P4132 No. 213 O7


Axes Coordinates Wyckoff positions
4a 4b 8c 12d 24e
I Maximal translationengleiche subgroups
[2] P213 (198) 4a 4a 2 × 4a 12b 2 × 12b
[4] R32 (155) x - (3/8), y - (3/8), z - (3/8) 1a; 3e 1b; 3d 2c; 6f 3d; 3e; 6f 4 × 6f
(rhombohedral axes)
a - b, b - c, (1/3)(2x - y - z), 3a; 9e 3b; 9d 6c; 18f 9d; 9e; 18f 4 × 18f
a + b + c (1/3)(x + y - 2z),
(hexagonal axes) (1/3)(x + y + z) - (3/8)
 conjugate: -a - b, b + c, (1/3)(-2x - y + z),
-a + b - c (1/3)(-x + y + 2z) - (1/2),
(hexagonal axes) (1/3)(-x + y - z) - (3/8)
 conjugate: a + b, -b + c, (1/3)(2x + y + z) - (1/2),
a - b - c (1/3)(x - y + 2z),
(hexagonal axes) (1/3)(x - y - z) - (3/8)
 conjugate: -a + b, -b - c, (1/3)(-2x + y - z) + (1/2),
-a - b + c (1/3)(-x - y - 2z) + (1/2),
(hexagonal axes) (1/3)(-x - y + z) - (3/8)
[3] P41212 (92) x + (1/4), y, z - (1/8) 4a 4a 8b 4a; 8b 3 × 8b
 conjugate: b, c, a y + (1/4), z, x - (1/8)
 conjugate: c, a, b z + (1/4), x, y - (1/8)
II Maximal klassengleiche subgroups
   Enlarged unit cell, isomorphic
[27] P4332 3a, 3b, 3c (1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); 4a; 8c; 2 × 12d; 4b; 8c; 2 × 12d; 3 × 8c; 8 × 24e 3 × 12d; 27 × 24e
  (212) ±(0, (1/3), 0); ±(0, 0, (1/3)); 3 × 24e 3 × 24e 12 × 24e
±(0, (1/3), (1/3)); ±((1/3), 0, (1/3));
±((1/3), (1/3), 0); ±(0, (1/3), (2/3));
±((1/3), 0, (2/3)); ±((1/3), (2/3), 0);
±((1/3), (1/3), (1/3)); ±((2/3), (1/3), (1/3));
±((1/3), (2/3), (1/3)); ±((1/3), (1/3), (2/3))
[p3] P4132 pa, pb, pc (1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) 4a(b*); ((p - 1)/2) × 8c; 4b(a*); ((p - 1)/2) × 8c; p × 8c; p × 12d; p3 × 24e
p = prime = 4n + 1; (p - 1) × 12d; (p - 1) × 12d; ((p(p2 - 1))/3) × 24e ((p(p2 - 1))/2) × 24e
u, v, w = 1, . . ., p - 1 ((p3 - 4p + 3)/6) × 24e ((p3 - 4p + 3)/6) × 24e
[p3] P4332 pa, pb, pc (1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) 4a(b† ); ((p - 1)/2) × 8c; 4b(a† ); ((p - 1)/2) × 8c; p × 8c; p × 12d; p3 × 24e
  (212) p = prime = 4n - 1; (p - 1) × 12d; (p - 1) × 12d; ((p(p2 - 1))/3) × 24e ((p(p2 - 1))/2) × 24e
u, v, w = 1, . . ., p - 1 ((p3 - 4p + 3)/6) × 24e ((p3 - 4p + 3)/6) × 24e
* p = 8n + 5 p = 8n - 1










































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