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