1. Space groups
Ex: 222 1 C2
C2'
C2"

2. Space groups G Ex: 222 1 C2 C2' C2" 1t 2t 3t
represents all elements in
the lattice translation group

3. Space groups G Ex: 222 1 C2 C2' C2" 1t 2t 3t
represents all elements in
the lattice translation group
C2 C2' = C2"
1t 2t = 3t

4. Space groups G Ex: 222 1 C2 C2' C2" 1t 2t 3t
represents all elements in
the lattice translation group
C2 C2' = C2"
1t 2t = 3t
C2,t1 C2',t2 = C2", t3

5. Space groups Ex: 222 C2 C2' = C2" 1t 2t = 3t C2,t1 C2',t2 = C2", t3 x
a/2
b/2
c/2 ––
a/2+b/2 y 2t z 3t
P222
P212'2
P2'212
P2'2'2
P22'21
P21212

6. Space groups Ex: 422 1 C4 C42 C43 C2 C2' C2" C2"' C2"' t4 C2" t3 C2'

7. Space groups G Ex: 422 tz tz2 tz3 t1 t2 t3 t4 C2"' t4 C2" t3 C2' t2 C2

8. Space groups G Ex: 422 tz tz2 tz3 t1 t2 t3 t4 tz = c/4, 2c/4, 3c/4, 0
[110] G tz
tz2
tz3 t1 t2 t3 t4
C2"' t4
C2" t3
[010]
C2' t2
[110] C2 t1
[100]
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t1 = t2

9. Space groups Ex: 422 P422 - tz = ti = 0 tz = c/4, 2c/4, 3c/4, 0
[110]
C2"' t4
C2" t3
[010]
C2' t2
[110] C2 t1
[100]
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t1 = t2

10. Space groups Ex: 422 P422 - tz = ti = 0 tz = c/4, 2c/4, 3c/4, 0
[110]
C2"' t4
C2" t3
[010]
C2' t2
[110] C2 t1
[100]
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t1 = t2

11. Space groups Ex: 422 P4212 - tz = 0, t1 = t2 = a/2 + b/2
[110]
C2"' t4
C2" t3
[010]
C2' t2
[110] C2 t1
[100]
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t1 = t2

12. Space groups Ex: 422 P4212 - tz = 0, t1 = t2 = a/2 + b/2
[110]
C2"' t4
C2" t3
[010]
C2' t2
[110] C2 t1
[100]
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t1 = t2

13. Space groups Ex: 422 P4122 - tz = c/4, t3 = 0, t4 = c/4
[110]
C2"' t4
C2" t3
[010]
C2' t2
[110] C2 t1
[100]
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t3 = t4 = c/4
tz t4 = t1 = c/2
tz t1 = t2 = 3c/4

14. Space groups Ex: 422 P4122 - tz = c/4, t3 = 0, t4 = c/4
[110]
C2"' t4
1/4
1/4
1/4
3/8
1/8
3/8
1/8
3/8
1/8
C2" t3
[010]
1/8
3/8
3/8
1/8
C2' t2
[110] C2 t1
1/8
3/8
[100]
3/8
1/8
tz = c/4, 2c/4, 3c/4, 0
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
1/8
3/8
1/8
3/8
1/8
3/8
1/4
1/4
1/4
tz t3 = t4 = c/4
tz t4 = t1 = c/2
tz t1 = t2 = 3c/4

15. Space groups G Pmm2 & Cmm2 1 xm ym z2 1t 2t 3t 32 zt Xm ym = z2
smt1 smt2 = Aπ, t3

16. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2
t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = 0
1t = 2t = 3t = Cmm2
1t 2t = 3t
G consists of translations for C lattice (includes a/2 + b/2)

17. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2 t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = 0
1t = a/2 3t = t = a/2 Cmm2
1t 2t = 3t

18. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2 t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = 0
1t = a/2 + b/2 3t = t = a/2 + b/ Cmm2
1t 2t = 3t

19. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2 t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = 0
1t = c/2 3t = t = c/ Ccc2
1t 2t = 3t

20. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2 t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = 0
1t = a/2 + c/2 3t = t = a/2 + c/ Ccc2
1t 2t = 3t

21. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2 t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = c/2
1t = t = c/ t = c/ Cmc21
1t 2t = 3t

22. Space groups t1 can only be 0, b/2, c/2, b/2 + c/2 t1 = 0, b/2, c/2
t2 = 0, a/2, c/2
t3 = 0, c/2
1t w/ 3t = c/2
1t = t = c/ t = c/ Cmc21
1t 2t = 3t
All other translation combinations equiv-alent to Cmm2, Ccc2, or Cmc21