ORIGIN CHOICE 1, Origin at -4, at 0, -1/4, -1/8 from -1
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(1/2, 1/2, 1/2); (2); (3); (5)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
|
| | (0, 0, 0)+ (1/2, 1/2, 1/2)+ |
| | (1) x, y, z | (2) -x + 1/2, -y + 1/2, z + 1/2 | (3) -y, x + 1/2, z + 1/4 | (4) y + 1/2, -x, z + 3/4 | | (5) -x, -y + 1/2, -z + 1/4 | (6) x + 1/2, y, -z + 3/4 | (7) y, -x, -z | (8) -y + 1/2, x + 1/2, -z + 1/2 |
|
I Maximal translationengleiche subgroups
| [2] I-4 (82) | (1; 2; 7; 8)+ |
|
|
| [2] I41 (80) | (1; 2; 3; 4)+ |
|
|
| [2] I2/a (15, A112/a) | (1; 2; 5; 6)+ | b, -a - b, c
| 0, 1/4, 1/8
|
II Maximal klassengleiche subgroups
- Loss of centring translations
[3] c' = 3c
 | I41/a (88) | <(2; 5) + (1, 0, 1); 3 + (1/2, -1/2, 1/2)> | a, b, 3c | 1/2, 0, 1/4 | | I41/a (88) | <2 + (1, 0, 1); 3 + (1/2, -1/2, 1/2); 5 + (1, 0, 3)> | a, b, 3c | 1/2, 0, 5/4 | | I41/a (88) | <2 + (1, 0, 1); 3 + (1/2, -1/2, 1/2); 5 + (1, 0, 5)> | a, b, 3c | 1/2, 0, 9/4 |
|
- Series of maximal isomorphic subgroups
[p] c' = pc
| I41/a (88) | <2 + (0, 0, p/2 - 1/2); 3 + (0, 0, p/4 - 1/4); 5 + (0, 0, p/4 - 1/4 + 2u)> | a, b, pc | 0, 0, u | | | p > 4; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 1 (mod 4) |
| I41/a (88) | <2 + (1, 0, p/2 - 1/2); 3 + (1/2, -1/2, p/4 - 1/4); 5 + (1, 0, p/4 + 1/4 + 2u)> | a, b, pc | 1/2, 0, 1/4 + u | | | p > 2; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 3 (mod 4) |
|
[p2] a' = pa, b' = pb
| I41/a (88) | <2 + (p/2 - 1/2 + 2u, p/2 - 1/2 + 2v, 0); 3 + (u + v, p/2 - 1/2 - u + v, 0); 5 + (2u, p/2 - 1/2 + 2v, 0)> | pa, pb, c | u, v, 0 | | | p > 2; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 3 (mod 4) |
|
[p = q2 + r2] a' = qa - rb, b' = ra + qb
| I41/a (88) | <2 + (q/2 + r/2 - 1/2 + 2u, q/2 - r/2 - 1/2, 0); 3 + (r/2 + u, q/2 - 1/2 - u, 0); 5 + (r/2 + 2u, q/2 - 1/2, 0)> | qa - rb, ra + qb, c | u, 0, 0 | | | q > 0; r > 1; p > 4; 0 ≤ u < p
p conjugate subgroups for odd q and prime p ≡ 1 (mod 4) |
| I41/a (88) | <2 + (q/2 + r/2 + 1/2 + 2u, q/2 - r/2 - 1/2, 0); 3 + (r/2 + 1/2 + u, q/2 - 1 - u, 0); 5 + (r/2 + 1 + 2u, q/2 - 1/2, 1/2)> | qa - rb, ra + qb, c | 1/2 + u, 0, 1/4 | | | q > 1; r > 0; p > 4; 0 ≤ u < p
p conjugate subgroups for even q and prime p ≡ 1 (mod 4) |
|
I Minimal translationengleiche supergroups
| [2] I41/amd (141); [2] I41/acd (142) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [2] c' = 1/2c C42/e (86, P42/n) |
ORIGIN CHOICE 2, Origin at -1 on glide plane b, at 0, 1/4, 1/8 from -4
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(1/2, 1/2, 1/2); (2); (3); (5)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates |
|
| | (0, 0, 0)+ (1/2, 1/2, 1/2)+ |
| | (1) x, y, z | (2) -x + 1/2, -y, z + 1/2 | (3) -y + 3/4, x + 1/4, z + 1/4 | (4) y + 3/4, -x + 3/4, z + 3/4 | | (5) -x, -y, -z | (6) x + 1/2, y, -z + 1/2 | (7) y + 1/4, -x + 3/4, -z + 3/4 | (8) -y + 1/4, x + 1/4, -z + 1/4 |
|
I Maximal translationengleiche subgroups
| [2] I-4 (82) | (1; 2; 7; 8)+ |
| 0, 1/4, 5/8
|
| [2] I41 (80) | (1; 2; 3; 4)+ |
| 1/2, 1/4, 0
|
| [2] I2/a (15, A112/a) | (1; 2; 5; 6)+ | b, -a - b, c
|
|
II Maximal klassengleiche subgroups
- Loss of centring translations
[3] c' = 3c
| I41/a (88) | <2 + (1, 0, 1); 3 + (1/2, -1/2, 1/2); 5 + (1, 0, 0)> | a, b, 3c | 1/2, 0, 0 | | I41/a (88) | <2 + (1, 0, 1); 3 + (1/2, -1/2, 1/2); 5 + (1, 0, 2)> | a, b, 3c | 1/2, 0, 1 | | I41/a (88) | <2 + (1, 0, 1); 3 + (1/2, -1/2, 1/2); 5 + (1, 0, 4)> | a, b, 3c | 1/2, 0, 2 |
|
- Series of maximal isomorphic subgroups
[p] c' = pc
| I41/a (88) | <2 + (0, 0, p/2 - 1/2); 3 + (0, 0, p/4 - 1/4); 5 + (0, 0, 2u)> | a, b, pc | 0, 0, u | | | p > 4; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 1 (mod 4) |
| I41/a (88) | <2 + (1, 0, p/2 - 1/2); 3 + (1/2, -1/2, p/4 - 1/4); 5 + (1, 0, 2u)> | a, b, pc | 1/2, 0, u | | | p > 2; 0 ≤ u < p
p conjugate subgroups for prime p ≡ 3 (mod 4) |
|
[p2] a' = pa, b' = pb
| I41/a (88) | <2 + (p/2 + 1/2 + 2u, 2v, 0); 3 + (3p/4 - 1/4 + u + v, p/4 - 3/4 - u + v, 0); 5 + (1 + 2u, 2v, 0)> | pa, pb, c | 1/2 + u, v, 0 | | | p > 2; 0 ≤ u < p; 0 ≤ v < p
p2 conjugate subgroups for prime p ≡ 3 (mod 4) |
|
[p = q2 + r2] a' = qa - rb, b' = ra + qb
| I41/a (88) | <2 + (q/2 + 1/2 + 2u, -r/2, 0); 3 + (3q/4 + r/4 - 1/4 + u, q/4 - 3r/4 - 3/4 - u, 0); 5 + (1 + 2u, 0, 0)> | qa - rb, ra + qb, c | 1/2 + u, 0, 0 | | | q > 0; r > 1; p > 4; 0 ≤ u < p; q odd; q + r = 3 (mod 4)
p conjugate subgroups for each pair of q and r |
| I41/a (88) | <2 + (q/2 - 1/2 + 2u, -r/2, 0); 3 + (3q/4 + r/4 - 3/4 + u, q/4 - 3r/4 - 1/4 - u, 0); 5 + (2u, 0, 0)> | qa - rb, ra + qb, c | u, 0, 0 | | | q > 0; r > 1; p > 12; 0 ≤ u < p; q odd; q + r = 1 (mod 4)
p conjugate subgroups for each pair of q and r |
| I41/a (88) | <2 + (q/2 + 1 + 2u, -r/2 + 1/2, 0); 3 + (3q/4 + r/4 + 1/4 + u, q/4 - 3r/4 - 3/4 - u, 0); 5 + (3/2 + 2u, 1/2, 1/2)> | qa - rb, ra + qb, c | 3/4 + u, 1/4, 1/4 | | | q > 1; r > 0; p > 4; 0 ≤ u < p; q even; q + r = 3 (mod 4)
p conjugate subgroups for each pair of q and r |
| I41/a (88) | <2 + (q/2 + 2u, -r/2 + 1/2, 0); 3 + (3q/4 + r/4 - 1/4 + u, q/4 - 3r/4 - 1/4 - u, 0); 5 + (1/2 + 2u, 1/2, 1/2)> | qa - rb, ra + qb, c | 1/4 + u, 1/4, 1/4 | | | q > 1; r > 0; p > 12; 0 ≤ u < p; q even; q + r = 1 (mod 4)
p conjugate subgroups for each pair of q and r |
|
I Minimal translationengleiche supergroups
| [2] I41/amd (141); [2] I41/acd (142) |
II Minimal non-isomorphic klassengleiche supergroups
- Additional centring translations
| [2] c' = 1/2c C42/e (86, P42/n) |
