C2 No. 5 C23


C121 A121

UNIQUE AXIS b

CELL CHOICE 1

CELL CHOICE 2
Axes Coordinates Wyckoff positions Axes Coordinates
2a 2b 4c
I Maximal translationengleiche subgroups
[2] P1 (1) (1/2)(a - b), b, c 2x, x + y, z 1a 1a 2 × 1a P1 a, b, (1/2)(-b + c) x, x + y, 2z
II Maximal klassengleiche subgroups
   Loss of centring translations
[2] P1211 (4) x + (1/4), y, z 2a 2a 2 × 2a P1211 x, y, z + (1/4)
[2] P121 (3) 1a; 1c 1b; 1d 2 × 2e
1a; 1b 1c; 1d 2 × 2e P121
   Enlarged unit cell, isomorphic
[3] C121 a, 3b, c x, (1/3)y, z; ±(0, (1/3), 0) 3 × 2a 3 × 2b 3 × 4c A121 a, 3b, c x, (1/3)y, z; ±(0, (1/3), 0)
[p] C121 a, pb, c x, (1/p)y, z; +(0, (u/p), 0) p × 2a p × 2b p × 4c A121 a, pb, c x, (1/p)y, z; +(0, (u/p), 0)
p = prime > 2; u = 1, . . ., p - 1 p = prime > 2; u = 1, . . ., p - 1
[2] C121 a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2a; 2b 4c 2 × 4c A121 2a, b, c (1/2)x, y, z; +((1/2), 0, 0)
[2] C121 a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 4c 2a; 2b 2 × 4c A121 2a, b, c (1/2)x - (1/4), y, z; +((1/2), 0, 0)
[2] I121 a, b, 2c x, y, (1/2)z; +(0, 0, (1/2)) 2a; 2b 4c 2 × 4c I121 2a, b, c (1/2)x, y, z; +((1/2), 0, 0)
[2] I121 a, b, 2c x, y, (1/2)z + (1/4); +(0, 0, (1/2)) 4c 2a; 2b 2 × 4c I121 2a, b, c (1/2)x - (1/4), y, z; +((1/2), 0, 0)
[3] C121 3a, b, c (1/3)x, y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c A121 a + c, b, 3c x, y, (1/3)(-x + z); ±(0, 0, (1/3))
[3] C121 3a, b, -a + c (1/3)(x + z), y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c A121 a - c, b, 3c x, y, (1/3)(x + z); ±(0, 0, (1/3))
[3] C121 3a, b, a + c (1/3)(x - z), y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c A121 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3))
[3] C121 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 2a; 4c 2b; 4c 3 × 4c A121 3a, b, c (1/3)x, y, z; ±((1/3), 0, 0)
[p] C121 a, b, pc x, y, (1/p)z; +(0, 0, (u/p)) 2a; ((p - 1)/2) × 2b; ((p - 1)/2) × p × 4c A121 pa, b, c (1/p)x, y, z; +((u/p), 0, 0)
p = prime > 2; u = 1, . . ., p - 1 4c 4c p = prime > 2; u = 1, . . ., p - 1
[p] C121 pa, b, qa + c (1/p)(x - qz), y, z; +((u/p), 0, 0) 2a; ((p - 1)/2) × 2b; ((p - 1)/2) × p × 4c A121 a + qc, b, pc x, y, (1/p)(-qx + z); +(0, 0, (u/p))
p = prime > 2; u = 1, . . ., p - 1; 4c 4c p = prime > 2; u = 1, . . ., p - 1;
-(1/2)(p - 1)≤ q≤ (1/2)(p - 1) -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1)

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3



I121 A112

UNIQUE AXIS b

UNIQUE AXIS c

CELL CHOICE 3

CELL CHOICE 1
Axes Coordinates Wyckoff positions Axes Coordinates
2a 2b 4c
I Maximal translationengleiche subgroups
[2] P1 (1) a, (1/2)(-a + b - c), c x + y, 2y, y + z 1a 1a 2 × 1a P1 (1) a, (1/2)(b - c), c x, 2y, y + z
II Maximal klassengleiche subgroups
   Loss of centring translations
[2] P1211 (4) x - (1/4), y, z - (1/4) 2a 2a 2 × 2a P1121 (4) x, y + (1/4), z
[2] P121 (3) 1a; 1d 1b; 1c 2 × 2e
1a; 1c 1b; 1d 2 × 2e P112 (3)
   Enlarged unit cell, isomorphic
[3] I121 a, 3b, c x, (1/3)y, z; ±(0, (1/3), 0) 3 × 2a 3 × 2b 3 × 4c A112 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3))
[p] I121 a, pb, c x, (1/p)y, z; +(0, (u/p), 0) p × 2a p × 2b p × 4c A112 a, b, pc x, y, (1/p)z; +(0, 0, (u/p))
p = prime > 2; u = 1, . . ., p - 1 p = prime > 2; u = 1, . . ., p - 1
[2] I121 2a, b, -a + c (1/2)(x + z), y, z; +((1/2), 0, 0) 2a; 2b 4c 2 × 4c A112 2a, b, c (1/2)x, y, z; +((1/2), 0, 0)
[2] I121 2a, b, -a + c (1/2)(x + z) + (1/4), y, z; +((1/2), 0, 0) 4c 2a; 2b 2 × 4c A112 2a, b, c (1/2)x + (1/4), y, z; +((1/2), 0, 0)
[2] I121 2a, b, a + c (1/2)(x - z), y, z; +((1/2), 0, 0) 2a; 2b 4c 2 × 4c I112 2a, b, c (1/2)x, y, z; +((1/2), 0, 0)
[2] I121 2a, b, a + c (1/2)(x - z) + (1/4), y, z; +((1/2), 0, 0) 4c 2a; 2b 2 × 4c I112 2a, b, c (1/2)x + (1/4), y, z; +((1/2), 0, 0)
[3] I121 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3)) 2a; 4c 2b; 4c 3 × 4c A112 a, 3b, c x, (1/3)y, z; ±(0, (1/3), 0)
[3] A121 3a, b, -a + c (1/3)(x + z), y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c A112 a - b, 3b, c x, (1/3)(x + y), z; ±(0, (1/3), 0)
[3] I121 3a, b, c (1/3)x, y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c A112 a + b, 3b, c x, (1/3)(-x + y), z; ±(0, (1/3), 0)
[3] A121 3a, b, a + c (1/3)(x - z), y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c A112 3a, b, c (1/3)x, y, z; ±((1/3), 0, 0)
[p] I121 pa, b, c (1/p)x, y, z; ±((u/p), 0, 0) 2a; ((p - 1)/2) × 2b; ((p - 1)/2) × p × 4c A112 pa, b, c (1/p)x, y, z; +((u/p), 0, 0)
p = prime > 2; u = 1, . . ., p - 1 4c 4c p = prime > 2; u = 1, . . ., p - 1
[p] I121 a + 2qc, b, pc x, y, (1/p)(-2qx + z); +(0, 0, (u/p)) 2a; ((p - 1)/2) × 2b; ((p - 1)/2) × p × 4c A112 a + qb, pb, c x, (1/p)(-qx + y), z; +(0, (u/p), 0)
p = prime > 2; u = 1, . . ., p - 1; 4c 4c p = prime > 2; u = 1, . . ., p - 1;
-(1/2)(p - 1)≤ q≤ (1/2)(p - 1) -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1)

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3



B112 I112

UNIQUE AXIS c

CELL CHOICE 2

CELL CHOICE 3
Axes Coordinates Wyckoff positions Axes Coordinates
2a 2b 4c
I Maximal translationengleiche subgroups
[2] P1 (1/2)(a - c), b, c 2x, y, y + z 1a 1a 2 × 1a P1 a, b, (1/2)(-a - b + c) x + z, y + z, 2z
II Maximal klassengleiche subgroups
   Loss of centring translations
[2] P1121 x + (1/4), y, z 2a 2a 2 × 2a P1121 x - (1/4), y - (1/4), z
[2] 1a; 1d 1b; 1c 2 × 2e P112
[2] P112 1a; 1b 1c; 1d 2 × 2e
   Enlarged unit cell, isomorphic
[3] B112 a, b, 3c x, y, (1/3), z; ±(0, 0, (1/3)) 3 × 2a 3 × 2b 3 × 4c I112 a, b, 3c x, y, (1/3)z; ±(0, 0, (1/3))
[p] B112 a, b, pc x, y, (1/p)z; +(0, 0, (u/p)) p × 2a p × 2b p × 4c I112 a, b, pc x, y, (1/p)z; +(0, 0, (u/p))
p = prime > 2; u = 1, . . ., p - 1 p = prime > 2; u = 1, . . ., p - 1
[2] B112 a, 2b, c x, (1/2)y, z; +(0, (1/2), 0) 2a; 2b 4c 2 × 4c I112 a - b, 2b, c x, (1/2)(x + y), z; +(0, (1/2), 0)
[2] B112 a, 2b, c x, (1/2)y - (1/4), z; +(0, (1/2), 0) 4c 2a; 2b 2 × 4c I112 a - b, 2b, c x, (1/2)(x + y) + (1/4), z; +(0, (1/2), 0)
[2] I112 a, 2b, c x, (1/2)y, z; +(0, (1/2), 0) 2a; 2b 4c 2 × 4c I112 a + b, 2b, c x, (1/2)(-x + y), z; +(0, (1/2), 0)
[2] I112 a, 2b, c x, (1/2)y - (1/4), z; +(0, (1/2), 0) 4c 2a; 2b 2 × 4c I112 a, 2b, c x, (1/2)y - (1/4), z; +(0, (1/2), 0)
[3] B112 3a, a + b, c (1/3)(x - y), y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c I112 3a, b, c (1/3)x, y, z; ±((1/3), 0, 0)
[3] B112 3a, -a + b, c (1/3)(x + y), y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c B112 a - b, 3b, c x, (1/3)(x + y), z; ±(0, (1/3), 0)
[3] B112 3a, b, c (1/3)x, y, z; ±((1/3), 0, 0) 2a; 4c 2b; 4c 3 × 4c I112 a, 3b, c x, (1/3)y, z; ±(0, (1/3), 0)
[3] B112 a, 3b, c x, (1/3)y, z; ±(0, (1/3), 0) 2a; 4c 2b; 4c 3 × 4c B112 a + b, 3b, c x, (1/3)(-x + y), z; ±(0, (1/3), 0)
[p] B112 a, pb, c x, (1/p)y, z; +(0, (u/p), 0) 2a; ((p - 1)/2) × 2b; ((p - 1)/2) × p × 4c I112 a, pb, c x, (1/p)y, z; ±(0, (u/p), 0)
p = prime > 2; u = 1, . . ., p - 1 4c 4c p = prime > 2; u = 1, . . ., p - 1
[p] B112 pa, qa + b, c (1/p)(x - qy), y, z; +(0, (u/p), 0) 2a; ((p - 1)/2) × 2b; ((p - 1)/2) × p × 4c I112 pa, 2qa + b, c (1/p)(x - 2qy), y, z; +((u/p), 0, 0)
p = prime > 2; u = 1, . . ., p - 1; 4c 4c p = prime > 2; u = 1, . . ., p - 1;
-(1/2)(p - 1)≤ q≤ (1/2)(p - 1) -(1/2)(p - 1)≤ q ≤ (1/2)(p - 1)

Space groups of the series of isomorphic subgroups appear in different sequences for cell choices 1, 2 and 3










































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