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

ORIGIN CHOICE 1, Origin at -4 3m, at -1/8, -1/8, -1/8 from centre (-3m)

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 1/21/2); t(1/2, 0, 1/2); (2); (3); (5); (13); (25)

General position

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates

 (0, 0, 0)+  (0, 1/21/2)+  (1/2, 0, 1/2)+  (1/21/2, 0)+  
192 i 1
(1) xyz(2) -x-y + 1/2z + 1/2(3) -x + 1/2y + 1/2-z(4) x + 1/2-y-z + 1/2
(5) zxy(6) z + 1/2-x-y + 1/2(7) -z-x + 1/2y + 1/2(8) -z + 1/2x + 1/2-y
(9) yzx(10) -y + 1/2z + 1/2-x(11) y + 1/2-z-x + 1/2(12) -y-z + 1/2x + 1/2
(13) y + 3/4x + 1/4-z + 3/4(14) -y + 1/4-x + 1/4-z + 1/4(15) y + 1/4-x + 3/4z + 3/4(16) -y + 3/4x + 3/4z + 1/4
(17) x + 3/4z + 1/4-y + 3/4(18) -x + 3/4z + 3/4y + 1/4(19) -x + 1/4-z + 1/4-y + 1/4(20) x + 1/4-z + 3/4y + 3/4
(21) z + 3/4y + 1/4-x + 3/4(22) z + 1/4-y + 3/4x + 3/4(23) -z + 3/4y + 3/4x + 1/4(24) -z + 1/4-y + 1/4-x + 1/4
(25) -x + 1/4-y + 1/4-z + 1/4(26) x + 1/4y + 3/4-z + 3/4(27) x + 3/4-y + 3/4z + 1/4(28) -x + 3/4y + 1/4z + 3/4
(29) -z + 1/4-x + 1/4-y + 1/4(30) -z + 3/4x + 1/4y + 3/4(31) z + 1/4x + 3/4-y + 3/4(32) z + 3/4-x + 3/4y + 1/4
(33) -y + 1/4-z + 1/4-x + 1/4(34) y + 3/4-z + 3/4x + 1/4(35) -y + 3/4z + 1/4x + 3/4(36) y + 1/4z + 3/4-x + 3/4
(37) -y + 1/2-xz + 1/2(38) yxz(39) -yx + 1/2-z + 1/2(40) y + 1/2-x + 1/2-z
(41) -x + 1/2-zy + 1/2(42) x + 1/2-z + 1/2-y(43) xzy(44) -xz + 1/2-y + 1/2
(45) -z + 1/2-yx + 1/2(46) -zy + 1/2-x + 1/2(47) z + 1/2-y + 1/2-x(48) zyx

I Maximal translationengleiche subgroups

[2] F-43m (216)(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48)+
[2] F4132 (210)(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24)+
[2] Fd-31 (203Fd-3)(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36)+
brace[3] F41/d12/m (141I41/amd)(1; 2; 3; 4; 13; 14; 15; 16; 25; 26; 27; 28; 37; 38; 39; 40)+1/2(a - b), 1/2(a + b), c
[3] F41/d12/m (141I41/amd)(1; 4; 2; 3; 18; 19; 17; 20; 25; 28; 26; 27; 42; 43; 41; 44)+1/2(b - c), 1/2(b + c), a
[3] F41/d12/m (141I41/amd)(1; 3; 4; 2; 22; 24; 23; 21; 25; 27; 28; 26; 46; 48; 47; 45)+1/2(-a + c), 1/2(a + c), b
brace[4] F1-32/m (166R-3m)(1; 5; 9; 14; 19; 24; 25; 29; 33; 38; 43; 48)+1/2(-a + b), 1/2(-b + c), a + b + c 1/81/81/8
[4] F1-32/m (166R-3m)(1; 6; 12; 13; 18; 24; 25; 30; 36; 37; 42; 48)+1/2(a + b), 1/2(-b - c), -a + b - c 3/81/83/8
[4] F1-32/m (166R-3m)(1; 7; 10; 13; 19; 22; 25; 31; 34; 37; 43; 46)+1/2(-a - b), 1/2(b - c), a - b - c 1/83/83/8
[4] F1-32/m (166R-3m)(1; 8; 11; 14; 18; 22; 25; 32; 35; 38; 42; 46)+1/2(a - b), 1/2(b + c), -a - b + c 3/83/81/8

II Maximal klassengleiche subgroups

none

none

[p3] a' = pa, b' = pb, c' = pc


Fd-3m (227)<2 + (1/2 + 2up/2 + 2vp/2 - 1/2); 3 + (p/2 + 2up/2 - 1/21/2 + 2w); 5 + (u - w, -u + v, -v + w); 13 + (3p/4 - 3/4 + u - vp/4 - 1/4 - u + v3p/4 - 1/4 + 2w); 25 + (p/4 + 1/4 + 2up/4 + 1/4 + 2vp/4 + 1/4 + 2w)>papbpcuvw
 p > 2; 0 ≤ u < p; 0 ≤ v < p; 0 ≤ w < p
p3 conjugate subgroups for prime p ≡ 3 (mod 4)
Fd-3m (227)<2 + (2up/2 - 1/2 + 2vp/2 - 1/2); 3 + (p/2 - 1/2 + 2up/2 - 1/2, 2w); 5 + (u - w, -u + v, -v + w); 13 + (3p/4 - 3/4 + u - vp/4 - 1/4 - u + v3p/4 - 3/4 + 2w); 25 + (p/4 - 1/4 + 2up/4 - 1/4 + 2vp/4 - 1/4 + 2w)>papbpcuvw
 p > 4; 0 ≤ u < p; 0 ≤ v < p; 0 ≤ w < p
p3 conjugate subgroups for prime p ≡ 1 (mod 4)

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

none
[2] a' = 1/2a, b' = 1/2b, c' = 1/2c  Pn-3m (224)
Fd-3m No. 227 F41/d-32/m Oh7

ORIGIN CHOICE 2, Origin at centre (-3 m), at 1/81/81/8 from -4 3 m

Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(0, 1/21/2); t(1/2, 0, 1/2); (2); (3); (5); (13); (25)

General position

Multiplicity, Wyckoff letter,
Site symmetry
Coordinates

 (0, 0, 0)+  (0, 1/21/2)+  (1/2, 0, 1/2)+  (1/21/2, 0)+  
192 i 1
(1) xyz(2) -x + 3/4-y + 1/4z + 1/2(3) -x + 1/4y + 1/2-z + 3/4(4) x + 1/2-y + 3/4-z + 1/4
(5) zxy(6) z + 1/2-x + 3/4-y + 1/4(7) -z + 3/4-x + 1/4y + 1/2(8) -z + 1/4x + 1/2-y + 3/4
(9) yzx(10) -y + 1/4z + 1/2-x + 3/4(11) y + 1/2-z + 3/4-x + 1/4(12) -y + 3/4-z + 1/4x + 1/2
(13) y + 3/4x + 1/4-z + 1/2(14) -y-x-z(15) y + 1/4-x + 1/2z + 3/4(16) -y + 1/2x + 3/4z + 1/4
(17) x + 3/4z + 1/4-y + 1/2(18) -x + 1/2z + 3/4y + 1/4(19) -x-z-y(20) x + 1/4-z + 1/2y + 3/4
(21) z + 3/4y + 1/4-x + 1/2(22) z + 1/4-y + 1/2x + 3/4(23) -z + 1/2y + 3/4x + 1/4(24) -z-y-x
(25) -x-y-z(26) x + 1/4y + 3/4-z + 1/2(27) x + 3/4-y + 1/2z + 1/4(28) -x + 1/2y + 1/4z + 3/4
(29) -z-x-y(30) -z + 1/2x + 1/4y + 3/4(31) z + 1/4x + 3/4-y + 1/2(32) z + 3/4-x + 1/2y + 1/4
(33) -y-z-x(34) y + 3/4-z + 1/2x + 1/4(35) -y + 1/2z + 1/4x + 3/4(36) y + 1/4z + 3/4-x + 1/2
(37) -y + 1/4-x + 3/4z + 1/2(38) yxz(39) -y + 3/4x + 1/2-z + 1/4(40) y + 1/2-x + 1/4-z + 3/4
(41) -x + 1/4-z + 3/4y + 1/2(42) x + 1/2-z + 1/4-y + 3/4(43) xzy(44) -x + 3/4z + 1/2-y + 1/4
(45) -z + 1/4-y + 3/4x + 1/2(46) -z + 3/4y + 1/2-x + 1/4(47) z + 1/2-y + 1/4-x + 3/4(48) zyx

I Maximal translationengleiche subgroups

[2] F-43m (216)(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48)+ 1/81/81/8
[2] F4132 (210)(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24)+ 3/83/83/8
[2] Fd-31 (203Fd-3)(1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36)+
brace[3] F41/d12/m (141I41/amd)(1; 2; 3; 4; 13; 14; 15; 16; 25; 26; 27; 28; 37; 38; 39; 40)+1/2(a - b), 1/2(a + b), c 1/41/4, 0
[3] F41/d12/m (141I41/amd)(1; 4; 2; 3; 18; 19; 17; 20; 25; 28; 26; 27; 42; 43; 41; 44)+1/2(b - c), 1/2(b + c), a 0, 1/41/4
[3] F41/d12/m (141I41/amd)(1; 3; 4; 2; 22; 24; 23; 21; 25; 27; 28; 26; 46; 48; 47; 45)+1/2(-a + c), 1/2(a + c), b 1/4, 0, 1/4
brace[4] F1-32/m (166R-3m)(1; 5; 9; 14; 19; 24; 25; 29; 33; 38; 43; 48)+1/2(-a + b), 1/2(-b + c), a + b + c
[4] F1-32/m (166R-3m)(1; 6; 12; 13; 18; 24; 25; 30; 36; 37; 42; 48)+1/2(a + b), 1/2(-b - c), -a + b - c 1/4, 0, 1/4
[4] F1-32/m (166R-3m)(1; 7; 10; 13; 19; 22; 25; 31; 34; 37; 43; 46)+1/2(-a - b), 1/2(b - c), a - b - c 0, 1/41/4
[4] F1-32/m (166R-3m)(1; 8; 11; 14; 18; 22; 25; 32; 35; 38; 42; 46)+1/2(a - b), 1/2(b + c), -a - b + c 1/41/4, 0

II Maximal klassengleiche subgroups

none

none

[p3] a' = pa, b' = pb, c' = pc


Fd-3m (227)<2 + (3p/4 - 3/4 + 2up/4 - 1/4 + 2vp/2 - 1/2); 3 + (p/4 - 1/4 + 2up/2 - 1/23p/4 - 3/4 + 2w); 5 + (u - w, -u + v, -v + w); 13 + (3p/4 - 3/4 + u - vp/4 - 1/4 - u + vp/2 - 1/2 + 2w); 25 + (2u, 2v, 2w)>papbpcuvw
 p > 2; 0 ≤ u < p; 0 ≤ v < p; 0 ≤ w < p
p3 conjugate subgroups for the prime p

I Minimal translationengleiche supergroups

none

II Minimal non-isomorphic klassengleiche supergroups

none
[2] a' = 1/2a, b' = 1/2b, c' = 1/2c  Pn-3m (224)








































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