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 ≤ 12 ;
0 ≤ z ≤ 12 ;
x≤y
Symmetry operations
(1)
(5)
(9)
(13)
1
2(0, 12 , 0) 14 , y, 14
1̄ 0, 0, 0
n( 12 , 0, 12 ) x, 14 , z
(2)
(6)
(10)
(14)
2 0, 0, z
2( 12 , 0, 0) x, 14 , 14
m x, y, 0
n(0, 12 , 12 ) 14 , y, z
(3)
(7)
(11)
(15)
Copyright © 2006 International Union of Crystallography
4+ (0, 0, 12 ) 0, 12 , z
2 x, x, 0
4̄+ 12 , 0, z; 12 , 0, 14
m x, x̄, z
468
(4)
(8)
(12)
(16)
4− (0, 0, 12 ) 12 , 0, z
2 x, x̄, 0
4̄− 0, 12 , z; 0, 12 , 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 + 12 (4) y + 12 , x̄ + 12 , z + 12
1
1
1
1
1
1
x̄ + 2 , y + 2 , z̄ + 2 (6) x + 2 , ȳ + 2 , z̄ + 2 (7) y, x, z̄
(8) ȳ, x̄, z̄
x̄, ȳ, z̄
(10) x, y, z̄
(11) y + 12 , x̄ + 12 , z̄ + 12 (12) ȳ + 12 , x + 12 , z̄ + 12
x + 12 , ȳ + 12 , z + 12 (14) x̄ + 12 , y + 12 , z + 12 (15) ȳ, x̄, z
(16) y, x, z
0kl : k + l = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
8
j
..m
x, x, z
x̄ + 12 , x + 12 , z̄ + 12
x̄ + 12 , x + 12 , z + 12
x, x, z̄
8
i
m..
x, y, 0
x̄ + 12 , y + 12 , 12
8
h
2..
0, 12 , z
0, 12 , z̄
0, 12 , z + 12
0, 12 , z̄ + 12
4
g
m . 2m
x, x̄, 0
x̄, x, 0
x + 12 , x + 12 , 12
x̄ + 12 , x̄ + 12 , 12
no extra conditions
4
f
m . 2m
x, x, 0
x̄, x̄, 0
x̄ + 12 , x + 12 , 12
x + 12 , x̄ + 12 , 12
no extra conditions
4
e
2 . mm
0, 0, z
1
2
0, 0, z̄
hkl : h + k + l = 2n
4
d
4̄ . .
0, 12 , 14
0, 12 , 34
1
2
, 0, 14
1
2
, 0, 34
hkl : h + k, l = 2n
4
c
2/m . .
0, 12 , 0
0, 12 , 12
1
2
, 0, 12
1
2
, 0, 0
hkl : h + k, l = 2n
2
b
m . mm
0, 0, 12
1
2
, 12 , 0
hkl : h + k + l = 2n
2
a
m . mm
0, 0, 0
1
2
, 12 , 12
hkl : h + k + l = 2n
x̄, x̄, z
x + 12 , x̄ + 12 , z̄ + 12
ȳ + 12 , x + 12 , 12
y, x, 0
x̄, ȳ, 0
x + 12 , ȳ + 12 , 12
1
2
1
2
, 12 , z + 12
1
2
, 0, z̄ + 12
, 0, z + 12
, 12 , z̄ + 12
1
2
1
2
x + 12 , x̄ + 12 , z + 12 no extra conditions
x̄, x̄, z̄
y + 12 , x̄ + 12 , 12
ȳ, x̄, 0
, 0, z̄
, 0, z
no extra conditions
hkl : h + k, l = 2n
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
a = 12 (−a + b)
b = c
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
none
I
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 = 12 c) (127)
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