c 2 m C -2c -2c
40:a-cb A m 2 a A -2a -2a
41 A b a 2 A 2 -2ac
41:ba-c B b a 2 B 2 -2bc
41:cab B 2 c b B -2bc 2
41:-cba C 2 c b C -2bc 2
41:bca C c 2 a C -2bc -2bc
41:a-cb A c 2 a A -2ac -2ac
42 F m m 2 F 2 -2
42:cab F 2 m m F -2 2
42:bca F m 2 m F -2 -2
43 F d d 2 F 2 -2d
43:cab F 2 d d F -2d 2
43:bca F d 2 d F -2d -2d
44 I m m 2 I 2 -2
44:cab I 2 m m I -2 2
44:bca I m 2 m I -2 -2
45 I b a 2 I 2 -2c
45:cab I 2 c b I -2a 2
45:bca I c 2 a I -2b -2b
46 I m a 2 I 2 -2a
46:ba-c I b m 2 I 2 -2b
46:cab I 2 m b I -2b 2
46:-cba I 2 c m I -2c 2
46:bca I c 2 m I -2c -2c
46:a-cb I m 2 a I -2a -2a
47 P m m m -P 2 2
48:1 P n n n:1 P 2 2 -1n
48:2 P n n n:2 -P 2ab 2bc
49 P c c m -P 2 2c
49:cab P m a a -P 2a 2
49:bca P b m b -P 2b 2b
50:1 P b a n:1 P 2 2 -1ab
50:2 P b a n:2 -P 2ab 2b
50:1cab P n c b:1 P 2 2 -1bc
50:2cab P n c b:2 -P 2b 2bc
50:1bca P c n a:1 P 2 2 -1ac
50:2bca P c n a:2 -P 2a 2c
51 P m m a -P 2a 2a
51:ba-c P m m b -P 2b 2
51:cab P b m m -P 2 2b
51:-cba P c m m -P 2c 2c
51:bca P m c m -P 2c 2
51:a-cb P m a m -P 2 2a
52 P n n a -P 2a 2bc
52:ba-c P n n b -P 2b 2n
52:cab P b n n -P 2n 2b
52:-cba P c n n -P 2ab 2c
52:bca P n c n -P 2ab 2n
52:a-cb P n a n -P 2n 2bc
53 P m n a -P 2ac 2
53:ba-c P n m b -P 2bc 2bc
53:cab P b m n -P 2ab 2ab
53:-cba P c n m -P 2 2ac
53:bca P n c m -P 2 2bc
53:a-cb P m a n -P 2ab 2
54 P c c a -P 2a 2ac
54:ba-c P c c b -P 2b 2c
54:cab P b a a -P 2a 2b
54:-cba P c a a -P 2ac 2c
54:bca P b c b -P 2bc 2b
54:a-cb P b a b -P 2b 2ab
55 P b a m -P 2 2ab
55:cab P m c b -P 2bc 2
55:bca P c m a -P 2ac 2ac
56 P c c n -P 2ab 2ac
56:cab P n a a -P 2ac 2bc
56:bca P b n b -P 2bc 2ab
57 P b c m -P 2c 2b
57:ba-c P c a m -P 2c 2ac
57:cab P m c a -P 2ac 2a
57:-cba P m a b -P 2b 2a
57:bca P b m a -P 2a 2ab
57:a-cb P c m b -P 2bc 2c
58 P n n m -P 2 2n
58:cab P m n n -P 2n 2
58:bca P n m n -P 2n 2n
59:1 P m m n:1 P 2 2ab -1ab
59:2 P m m n:2 -P 2ab 2a
59:1cab P n m m:1 P 2bc 2 -1bc
59:2cab P n m m:2 -P 2c 2bc
59:1bca P m n m:1 P 2ac 2ac -1ac
59:2bca P m n m:2 -P 2c 2a
60 P b c n -P 2n 2ab
60:ba-c P c a n -P 2n 2c
60:cab P n c a -P 2a 2n
60:-cba P n a b -P 2bc 2n
60:bca P b n a -P 2ac 2b
60:a-cb P c n b -P 2b 2ac
61 P b c a -P 2ac 2ab
61:ba-c P c a b -P 2bc 2ac
62 P n m a -P 2ac 2n
62:ba-c P m n b -P 2bc 2a
62:cab P b n m -P 2c 2ab
62:-cba P c m n -P 2n 2ac
62:bca P m c n -P 2n 2a
62:a-cb P n a m -P 2c 2n
63 C m c m -C 2c 2
63:ba-c C c m m -C 2c 2c
63:cab A m m a -A 2a 2a
63:-cba A m a m -A 2 2a
63:bca B b m m -B 2 2b
63:a-cb B m m b -B 2b 2
64 C m c a -C 2bc 2
64:ba-c C c m b -C 2bc 2bc
64:cab A b m a -A 2ac 2ac
64:-cba A c a m -A 2 2ac
64:bca B b c m -B 2 2bc
64:a-cb B m a b -B 2bc 2
65 C m m m -C 2 2
65:cab A m m m -A 2 2
65:bca B m m m -B 2 2
66 C c c m -C 2 2c
66:cab A m a a -A 2a 2
66:bca B b m b -B 2b 2b
67 C m m a -C 2b 2
67:ba-c C m m b -C 2b 2b
67:cab A b m m -A 2c 2c
67:-cba A c m m -A 2 2c
67:bca B m c m -B 2 2c
67:a-cb B m a m -B 2c 2
68:1 C c c a:1 C 2 2 -1bc
68:2 C c c a:2 -C 2b 2bc
68:1ba-c C c c b:1 C 2 2 -1bc
68:2ba-c C c c b:2 -C 2b 2c
68:1cab A b a a:1 A 2 2 -1ac
68:2cab A b a a:2 -A 2a 2c
68:1-cba A c a a:1 A 2 2 -1ac
68:2-cba A c a a:2 -A 2ac 2c
68:1bca B b c b:1 B 2 2 -1bc
68:2bca B b c b:2 -B 2bc 2b
68:1a-cb B b a b:1 B 2 2 -1bc
68:2a-cb B b a b:2 -B 2b 2bc
69 F m m m -F 2 2
70:1 F d d d:1 F 2 2 -1d
70:2 F d d d:2 -F 2uv 2vw
71 I m m m -I 2 2
72 I b a m -I 2 2c
72:cab I m c b -I 2a 2
72:bca I c m a -I 2b 2b
73 I b c a -I 2b 2c
73:ba-c I c a b -I 2a 2b
74 I m m a -I 2b 2
74:ba-c I m m b -I 2a 2a
74:cab I b m m -I 2c 2c
74:-cba I c m m -I 2 2b
74:bca I m c m -I 2 2a
74:a-cb I m a m -I 2c 2
75 P 4 P 4
76 P 41 P 4w
77 P 42 P 4c
78 P 43 P 4cw
79 I 4 I 4
80 I 41 I 4bw
81 P -4 P -4
82 I -4 I -4
83 P 4/m -P 4
84 P 42/m -P 4c
85:1 P 4/n:1 P 4ab -1ab
85:2 P 4/n:2 -P 4a
86:1 P 42/n:1 P 4n -1n
86:2 P 42/n:2 -P 4bc
87 I 4/m -I 4
88:1 I 41/a:1 I 4bw -1bw
88:2 I 41/a:2 -I 4ad
89 P 4 2 2 P 4 2
90 P 42 1 2 P 4ab 2ab
91 P 41 2 2 P 4w 2c
92 P 41 21 2 P 4abw 2nw
93 P 42 2 2 P 4c 2
94 P 42 21 2 P 4n 2n
95 P 43 2 2 P 4cw 2c
96 P 43 21 2 P 4nw 2abw
97 I 4 2 2 I 4 2
98 I 41 2 2 I 4bw 2bw
99 P 4 m m P 4 -2
100 P 4 b m P 4 -2ab
101 P 42 c m P 4c -2c
102 P 42 n m P 4n -2n
103 P 4 c c P 4 -2c
104 P 4 n c P 4 -2n
105 P 42 m c P 4c -2
106 P 42 b c P 4c -2ab
107 I 4 m m I 4 -2
108 I 4 c m I 4 -2c
109 I 41 m d I 4bw -2
110 I 41 c d I 4bw -2c
111 P -4 2 m P -4 2
112 P -4 2 c P -4 2c
113 P -4 21 m P -4 2ab
114 P -4 21 c P -4 2n
115 P -4 m 2 P -4 -2
116 P -4 c 2 P -4 -2c
117 P -4 b 2 P -4 -2ab
118 P -4 n 2 P -4 -2n
119 I -4 m 2 I -4 -2
120 I -4 c 2 I -4 -2c
121 I -4 2 m I -4 2
122 I -4 2 d I -4 2bw
123 P 4/m m m -P 4 2
124 P 4/m c c -P 4 2c
125:1 P 4/n b m:1 P 4 2 -1ab
125:2 P 4/n b m:2 -P 4a 2b
126:1 P 4/n n c:1 P 4 2 -1n
126:2 P 4/n n c:2 -P 4a 2bc
127 P 4/m b m -P 4 2ab
128 P 4/m n c -P 4 2n
129:1 P 4/n m m:1 P 4ab 2ab -1ab
129:2 P 4/n m m:2 -P 4a 2a
130:1 P 4/n c c:1 P 4ab 2n -1ab
130:2 P 4/n c c:2 -P 4a 2ac
131 P 42/m m c -P 4c 2
132 P 42/m c m -P 4c 2c
133:1 P 42/n b c:1 P 4n 2c -1n
133:2 P 42/n b c:2 -P 4ac 2b
134:1 P 42/n n m:1 P 4n 2 -1n
134:2 P 42/n n m:2 -P 4ac 2bc
135 P 42/m b c -P 4c 2ab
136 P 42/m n m -P 4n 2n
137:1 P 42/n m c:1 P 4n 2n -1n
137:2 P 42/n m c:2 -P 4ac 2a
138:1 P 42/n c m:1 P 4n 2ab -1n
138:2 P 42/n c m:2 -P 4ac 2ac
139 I 4/m m m -I 4 2
140 I 4/m c m -I 4 2c
141:1 I 41/a m d:1 I 4bw 2bw -1bw
141:2 I 41/a m d:2 -I 4bd 2
142:1 I 41/a c d:1 I 4bw 2aw -1bw
142:2 I 41/a c d:2 -I 4bd 2c
143 P 3 P 3
144 P 31 P 31
145 P 32 P 32
146:H R 3:H R 3
146:R R 3:R P 3*
147 P -3 -P 3
148:H R -3:H -R 3
148:R R -3:R -P 3*
149 P 3 1 2 P 3 2
150 P 3 2 1 P 3 2"
151 P 31 1 2 P 31 2c (0 0 1)
152 P 31 2 1 P 31 2"
153 P 32 1 2 P 32 2c (0 0 -1)
154 P 32 2 1 P 32 2"
155:H R 32:H R 3 2"
155:R R 32:R P 3* 2
156 P 3 m 1 P 3 -2"
157 P 3 1 m P 3 -2
158 P 3 c 1 P 3 -2"c
159 P 3 1 c P 3 -2c
160:H R 3 m:H R 3 -2"
160:R R 3 m:R P 3* -2
161:H R 3 c:H R 3 -2"c
161:R R 3 c:R P 3* -2n
162 P -3 1 m -P 3 2
163 P -3 1 c -P 3 2c
164 P -3 m 1 -P 3 2"
165 P -3 c 1 -P 3 2"c
166:H R -3 m:H -R 3 2"
166:R R -3 m:R -P 3* 2
167:H R -3 c:H -R 3 2"c
167:R R -3 c:R -P 3* 2n
168 P 6 P 6
169 P 61 P 61
170 P 65 P 65
171 P 62 P 62
172 P 64 P 64
173 P 63 P 6c
174 P -6 P -6
175 P 6/m -P 6
176 P 63/m -P 6c
177 P 6 2 2 P 6 2
178 P 61 2 2 P 61 2 (0 0 -1)
179 P 65 2 2 P 65 2 (0 0 1)
180 P 62 2 2 P 62 2c (0 0 1)
181 P 64 2 2 P 64 2c (0 0 -1)
182 P 63 2 2 P 6c 2c
183 P 6 m m P 6 -2
184 P 6 c c P 6 -2c
185 P 63 c m P 6c -2
186 P 63 m c P 6c -2c
187 P -6 m 2 P -6 2
188 P -6 c 2 P -6c 2
189 P -6 2 m P -6 -2
190 P -6 2 c P -6c -2c
191 P 6/m m m -P 6 2
192 P 6/m c c -P 6 2c
193 P 63/m c m -P 6c 2
194 P 63/m m c -P 6c 2c
195 P 2 3 P 2 2 3
196 F 2 3 F 2 2 3
197 I 2 3 I 2 2 3
198 P 21 3 P 2ac 2ab 3
199 I 21 3 I 2b 2c 3
200 P m -3 -P 2 2 3
201:1 P n -3:1 P 2 2 3 -1n
201:2 P n -3:2 -P 2ab 2bc 3
202 F m -3 -F 2 2 3
203:1 F d -3:1 F 2 2 3 -1d
203:2 F d -3:2 -F 2uv 2vw 3
204 I m -3 -I 2 2 3
205 P a -3 -P 2ac 2ab 3
206 I a -3 -I 2b 2c 3
207 P 4 3 2 P 4 2 3
208 P 42 3 2 P 4n 2 3
209 F 4 3 2 F 4 2 3
210 F 41 3 2 F 4d 2 3
211 I 4 3 2 I 4 2 3
212 P 43 3 2 P 4acd 2ab 3
213 P 41 3 2 P 4bd 2ab 3
214 I 41 3 2 I 4bd 2c 3
215 P -4 3 m P -4 2 3
216 F -4 3 m F -4 2 3
217 I -4 3 m I -4 2 3
218 P -4 3 n P -4n 2 3
219 F -4 3 c F -4c 2 3
220 I -4 3 d I -4bd 2c 3
221 P m -3 m -P 4 2 3
222:1 P n -3 n:1 P 4 2 3 -1n
222:2 P n -3 n:2 -P 4a 2bc 3
223 P m -3 n -P 4n 2 3
224:1 P n -3 m:1 P 4n 2 3 -1n
224:2 P n -3 m:2 -P 4bc 2bc 3
225 F m -3 m -F 4 2 3
226 F m -3 c -F 4c 2 3
227:1 F d -3 m:1 F 4d 2 3 -1d
227:2 F d -3 m:2 -F 4vw 2vw 3
228:1 F d -3 c:1 F 4d 2 3 -1cd
228:2 F d -3 c:2 -F 4cvw 2vw 3
229 I m -3 m -I 4 2 3
230 I a -3 d -I 4bd 2c 3