Fd-3m No. 227 F41/d-32/m Oh7


Axes Coordinates Wyckoff positions
origin 1 origin 2 8a 8b 16c 16d 32e 48f
96g 96h 192i
I Maximal translationengleiche subgroups
[2] F-43m x + (1/8), y + (1/8), 4a; 4d 4b; 4c 16e 16e 2 × 16e 24f; 24g
  (216) z + (1/8) 2 × 48h 96i 2 × 96i
[2] F4132 x + (1/8), y + (1/8), 8a 8b 16c 16d 32e 48f
  (210) z + (1/8) 96h 2 × 48g 2 × 96h
[2] Fd-3 8a 8b 16c 16d 32e 48f
  (203) 96g 96g 2 × 96g
[4] R-3m (1/2)(b + c), (1/2)(a + c), - x + y + z - (1/8), -x + y + z, 2c 2c 1a; 3d 1b; 3e 2c; 6h 2 × 6h
  (166) (1/2)(a + b) x - y + z - (1/8), x - y + z, 2 × 6h; 12i 6f; 6g; 12i 4 × 12i
(rhombohedral axes) x + y - z - (1/8) x + y - z
(1/2)(-a + b), (1/2)(-b + c), (2/3)(-2x + y + z), (2/3)(-2x + y + z), 6c 6c 3a; 9d 3b; 9e 6c; 18h 2 × 18h
a + b + c (2/3)(-x - y + 2z), (2/3)(-x - y + 2z), 2 × 18h; 36i 18f; 18g; 36i 4 × 36i
(hexagonal axes) (1/3)(x + y + z) - (1/8) (1/3)(x + y + z)
 conjugate: (1/2)(a + b), (1/2)(-b - c), (2/3)(2x + y - z), (2/3)(2x + y - z) + (1/2),
-a + b - c (2/3)(x - y - 2z), (2/3)(x - y - 2z) + (1/2),
(hexagonal axes) (1/3)(-x + y - z) - (1/8) (1/3)(-x + y - z) + (1/2)
 conjugate: (1/2)(-a - b), (1/2)(b - c), (2/3)(-2x - y - z), (2/3)(-2x - y - z),
a - b - c (2/3)(-x + y - 2z), (2/3)(-x + y - 2z) + (1/2),
(hexagonal axes) (1/3)(x - y - z) - (1/8) (1/3)(x - y - z) + (1/2)
 conjugate: (1/2)(a - b), (1/2)(b + c), (2/3)(2x - y + z), (2/3)(2x - y + z) + (1/2),
-a - b + c (2/3)(x + y + 2z), (2/3)(x + y + 2z),
(hexagonal axes) (1/3)(-x - y + z) - (1/8) (1/3)(-x - y + z) + (1/2)
[3] I41/amd (1/2)(a - b), (1/2)(a + b), c x - y, x + y, z x - y, x + y + (1/2), z 4a 4b 8c 8d 16h 8e; 16g
  (141) 16h; 32i 16f; 32i 3 × 32i
 conjugate: (1/2)(b - c), (1/2)(b + c), a y - z, y + z, x y - z, y + z + (1/2), x
 conjugate: (1/2)(-a + c), (1/2)(a + c), b -x + z, x + z, y -x + z, x + z + (1/2), y
II Maximal klassengleiche subgroups
   Enlarged unit cell, isomorphic
[27] Fd-3m 3a, 3b, 3c (1/3)x - (1/4), (1/3)y - (1/4), (1/3)x, (1/3)y, (1/3)z; 8b; 8a; 16c; 32e; 16d; 32e; 3 × 32e; 3 × 48f;
(1/3)z - (1/4) 2 × 32e; 2 × 32e; 3 × 96g; 3 × 96g; 6 × 96g; 6 × 96g;
±((1/3), 0, 0); ±(0, (1/3), 0); ±(0, 0, (1/3)); 48f; 96g 48f; 96g 96h 96h 192i 3 × 192i
±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); ±((1/3), (1/3), 0); 9 × 96g; 3 × 96h; 27 × 192i
±(0, (1/3), (2/3)); ±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); 9 × 192i 12 × 192i
±((1/3), (1/3), (1/3)); ±((2/3), (1/3), (1/3)); ±((1/3), (2/3), (1/3));
±((1/3), (1/3), (2/3))


Axes Coordinates Wyckoff positions
origin 1 origin 2 8a 8b 16c 16d 32e
48f 96g 96h 192i
[p3] Fd-3m pa, pb, pc (1/p)x + s, (1/p)y + s, (1/p)x, (1/p)y, (1/p)z; 8a(b*); 8b(a*); 16c; 16d; p × 32e;
(1/p)z + s; +((u/p), (v/p), (w/p)) (p - 1) × 32e; (p - 1) × 32e; ((p - 1)/2) × 32e; ((p - 1)/2) × 32e; p(p - 1) × 96g;
+((u/p), (v/p), (w/p)) ((p - 1)/2) × 48f; ((p - 1)/2) × 48f; ((p(p - 1))/2) × 96g; ((p(p - 1))/2) × 96g; ((p(p - 1)(p - 2))/6) ×
s = 0  if p = 8n + 1; (((p - 1)(p - 2))/2) × 96g; (((p - 1)(p - 2))/2) × 96g; ((p - 1)/2) × 96h; ((p - 1)/2) × 96h; 192i
s = - (1/4) if p = 8n + 3; (((p - 1)(p - 2)(p - 3))/24) × (((p - 1)(p - 2)(p - 3))/24) × (((p2 - 1)(p - 3))/12) × (((p2 - 1)(p - 3))/12) ×
s = (1/2) if p = 8n + 5; 192i 192i 192i 192i
s = (1/4) if p = 8n + 7 p × 48f; p2 × 96g; p × 96h; p3 × 192i
p = prime > 2; p(p - 1) × 96g; ((p2(p - 1))/2) × 192i ((p(p2 - 1))/2) × 192i
u, v, w = 1, . . ., p - 1 ((p(p - 1)2)/4) × 192i
* p = 8n ± 3









































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