International Tables for Crystallography (2006). Vol. A, Space group 136, pp. 468–469.
P 42/m n m
D144h
No. 136
P 42/m 21/n 2/m
4/m m m
Tetragonal
Patterson symmetry P 4/m m m
Origin at centre (m m m) at 2/m 1 2/m
Asymmetric unit
0 ≤ x ≤ 12 ;
0 ≤ y ≤ 21 ;
0 ≤ z ≤ 12 ;
x≤y
Symmetry operations
(1)
(5)
(9)
(13)
1
2(0, 12 , 0) 14 , y, 41
1̄ 0, 0, 0
n( 12 , 0, 21 ) x, 14 , z
(2)
(6)
(10)
(14)
2 0, 0, z
2( 12 , 0, 0) x, 41 , 14
m x, y, 0
n(0, 21 , 12 ) 41 , y, z
(3)
(7)
(11)
(15)
Copyright  2006 International Union of Crystallography
4+ (0, 0, 21 ) 0, 21 , z
2 x, x, 0
4̄+ 21 , 0, z; 12 , 0, 41
m x, x̄, z
468
(4)
(8)
(12)
(16)
4− (0, 0, 21 ) 21 , 0, z
2 x, x̄, 0
4̄− 0, 21 , z; 0, 21 , 14
m x, x, z
P 42/m n m
No. 136
CONTINUED
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5); (9)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
16
k
1
Reflection conditions
General:
(1)
(5)
(9)
(13)
x, y, z
(2) x̄, ȳ, z
(3) ȳ + 12 , x + 12 , z + 21 (4) y + 12 , x̄ + 12 , z + 12
1
1
1
1
1
1
(8) ȳ, x̄, z̄
x̄ + 2 , y + 2 , z̄ + 2 (6) x + 2 , ȳ + 2 , z̄ + 2 (7) y, x, z̄
x̄, ȳ, z̄
(10) x, y, z̄
(11) y + 12 , x̄ + 12 , z̄ + 21 (12) ȳ + 12 , x + 12 , z̄ + 21
(16) y, x, z
x + 12 , ȳ + 12 , z + 21 (14) x̄ + 12 , y + 12 , z + 12 (15) ȳ, x̄, z
0kl : k + l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8
j
..m
x, x, z
x̄ + 21 , x + 12 , z̄ + 21
x̄ + 12 , x + 12 , z + 21
x, x, z̄
8
i
m..
x, y, 0
x̄ + 12 , y + 21 , 12
8
h
2..
0, 21 , z
0, 12 , z̄
0, 21 , z + 12
0, 21 , z̄ + 21
4
g
m . 2m
x, x̄, 0
x̄, x, 0
x + 12 , x + 12 , 12
x̄ + 12 , x̄ + 12 , 21
no extra conditions
4
f
m . 2m
x, x, 0
x̄, x̄, 0
x̄ + 12 , x + 12 , 12
x + 12 , x̄ + 12 , 21
no extra conditions
4
e
2 . mm
0, 0, z
1
2
0, 0, z̄
hkl : h + k + l = 2n
4
d
4̄ . .
0, 21 , 14
0, 21 , 34
1
2
, 0, 41
1
2
, 0, 43
hkl : h + k, l = 2n
4
c
2/m . .
0, 12 , 0
0, 12 , 12
1
2
, 0, 21
1
2
, 0, 0
hkl : h + k, l = 2n
2
b
m . mm
0, 0, 12
1
2
, 21 , 0
hkl : h + k + l = 2n
2
a
m . mm
0, 0, 0
1
2
, 21 , 12
hkl : h + k + l = 2n
x̄, x̄, z
x + 12 , x̄ + 12 , z̄ + 21
ȳ + 21 , x + 12 , 12
y, x, 0
x̄, ȳ, 0
x + 12 , ȳ + 21 , 21
1
2
1
2
, 21 , z + 12
1
2
, 0, z̄ + 21
, 0, z + 21
, 21 , z̄ + 12
1
2
1
2
x + 12 , x̄ + 12 , z + 21 no extra conditions
x̄, x̄, z̄
y + 12 , x̄ + 21 , 21
ȳ, x̄, 0
no extra conditions
hkl : h + k, l = 2n
, 0, z̄
, 0, z
Symmetry of special projections
Along [001] p 4 g m
a′ = a
b′ = b
Origin at 0, 12 , z
Along [110] p 2 m m
b′ = c
a′ = 21 (−a + b)
Origin at x, x, 0
Along [100] c 2 m m
b′ = c
a′ = b
Origin at x, 0, 0
Maximal non-isomorphic subgroups
I
[2] P 4̄ n 2 (118)
1; 2; 7; 8; 11; 12; 13; 14
IIa
IIb
[2] P 4̄ 21 m (113)
[2] P 42 n m (102)
[2] P 42 21 2 (94)
[2] P 42 /m 1 1 (P42 /m, 84)
[2] P 2/m 1 2/m (C m m m, 65)
[2] P 2/m 21/n 1 (Pn n m, 58)
none
none
1;
1;
1;
1;
1;
1;
2;
2;
2;
2;
2;
2;
5;
3;
3;
3;
7;
5;
6;
4;
4;
4;
8;
6;
11; 12; 15; 16
13; 14; 15; 16
5; 6; 7; 8
9; 10; 11; 12
9; 10; 15; 16
9; 10; 13; 14
Maximal isomorphic subgroups of lowest index
IIc [3] P 42 /m n m (c′ = 3c) (136); [9] P 42 /m n m (a′ = 3a, b′ = 3b) (136)
Minimal non-isomorphic supergroups
I
none
II
[2] C 42 /m c m (P42 /m m c, 131); [2] I 4/m m m (139); [2] P 4/m b m (c′ = 21 c) (127)
469
International Tables for Crystallography (2006). Vol. A, Space group 164, pp. 540–541.
P 3̄ m 1
D33d
No. 164
P 3̄ 2/m 1
3̄ m 1
Patterson symmetry P 3̄ m 1
Origin at centre (3̄ m 1)
Asymmetric unit
Vertices
0 ≤ x ≤ 23 ; 0 ≤ y ≤ 31 ;
0, 0, 0 12 , 0, 0 32 , 13 , 0
0, 0, 1 12 , 0, 1 32 , 13 , 1
0 ≤ z ≤ 1;
x ≤ (1 + y)/2;
Symmetry operations
(1)
(4)
(7)
(10)
1
2 x, x, 0
1̄ 0, 0, 0
m x, x̄, z
(2)
(5)
(8)
(11)
3+
2
3̄+
m
0, 0, z
x, 0, 0
0, 0, z; 0, 0, 0
x, 2x, z
(3)
(6)
(9)
(12)
Trigonal
3−
2
3̄−
m
Copyright  2006 International Union of Crystallography
0, 0, z
0, y, 0
0, 0, z; 0, 0, 0
2x, x, z
540
y ≤ x/2
No. 164
CONTINUED
P 3̄ m 1
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4); (7)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
12
j
1
Reflection conditions
General:
(1)
(4)
(7)
(10)
x, y, z
y, x, z̄
x̄, ȳ, z̄
ȳ, x̄, z
(2)
(5)
(8)
(11)
(3)
(6)
(9)
(12)
ȳ, x − y, z
x − y, ȳ, z̄
y, x̄ + y, z̄
x̄ + y, y, z
x̄ + y, x̄, z
x̄, x̄ + y, z̄
x − y, x, z̄
x, x − y, z
no conditions
Special: no extra conditions
6
i
.m.
x, x̄, z
x, 2x, z
2x̄, x̄, z
6
h
.2.
x, 0, 12
0, x, 21
x̄, x̄, 12
x̄, 0, 12
0, x̄, 21
x, x, 12
6
g
.2.
x, 0, 0
0, x, 0
x̄, x̄, 0
x̄, 0, 0
0, x̄, 0
x, x, 0
3
f
. 2/m .
1
2
, 0, 21
0, 12 , 21
1
2
, 12 , 12
3
e
. 2/m .
1
2
, 0, 0
0, 12 , 0
1
2
, 21 , 0
2
d
3m.
1
3
, 23 , z
2
3
2
c
3m.
0, 0, z
1
b
3̄ m .
0, 0, 21
1
a
3̄ m .
0, 0, 0
2x, x, z̄
x̄, x, z̄
x̄, 2x̄, z̄
, 13 , z̄
0, 0, z̄
Symmetry of special projections
Along [001] p 6 m m
a′ = a
b′ = b
Origin at 0, 0, z
Along [100] p 2
a′ = 12 (a + 2b)
Origin at x, 0, 0
b′ = c
Maximal non-isomorphic subgroups
I
[2] P 3 m 1 (156)
1; 2; 3; 10; 11; 12
[2] P 3 2 1 (150)
1; 2; 3; 4; 5; 6
[2] P 3̄ 1 1 (P 3̄, 147)
1; 2; 3; 7; 8; 9
[3] P 1 2/m 1 (C 2/m, 12) 1; 4; 7; 10
[3] P 1 2/m 1 (C 2/m, 12) 1; 5; 7; 11
[3] P 1 2/m 1 (C 2/m, 12) 1; 6; 7; 12
IIa none
IIb [2] P 3̄ c 1 (c′ = 2c) (165); [3] H 3̄ m 1 (a′ = 3a, b′ = 3b) (P 3̄ 1 m, 162)
Maximal isomorphic subgroups of lowest index
IIc [2] P 3̄ m 1 (c′ = 2c) (164); [4] P 3̄ m 1 (a′ = 2a, b′ = 2b) (164)
Minimal non-isomorphic supergroups
I
[2] P 6/m m m (191); [2] P 63 /m m c (194)
II
[3] H 3̄ m 1 (P 3̄ 1 m, 162); [3] R 3̄ m (obverse) (166); [3] R 3̄ m (reverse) (166)
541
Along [210] p 2 m m
a′ = 21 b
b′ = c
Origin at x, 12 x, 0
International Tables for Crystallography (2006). Vol. A, Space group 166, pp. 544–547.
R 3̄ m
D53d
No. 166
R 3̄ 2/m
3̄ m
Patterson symmetry R 3̄ m
HEXAGONAL AXES
Origin at centre (3̄ m)
Asymmetric unit
Vertices
0 ≤ x ≤ 23 ; 0 ≤ y ≤ 32 ;
0, 0, 0 23 , 31 , 0 31 , 23 , 0
0, 0, 61 23 , 31 , 16 31 , 23 , 16
0 ≤ z ≤ 16 ;
Copyright  2006 International Union of Crystallography
Trigonal
x ≤ 2y;
544
y ≤ min(1 − x, 2x)
No. 166
CONTINUED
R 3̄ m
Symmetry operations
For (0, 0, 0)+ set
(1) 1
(4) 2 x, x, 0
(7) 1̄ 0, 0, 0
(10) m x, x̄, z
(2)
(5)
(8)
(11)
3+
2
3̄+
m
0, 0, z
x, 0, 0
0, 0, z; 0, 0, 0
x, 2x, z
(3)
(6)
(9)
(12)
3−
2
3̄−
m
For ( 23 , 13 , 13 )+ set
(1) t( 32 , 13 , 31 )
(4) 2( 21 , 12 , 0) x, x − 16 , 61
(7) 1̄ 13 , 16 , 16
(10) g( 16 , − 61 , 31 ) x + 12 , x̄, z
(2)
(5)
(8)
(11)
3+ (0, 0, 13 ) 31 , 13 , z
2( 12 , 0, 0) x, 16 , 61
3̄+ 13 , − 31 , z; 13 , − 31 , 16
g( 16 , 31 , 13 ) x + 41 , 2x, z
(3)
(6)
(9)
(12)
3− (0, 0, 13 ) 31 , 0, z
2 13 , y, 61
3̄− 31 , 23 , z; 13 , 32 , 16
g( 23 , 13 , 31 ) 2x, x, z
For ( 13 , 23 , 23 )+ set
(1) t( 31 , 23 , 32 )
(4) 2( 12 , 12 , 0) x, x + 16 , 31
(7) 1̄ 61 , 13 , 13
(10) g(− 61 , 61 , 32 ) x + 12 , x̄, z
(2)
(5)
(8)
(11)
3+ (0, 0, 23 ) 0, 13 , z
2 x, 31 , 13
3̄+ 23 , 31 , z; 32 , 13 , 13
g( 13 , 32 , 23 ) x, 2x, z
(3)
(6)
(9)
(12)
3− (0, 0, 23 ) 31 , 31 , z
2(0, 21 , 0) 16 , y, 31
3̄− − 31 , 13 , z; − 13 , 31 , 13
g( 13 , 16 , 32 ) 2x − 12 , x, z
0, 0, z
0, y, 0
0, 0, z; 0, 0, 0
2x, x, z
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 32 , 13 , 31 ); (2); (4); (7)
Positions
36
i
1
Reflection conditions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
(0, 0, 0)+ ( 32 , 13 , 31 )+
(1)
(4)
(7)
(10)
x, y, z
y, x, z̄
x̄, ȳ, z̄
ȳ, x̄, z
(2)
(5)
(8)
(11)
( 13 , 32 , 23 )+
(3)
(6)
(9)
(12)
ȳ, x − y, z
x − y, ȳ, z̄
y, x̄ + y, z̄
x̄ + y, y, z
General:
x̄ + y, x̄, z
x̄, x̄ + y, z̄
x − y, x, z̄
x, x − y, z
hkil :
hki0 :
hh2hl :
hh̄0l :
000l :
hh̄00 :
−h + k + l = 3n
−h + k = 3n
l = 3n
h + l = 3n
l = 3n
h = 3n
Special: no extra conditions
18
h
.m
x, x̄, z
x, 2x, z
2x̄, x̄, z
18
g
.2
x, 0, 12
0, x, 21
x̄, x̄, 12
x̄, 0, 12
0, x̄, 21
x, x, 12
18
f
.2
x, 0, 0
0, x, 0
x̄, x̄, 0
x̄, 0, 0
0, x̄, 0
x, x, 0
9
e
. 2/m
1
2
, 0, 0
0, 12 , 0
1
2
, 21 , 0
9
d
. 2/m
1
2
, 0, 21
0, 21 , 21
1
2
, 21 , 12
6
c
3m
0, 0, z
0, 0, z̄
3
b
3̄ m
0, 0, 21
3
a
3̄ m
0, 0, 0
2x, x, z̄
x̄, x, z̄
x̄, 2x̄, z̄
Symmetry of special projections
Along [001] p 6 m m
a′ = 31 (2a + b)
b′ = 31 (−a + b)
Origin at 0, 0, z
Along [100] p 2
a′ = 12 (a + 2b)
Origin at x, 0, 0
b′ = 31 (−a − 2b + c)
545
Along [210] p 2 m m
a′ = 12 b
b′ = 31 c
1
Origin at x, 2 x, 0
R 3̄ m
No. 166
CONTINUED
HEXAGONAL AXES
Maximal non-isomorphic subgroups
I
[2] R 3 m (160)
(1; 2; 3; 10; 11; 12)+
[2] R 3 2 (155)
[2] R 3̄ 1 (R 3̄, 148)
[3] R 1 2/m (C 2/m, 12)
[3] R 1 2/m (C 2/m, 12)
[3] R 1 2/m (C 2/m, 12)
⎧
IIa ⎨ [3] P 3̄ m 1 (164)
[3] P 3̄ m 1 (164)
⎩
[3] P 3̄ m 1 (164)
IIb
(1;
(1;
(1;
(1;
(1;
2;
2;
4;
5;
6;
3;
3;
7;
7;
7;
4; 5; 6)+
7; 8; 9)+
10)+
11)+
12)+
1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
1; 2; 3; 10; 11; 12; (4; 5; 6; 7; 8; 9) + ( 32 , 13 , 13 )
1; 2; 3; 10; 11; 12; (4; 5; 6; 7; 8; 9) + ( 13 , 23 , 23 )
[2] R 3̄ c (a′ = −a, b′ = −b, c′ = 2c) (167)
Maximal isomorphic subgroups of lowest index
IIc [2] R 3̄ m (a′ = −a, b′ = −b, c′ = 2c) (166); [4] R 3̄ m (a′ = −2a, b′ = −2b) (166)
Minimal non-isomorphic supergroups
I
[4] P m 3̄ m (221); [4] P n 3̄ m (224); [4] F m 3̄ m (225); [4] F d 3̄m (227); [4] I m 3̄ m (229)
II
[3] P 3̄ 1 m (a′ = 31 (2a + b), b′ = 31 (−a + b), c′ = 13 c) (162)
RHOMBOHEDRAL AXES
Maximal non-isomorphic subgroups
I
[2] R 3 m (160)
1; 2; 3; 10; 11; 12
[2] R 3 2 (155)
1; 2; 3; 4; 5; 6
[2] R 3̄ 1 (R 3̄, 148)
1; 2; 3; 7; 8; 9
[3] R 1 2/m (C 2/m, 12) 1; 4; 7; 10
[3] R 1 2/m (C 2/m, 12) 1; 5; 7; 11
[3] R 1 2/m (C 2/m, 12) 1; 6; 7; 12
IIa none
IIb [2] F 3̄ c (a′ = 2a, b′ = 2b, c′ = 2c) (R 3̄ c, 167); [3] P 3̄ m 1 (a′ = a − b, b′ = b − c, c′ = a + b + c) (164)
Maximal isomorphic subgroups of lowest index
IIc [2] R 3̄ m (a′ = b + c, b′ = a + c, c′ = a + b) (166); [4] R 3̄ m (a′ = −a + b + c, b′ = a − b + c, c′ = a + b − c) (166)
Minimal non-isomorphic supergroups
I
[4] P m 3̄ m (221); [4] P n 3̄ m (224); [4] F m 3̄ m (225); [4] F d 3̄m (227); [4] I m 3̄ m (229)
II
[3] P 3̄ 1 m (a′ = 31 (2a − b − c), b′ = 13 (−a + 2b − c), c′ = 31 (a + b + c)) (162)
546
Trigonal
3̄ m
Patterson symmetry R 3̄ m
D53d
R 3̄ m
R 3̄ 2/m
No. 166
RHOMBOHEDRAL AXES
(For drawings see hexagonal axes)
Origin at centre (3̄ m)
Asymmetric unit
0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 12 ;
0, 0, 0 1, 0, 0 1, 1, 0 12 , 21 , 12
Vertices
y ≤ x;
z ≤ min(y, 1 − x)
Symmetry operations
(1)
(4)
(7)
(10)
1
2 x̄, 0, x
1̄ 0, 0, 0
m x, y, x
(2)
(5)
(8)
(11)
3+
2
3̄+
m
(3)
(6)
(9)
(12)
x, x, x
x, x̄, 0
x, x, x; 0, 0, 0
x, x, z
3−
2
3̄−
m
x, x, x
0, y, ȳ
x, x, x; 0, 0, 0
x, y, y
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4); (7)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
12
i
1
Reflection conditions
General:
(1)
(4)
(7)
(10)
x, y, z
z̄, ȳ, x̄
x̄, ȳ, z̄
z, y, x
(2)
(5)
(8)
(11)
(3)
(6)
(9)
(12)
z, x, y
ȳ, x̄, z̄
z̄, x̄, ȳ
y, x, z
no conditions
y, z, x
x̄, z̄, ȳ
ȳ, z̄, x̄
x, z, y
Special: no extra conditions
6
h
.m
x, x, z
6
g
.2
x, x̄, 12
1
2
6
f
.2
x, x̄, 0
3
e
. 2/m
0, 21 , 12
3
d
. 2/m
1
2
, 0, 0
0, 12 , 0
2
c
3m
x, x, x
x̄, x̄, x̄
1
b
3̄ m
1
2
1
a
3̄ m
0, 0, 0
z, x, x
x, z, x
z̄, x̄, x̄
, x, x̄
x̄, 21 , x
x̄, x, 12
1
2
, x̄, x
x, 12 , x̄
0, x, x̄
x̄, 0, x
x̄, x, 0
0, x̄, x
x, 0, x̄
1
2
, 0, 21
1
2
x̄, x̄, z̄
x̄, z̄, x̄
, 21 , 0
0, 0, 21
, 21 , 12
Symmetry of special projections
Along [111] p 6 m m
a′ = 31 (2a − b − c)
Origin at x, x, x
b′ = 13 (−a + 2b − c)
Along [11̄0] p 2
a′ = 21 (a + b − 2c)
Origin at x, x̄, 0
(Continued on preceding page)
547
b′ = c
Along [21̄1̄] p 2 m m
a′ = 12 (b − c)
b′ = 31 (a + b + c)
Origin at 2x, x̄, x̄
International Tables for Crystallography (2006). Vol. A, Space group 167, pp. 548–551.
R 3̄ c
D63d
No. 167
R 3̄ 2/c
3̄ m
Patterson symmetry R 3̄ m
HEXAGONAL AXES
Origin at centre (3̄) at 3̄ c
Asymmetric unit
Vertices
0 ≤ x ≤ 23 ; 0 ≤ y ≤ 32 ;
0, 0, 0 12 , 0, 0 32 , 13 , 0
0, 0, 121 12 , 0, 121 32 , 13 , 121
0 ≤ z ≤ 121 ; x ≤ (1 + y)/2;
1 2
0, 21 , 0
3, 3,0
1 2 1
0, 21 , 121
3 , 3 , 12
Copyright  2006 International Union of Crystallography
Trigonal
548
y ≤ min(1 − x, (1 + x)/2)
No. 167
CONTINUED
R 3̄ c
Symmetry operations
For (0, 0, 0)+ set
(1) 1
(4) 2 x, x, 14
(7) 1̄ 0, 0, 0
(10) c x, x̄, z
(2)
(5)
(8)
(11)
3+
2
3̄+
c
0, 0, z
x, 0, 14
0, 0, z; 0, 0, 0
x, 2x, z
(3)
(6)
(9)
(12)
3−
2
3̄−
c
For ( 23 , 13 , 13 )+ set
(1) t( 32 , 13 , 31 )
(4) 2( 21 , 12 , 0) x, x − 16 , 125
(7) 1̄ 13 , 16 , 16
(10) g( 16 , − 61 , 65 ) x + 12 , x̄, z
(2)
(5)
(8)
(11)
3+ (0, 0, 13 ) 31 , 13 , z
2( 12 , 0, 0) x, 16 , 125
3̄+ 13 , − 31 , z; 13 , − 31 , 16
g( 16 , 31 , 56 ) x + 41 , 2x, z
(3)
(6)
(9)
(12)
3− (0, 0, 13 ) 31 , 0, z
2 13 , y, 125
3̄− 31 , 23 , z; 13 , 32 , 16
g( 23 , 13 , 65 ) 2x, x, z
For ( 13 , 23 , 23 )+ set
(1) t( 31 , 23 , 32 )
(4) 2( 12 , 12 , 0) x, x + 16 , 121
(7) 1̄ 61 , 13 , 13
(10) g(− 61 , 61 , 61 ) x + 12 , x̄, z
(2)
(5)
(8)
(11)
3+ (0, 0, 23 ) 0, 13 , z
2 x, 31 , 121
3̄+ 23 , 31 , z; 32 , 13 , 13
g( 13 , 32 , 16 ) x, 2x, z
(3)
(6)
(9)
(12)
3− (0, 0, 23 ) 31 , 31 , z
2(0, 21 , 0) 16 , y, 121
3̄− − 31 , 13 , z; − 13 , 31 , 13
g( 13 , 16 , 61 ) 2x − 12 , x, z
0, 0, z
0, y, 41
0, 0, z; 0, 0, 0
2x, x, z
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t( 32 , 13 , 31 ); (2); (4); (7)
Positions
36
f
1
Reflection conditions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
(0, 0, 0)+ ( 32 , 13 , 31 )+
(1)
(4)
(7)
(10)
(2)
(5)
(8)
(11)
x, y, z
y, x, z̄ + 21
x̄, ȳ, z̄
ȳ, x̄, z + 21
( 13 , 32 , 23 )+
General:
(3)
(6)
(9)
(12)
ȳ, x − y, z
x − y, ȳ, z̄ + 21
y, x̄ + y, z̄
x̄ + y, y, z + 21
x̄ + y, x̄, z
x̄, x̄ + y, z̄ + 12
x − y, x, z̄
x, x − y, z + 12
hkil :
hki0 :
hh2hl :
hh̄0l :
000l :
hh̄00 :
−h + k + l = 3n
−h + k = 3n
l = 3n
h + l = 3n, l = 2n
l = 6n
h = 3n
Special: as above, plus
18
e
.2
x, 0, 14
0, x, 14
18
d
1̄
1
2
, 0, 0
0, 21 , 0
12
c
3.
0, 0, z
0, 0, z̄ + 12
6
b
3̄ .
0, 0, 0
0, 0, 12
hkil : l = 2n
6
a
32
0, 0, 14
0, 0, 34
hkil : l = 2n
x̄, x̄, 41
1
2
, 12 , 0
0, 0, z̄
x̄, 0, 34
0, x̄, 34
0, 12 , 21
1
2
, 0, 21
x, x, 43
1
2
, 12 , 21
0, 0, z + 12
no extra conditions
hkil : l = 2n
hkil : l = 2n
Symmetry of special projections
Along [001] p 6 m m
a′ = 13 (2a + b)
b′ = 31 (−a + b)
Origin at 0, 0, z
Along [100] p 2
a′ = 61 (2a + 4b + c)
Origin at x, 0, 0
549
b′ = 16 (−a − 2b + c)
Along [210] p 2 g m
a′ = 21 b
b′ = 13 c
1
Origin at x, 2 x, 0
R 3̄ c
No. 167
CONTINUED
HEXAGONAL AXES
Maximal non-isomorphic subgroups
I
[2] R 3 c (161)
(1; 2; 3; 10; 11; 12)+
[2] R 3 2 (155)
[2] R 3̄ 1 (R 3̄, 148)
[3] R 1 2/c (C 2/c, 15)
[3] R 1 2/c (C 2/c, 15)
[3] R 1 2/c (C 2/c, 15)
⎧
IIa ⎨ [3] P 3̄ c 1 (165)
[3] P 3̄ c 1 (165)
⎩
[3] P 3̄ c 1 (165)
IIb none
(1;
(1;
(1;
(1;
(1;
2;
2;
4;
5;
6;
3;
3;
7;
7;
7;
4; 5; 6)+
7; 8; 9)+
10)+
11)+
12)+
1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12
1; 2; 3; 10; 11; 12; (4; 5; 6; 7; 8; 9) + ( 32 , 31 , 13 )
1; 2; 3; 10; 11; 12; (4; 5; 6; 7; 8; 9) + ( 31 , 32 , 23 )
Maximal isomorphic subgroups of lowest index
IIc [4] R 3̄ c (a′ = −2a, b′ = −2b) (167); [5] R 3̄ c (a′ = −a, b′ = −b, c′ = 5c) (167)
Minimal non-isomorphic supergroups
I
[4] P n 3̄ n (222); [4] P m 3̄ n (223); [4] F m 3̄ c (226); [4] F d 3̄c (228); [4] I a 3̄ d (230)
II
[2] R 3̄ m (a′ = −a, b′ = −b, c′ = 12 c) (166); [3] P 3̄ 1 c (a′ = 13 (2a + b), b′ = 31 (−a + b), c′ = 13 c) (163)
RHOMBOHEDRAL AXES
Maximal non-isomorphic subgroups
I
[2] R 3 c (161)
1; 2; 3; 10; 11; 12
[2] R 3 2 (155)
1; 2; 3; 4; 5; 6
[2] R 3̄ 1 (R 3̄, 148)
1; 2; 3; 7; 8; 9
[3] R 1 2/c (C 2/c, 15) 1; 4; 7; 10
[3] R 1 2/c (C 2/c, 15) 1; 5; 7; 11
[3] R 1 2/c (C 2/c, 15) 1; 6; 7; 12
IIa none
IIb [3] P 3̄ c 1 (a′ = a − b, b′ = b − c, c′ = a + b + c) (165)
Maximal isomorphic subgroups of lowest index
IIc [4] R 3̄ c (a′ = −a + b + c, b′ = a − b + c, c′ = a + b − c) (167); [5] R 3̄ c (a′ = a + 2b + 2c, b′ = 2a + b + 2c, c′ = 2a + 2b + c) (167)
Minimal non-isomorphic supergroups
I
[4] P n 3̄ n (222); [4] P m 3̄ n (223); [4] F m 3̄ c (226); [4] F d 3̄c (228); [4] I a 3̄ d (230)
II
[2] R 3̄ m (a′ = 12 (−a + b + c), b′ = 21 (a − b + c), c′ = 12 (a + b − c)) (166);
[3] P 3̄ 1 c (a′ = 31 (2a − b − c), b′ = 31 (−a + 2b − c), c′ = 31 (a + b + c)) (163)
550
Trigonal
D63d
3̄ m
R 3̄ c
R 3̄ 2/c
Patterson symmetry R 3̄ m
No. 167
RHOMBOHEDRAL AXES
(For drawings see hexagonal axes)
Origin at centre (3̄) at 3̄c
Asymmetric unit
1
4
≤ x ≤ 54 ; 41 ≤ y ≤ 45 ; 41 ≤ z ≤ 34 ;
5 1 1
5 5 1
3 3 3
1 1 1
4, 4, 4
4, 4, 4
4, 4, 4
4, 4, 4
Vertices
y ≤ x;
z ≤ min(y, 23 − x)
Symmetry operations
(1)
(4)
(7)
(10)
1
2 x̄ + 12 , 14 , x
1̄ 0, 0, 0
n( 21 , 12 , 21 ) x, y, x
3+ x, x, x
2 x, x̄ + 12 , 41
3̄+ x, x, x; 0, 0, 0
n( 12 , 21 , 12 ) x, x, z
(2)
(5)
(8)
(11)
(3)
(6)
(9)
(12)
3− x, x, x
2 14 , y + 21 , ȳ
3̄− x, x, x; 0, 0, 0
n( 21 , 12 , 21 ) x, y, y
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4); (7)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
12
f
1
Reflection conditions
General:
(1)
(4)
(7)
(10)
x, y, z
z̄ + 12 , ȳ + 21 , x̄ + 12
x̄, ȳ, z̄
z + 12 , y + 21 , x + 12
(2)
(5)
(8)
(11)
(3)
(6)
(9)
(12)
z, x, y
ȳ + 21 , x̄ + 12 , z̄ + 12
z̄, x̄, ȳ
y + 21 , x + 12 , z + 12
y, z, x
x̄ + 12 , z̄ + 12 , ȳ + 12
ȳ, z̄, x̄
x + 12 , z + 12 , y + 12
hhl : l = 2n
hhh : h = 2n
Special: as above, plus
6
e
.2
x, x̄ + 21 , 41
x̄, x + 21 , 43
6
d
1̄
1
2
4
c
3.
x, x, x
2
b
3̄ .
0, 0, 0
2
a
32
1
4
, 0, 0
, 14 , 41
1
4
3
4
, x, x̄ + 21
, x̄, x + 21
0, 21 , 0
x̄ + 12 , 14 , x
x + 12 , 34 , x̄
0, 0, 12
x̄ + 12 , x̄ + 21 , x̄ + 12
1
2
no extra conditions
, 12 , 0
x̄, x̄, x̄
1
2
, 0, 21
0, 12 , 21
x + 21 , x + 12 , x + 12
hkl : h + k + l = 2n
hkl : h + k + l = 2n
1
2
, 21 , 12
hkl : h + k + l = 2n
3
4
, 43 , 34
hkl : h + k + l = 2n
Symmetry of special projections
Along [111] p 6 m m
a′ = 31 (2a − b − c)
Origin at x, x, x
b′ = 13 (−a + 2b − c)
Along [11̄0] p 2
a′ = 21 (a + b − 2c)
Origin at x, x̄, 0
(Continued on preceding page)
551
b′ = 12 c
Along [21̄1̄] p 2 g m
a′ = 21 (b − c)
b′ = 13 (a + b + c)
Origin at 2x, x̄, x̄
International Tables for Crystallography (2006). Vol. A, Space group 186, pp. 584–585.
P 63 m c
C6v4
No. 186
P 63 m c
6mm
Patterson symmetry P 6/m m m
Origin on 3 m 1 on 63 m c
Asymmetric unit
Vertices
0 ≤ x ≤ 23 ; 0 ≤ y ≤ 31 ;
0, 0, 0 12 , 0, 0 32 , 13 , 0
0, 0, 1 12 , 0, 1 32 , 13 , 1
x ≤ (1 + y)/2;
0 ≤ z ≤ 1;
y ≤ x/2
Symmetry operations
(1)
(4)
(7)
(10)
1
2(0, 0, 21 ) 0, 0, z
m x, x̄, z
c x, x, z
(2)
(5)
(8)
(11)
3+ 0, 0, z
6− (0, 0, 12 ) 0, 0, z
m x, 2x, z
c x, 0, z
Hexagonal
(3)
(6)
(9)
(12)
Copyright  2006 International Union of Crystallography
3− 0, 0, z
6+ (0, 0, 21 ) 0, 0, z
m 2x, x, z
c 0, y, z
584
No. 186
CONTINUED
P 63 m c
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4); (7)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
12
Reflection conditions
General:
d
1
(1)
(4)
(7)
(10)
6
c
.m.
x, x̄, z
2
b
3m.
1
3
2
a
3m.
0, 0, z
(2)
(5)
(8)
(11)
x, y, z
x̄, ȳ, z + 12
ȳ, x̄, z
y, x, z + 12
(3)
(6)
(9)
(12)
ȳ, x − y, z
y, x̄ + y, z + 21
x̄ + y, y, z
x − y, ȳ, z + 21
x̄ + y, x̄, z
x − y, x, z + 12
x, x − y, z
x̄, x̄ + y, z + 12
hh2hl : l = 2n
000l : l = 2n
Special: as above, plus
, 32 , z
x, 2x, z
2
3
2x̄, x̄, z
x̄, x, z +
x̄, 2x̄, z +
1
2
, 31 , z + 21
1
2
2x, x, z +
1
2
no extra conditions
hkil : l = 2n
or h − k = 3n + 1
or h − k = 3n + 2
hkil : l = 2n
0, 0, z + 21
Symmetry of special projections
Along [001] p 6 m m
a′ = a
b′ = b
Origin at 0, 0, z
Along [100] p 1 g 1
a′ = 12 (a + 2b)
b′ = c
Origin at x, 0, 0
Maximal non-isomorphic subgroups
I
[2] P 63 1 1 (P 63 , 173)
1; 2; 3; 4; 5; 6
[2] P 3 1 c (159)
1; 2; 3; 10; 11; 12
[2] P 3 m 1 (156)
1; 2; 3; 7; 8; 9
[3] P 21 m c (C m c 21 , 36) 1; 4; 7; 10
[3] P 21 m c (C m c 21 , 36) 1; 4; 8; 11
[3] P 21 m c (C m c 21 , 36) 1; 4; 9; 12
IIa none
IIb [3] H 63 m c (a′ = 3a, b′ = 3b) (P 63 c m, 185)
Maximal isomorphic subgroups of lowest index
IIc [3] P 63 m c (c′ = 3c) (186); [4] P 63 m c (a′ = 2a, b′ = 2b) (186)
Minimal non-isomorphic supergroups
I
[2] P 63 /m m c (194)
II
[3] H 63 m c (P 63 c m, 185); [2] P 6 m m (c′ = 21 c) (183)
585
Along [210] p 1 m 1
a′ = 12 b
b′ = 21 c
1
Origin at x, 2 x, 0
International Tables for Crystallography (2006). Vol. A, Space group 216, pp. 658–660.
F 4̄ 3 m
Td2
No. 216
F 4̄ 3 m
4̄ 3 m
Patterson symmetry F m 3̄ m
Origin at 4̄ 3 m
Asymmetric unit
Vertices
0 ≤ x ≤ 12 ; 0 ≤ y ≤ 41 ; − 41 ≤ z ≤ 14 ;
0, 0, 0 12 , 0, 0 41 , 14 , 14 41 , 41 , − 41
Copyright  2006 International Union of Crystallography
Cubic
y ≤ min(x, 12 − x);
658
−y ≤ z ≤ y
No. 216
CONTINUED
Symmetry operations
For (0,0,0)+ set
(1) 1
(5) 3+ x,x,x
(9) 3− x,x,x
(13) m x,x,z
(17) m x,y,y
(21) m x,y,x
(2)
(6)
(10)
(14)
(18)
(22)
2
3+
3−
m
4̄+
4̄−
For (0, 12 , 12 )+ set
(1) t(0, 21 , 12 )
(5) 3+ ( 31 , 13 , 13 ) x− 31 ,x− 16 ,x
(9) 3− ( 31 , 13 , 13 ) x− 61 ,x+ 16 ,x
(13) g( 14 , 41 , 12 ) x− 41 ,x,z
(17) g(0, 21 , 12 ) x,y,y
(21) g( 41 , 21 , 14 ) x− 41 ,y,x
(2)
(6)
(10)
(14)
(18)
(22)
For ( 21 ,0, 21 )+ set
(1) t( 12 ,0, 21 )
(5) 3+ ( 31 , 13 , 13 ) x+ 61 ,x− 16 ,x
(9) 3− ( 13 , 13 , 13 ) x− 61 ,x− 13 ,x
(13) g( 41 , 41 , 12 ) x+ 41 ,x,z
(17) g( 12 , 41 , 14 ) x,y− 41 ,y
(21) g( 21 ,0, 12 ) x,y,x
For ( 21 , 12 ,0)+ set
(1) t( 21 , 21 ,0)
(5) 3+ ( 31 , 13 , 13 ) x+ 61 ,x+ 13 ,x
(9) 3− ( 13 , 13 , 13 ) x+ 31 ,x+ 16 ,x
(13) g( 12 , 21 ,0) x,x,z
(17) g( 12 , 41 , 14 ) x,y+ 41 ,y
(21) g( 41 , 21 , 14 ) x+ 41 ,y,x
0,0,z
x̄,x, x̄
x, x̄, x̄
x, x̄,z
x,0,0; 0,0,0
0,y,0; 0,0,0
(3)
(7)
(11)
(15)
(19)
(23)
2
3+
3−
4̄+
4̄−
m
2(0,0, 21 ) 0, 41 ,z
3+ x̄,x+ 12 , x̄
3− (− 13 , 31 , 13 ) x+ 16 , x̄+ 61 , x̄
g(− 41 , 41 , 12 ) x+ 14 , x̄,z
4̄+ x, 12 ,0; 0, 21 ,0
4̄− 14 ,y, 41 ; 14 , 41 , 14
(3)
(7)
(11)
(15)
(19)
(23)
2(0, 21 ,0) 0,y, 41
3+ (− 13 , 31 , 31 ) x+ 13 , x̄− 61 , x̄
3− x̄+ 12 , x̄+ 12 ,x
4̄+ 41 , 14 ,z; 41 , 14 , 41
4̄− x,0, 12 ; 0,0, 21
g(− 41 , 12 , 41 ) x̄+ 41 ,y,x
(2)
(6)
(10)
(14)
(18)
(22)
2(0,0, 21 ) 14 ,0,z
3+ ( 13 ,− 31 , 13 ) x̄+ 61 ,x+ 61 , x̄
3− x+ 21 , x̄, x̄
g( 14 ,− 41 , 12 ) x+ 14 , x̄,z
4̄+ x, 14 , 41 ; 14 , 41 , 14
4̄− 21 ,y,0; 21 ,0,0
(3)
(7)
(11)
(15)
(19)
(23)
2
3+
3−
4̄+
4̄−
m
(2)
(6)
(10)
(14)
(18)
(22)
2
3+
3−
m
4̄+
4̄−
(3)
(7)
(11)
(15)
(19)
(23)
2(0, 21 ,0) 14 ,y,0
3+ x+ 12 , x̄, x̄
3− ( 13 , 13 ,− 31 ) x̄+ 13 , x̄+ 61 ,x
4̄+ 21 ,0,z; 12 ,0,0
4̄− x, 14 , 14 ; 41 , 14 , 41
g( 41 , 21 ,− 41 ) x̄+ 41 ,y,x
1
4
, 41 ,z
x̄+ 21 ,x, x̄
x, x̄+ 12 , x̄
x+ 21 , x̄,z
x, 14 ,− 41 ; 41 , 14 ,− 41
1
1 1
1
1
4 ,y,− 4 ; 4 , 4 ,− 4
0,y,0
x, x̄, x̄
x̄, x̄,x
0,0,z; 0,0,0
x,0,0; 0,0,0
x̄,y,x
1
4
,y, 14
x+ 12 , x̄− 12 , x̄
x̄+ 12 , x̄,x
1
1
1 1
1
4 ,− 4 ,z; 4 ,− 4 , 4
1 1
1
x,− 4 , 4 ; 4 ,− 41 , 41
x̄+ 12 ,y,x
F 4̄ 3 m
(4)
(8)
(12)
(16)
(20)
(24)
2
3+
3−
4̄−
m
4̄+
x,0,0
x̄, x̄,x
x̄,x, x̄
0,0,z; 0,0,0
x,y,ȳ
0,y,0; 0,0,0
(4)
(8)
(12)
(16)
(20)
(24)
2
3+
3−
4̄−
m
4̄+
x, 14 , 14
x̄, x̄+ 12 ,x
x̄− 21 ,x+ 21 , x̄
− 41 , 14 ,z; − 14 , 41 , 14
x,y+ 21 ,ȳ
− 41 ,y, 41 ; − 14 , 41 , 14
(4)
(8)
(12)
(16)
(20)
(24)
2( 21 ,0,0) x,0, 14
3+ x̄+ 21 , x̄+ 21 ,x
3− ( 13 ,− 31 , 31 ) x̄− 61 ,x+ 13 , x̄
4̄− 41 , 14 ,z; 41 , 14 , 14
g( 12 ,− 41 , 14 ) x,y+ 41 ,ȳ
4̄+ 0,y, 21 ; 0,0, 21
(4)
(8)
(12)
(16)
(20)
(24)
2( 12 ,0,0) x, 41 ,0
3+ ( 13 , 31 ,− 31 ) x̄+ 61 , x̄+ 13 ,x
3− x̄,x+ 12 , x̄
4̄− 0, 21 ,z; 0, 21 ,0
g( 12 , 41 ,− 14 ) x,y+ 41 ,ȳ
4̄+ 41 ,y, 14 ; 41 , 14 , 14
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 12 , 21 ); t( 12 , 0, 21 ); (2); (3); (5); (13)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
96
i
1
Coordinates
(0, 0, 0)+ (0, 12 , 21 )+
(1)
(5)
(9)
(13)
(17)
(21)
x, y, z
z, x, y
y, z, x
y, x, z
x, z, y
z, y, x
(2)
(6)
(10)
(14)
(18)
(22)
Reflection conditions
( 12 , 0, 21 )+ ( 12 , 21 , 0)+
(3)
(7)
(11)
(15)
(19)
(23)
x̄, ȳ, z
z, x̄, ȳ
ȳ, z, x̄
ȳ, x̄, z
x̄, z, ȳ
z, ȳ, x̄
h, k, l permutable
General:
(4)
(8)
(12)
(16)
(20)
(24)
x̄, y, z̄
z̄, x̄, y
y, z̄, x̄
y, x̄, z̄
x̄, z̄, y
z̄, y, x̄
hkl :
0kl :
hhl :
h00 :
x, ȳ, z̄
z̄, x, ȳ
ȳ, z̄, x
ȳ, x, z̄
x, z̄, ȳ
z̄, ȳ, x
h + k, h + l, k + l = 2n
k, l = 2n
h + l = 2n
h = 2n
Special: no extra conditions
48
h
..m
x, x, z
z̄, x̄, x
x̄, x̄, z
z̄, x, x̄
24
g
2 . mm
x, 14 , 41
x̄, 34 , 41
1
4
24
f
2 . mm
x, 0, 0
x̄, 0, 0
0, x, 0
16
e
.3m
x, x, x
x̄, x̄, x
x̄, x, x̄
4
d
4̄ 3 m
3
4
, 34 , 43
4
c
4̄ 3 m
1
4
, 14 , 41
4
b
4̄ 3 m
1
2
, 12 , 21
4
a
4̄ 3 m
0, 0, 0
x̄, x, z̄
x, z, x
, x, 41
x, x̄, z̄
x̄, z, x̄
1
4
z, x, x
x, z̄, x̄
, x̄, 34
1
4
0, x̄, 0
x, x̄, x̄
659
, 14 , x
0, 0, x
z, x̄, x̄
x̄, z̄, x
3
4
, 41 , x̄
0, 0, x̄
F 4̄ 3 m
No. 216
CONTINUED
Symmetry of special projections
Along [001] p 4 m m
a′ = 12 a
b′ = 12 b
Origin at 0, 0, z
Along [111] p 3 1 m
a′ = 61 (2a − b − c)
Origin at x, x, x
b′ = 16 (−a + 2b − c)
Along [110] c 1 m 1
a′ = 12 (−a + b)
b′ = c
Origin at x, x, 0
Maximal non-isomorphic subgroups
I  [2] F 2 3 1 (F 2 3, 196)
(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+
 [3] F 4̄ 1 m (I 4̄ m 2, 119)
[3] F 4̄ 1 m (I 4̄ m 2, 119)

[3] F 4̄ 1 m (I 4̄ m 2, 119)

[4] F 1 3 m (R 3 m, 160)


[4] F 1 3 m (R 3 m, 160)

 [4] F 1 3 m (R 3 m, 160)
[4] F 1 3 m (R 3 m, 160)

[4] P 4̄ 3 m (215)
IIa 



[4]
P 4̄ 3 m (215)




[4] P 4̄ 3 m (215)






 [4] P 4̄ 3 m (215)


[4] P 4̄ 3 m (215)








[4] P 4̄ 3 m (215)


IIb
[4] P 4̄ 3 m (215)









 [4] P 4̄ 3 m (215)
(1;
(1;
(1;
(1;
(1;
(1;
(1;
2;
2;
2;
5;
6;
7;
8;
3; 4; 13; 14; 15; 16)+
3; 4; 17; 18; 19; 20)+
3; 4; 21; 22; 23; 24)+
9; 13; 17; 21)+
12; 14; 20; 21)+
10; 14; 17; 23)+
11; 13; 20; 23)+
1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24
1; 2; 3; 4; 13; 14; 15; 16; (9; 10; 11; 12; 17; 18; 19; 20) + (0, 21 , 21 ); (5; 6; 7; 8; 21; 22; 23;
24) + ( 21 , 0, 21 )
1; 2; 3; 4; 17; 18; 19; 20; (9; 10; 11; 12; 21; 22; 23; 24) + ( 21 , 0, 21 ); (5; 6; 7; 8; 13; 14; 15;
16) + ( 21 , 12 , 0)
1; 2; 3; 4; 21; 22; 23; 24; (5; 6; 7; 8; 17; 18; 19; 20) + (0, 21 , 21 ); (9; 10; 11; 12; 13; 14; 15;
16) + ( 21 , 12 , 0)
1; 5; 9; 13; 17; 21; (4; 6; 11; 15; 20; 22) + (0, 21 , 12 ); (3; 8; 10; 16; 18; 23) + ( 21 , 0, 21 ); (2; 7; 12;
14; 19; 24) + ( 21 , 12 , 0)
1; 6; 12; 14; 20; 21; (4; 5; 10; 16; 17; 22) + (0, 12 , 21 ); (3; 7; 11; 15; 19; 23) + ( 21 , 0, 21 ); (2; 8; 9;
13; 18; 24) + ( 21 , 12 , 0)
1; 7; 10; 14; 17; 23; (4; 8; 12; 16; 20; 24) + (0, 12 , 21 ); (3; 6; 9; 15; 18; 21) + ( 12 , 0, 21 ); (2; 5; 11;
13; 19; 22) + ( 21 , 12 , 0)
1; 8; 11; 13; 20; 23; (4; 7; 9; 15; 17; 24) + (0, 12 , 12 ); (3; 5; 12; 16; 19; 21) + ( 21 , 0, 21 ); (2; 6; 10;
14; 18; 22) + ( 21 , 12 , 0)
none
Maximal isomorphic subgroups of lowest index
IIc [27] F 4̄ 3 m (a′ = 3a, b′ = 3b, c′ = 3c) (216)
Minimal non-isomorphic supergroups
I
[2] F m 3̄ m (225); [2] F d 3̄ m (227)
II
[2] P 4̄ 3 m (a′ = 21 a, b′ = 12 b, c′ = 12 c) (215)
660