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Classification of the subgroups of type R3(146) of group Fm-3m(225) with index 64

For the group G = Fm-3m there are 16 different subgroups Hj isomorphic to H = R3 of index 64. These subgroups are distributed in 1 class of conjugate subgroups with respect to the group G.

In the tables below each table corresponds to one class of conjugate subgroups. Note that the subgroups are specified by their space-group types and transformation matrices. The listed transformation matrices are obtained combining the maximal subgroups transformation matrices from the MAXSUB database for each pair group-maximal subgroup in the chain indicated for each subgroup. Different transformation matrices (e.g. calculated following different chains) might specify the same (identical) subgroup if these transformation matrices are related by an element of the normalizer of the subgroup. The transformations by such transformation matrices of the subgroup operations to the basis of G result in the identical sets of elements of G.

The list of chains and the corresponding transformation matrices that specify the same subgroups can be seen by clicking on the button in the column Identical of the table.

Class 1

Check Chain [indices] Chain with HM symbols Transformation Identical
1 225 221 215 160 146 [4 2 4 2] Fm-3m > Pm-3m > P-43m > R3m > R3
(
  -1   0  -1    0
   1  -1  -1    0
   0  -1   1    0
)
2 225 221 215 160 146 [4 2 4 2] Fm-3m > Pm-3m > P-43m > R3m > R3
(
   1   0   1   1/2
  -1   1   1   1/2
   0  -1   1    0
)
3 225 221 215 160 146 [4 2 4 2] Fm-3m > Pm-3m > P-43m > R3m > R3
(
  -1   0  -1   1/2
   1  -1  -1   1/2
   0  -1   1    0
)
4 225 221 207 195 146 [4 2 2 4] Fm-3m > Pm-3m > P432 > P23 > R3
(
   1   0   1    0
  -1   1   1   1/2
   0  -1   1   1/2
)
5 225 209 196 195 146 [2 2 4 4] Fm-3m > F432 > F23 > P23 > R3
(
  -1   0  -1   1/4
  -1   1   1   3/4
   0   1  -1   3/4
)
6 225 209 196 195 146 [2 2 4 4] Fm-3m > F432 > F23 > P23 > R3
(
  -1   0  -1   1/4
   1  -1  -1   3/4
   0  -1   1   3/4
)
7 225 209 207 155 146 [2 4 4 2] Fm-3m > F432 > P432 > R32 > R3
(
   1   0   1   1/2
  -1   1   1    0
   0  -1   1   1/2
)
8 225 209 207 155 146 [2 4 4 2] Fm-3m > F432 > P432 > R32 > R3
(
  -1   0  -1   1/2
  -1   1   1    0
   0   1  -1   1/2
)
9 225 202 196 195 146 [2 2 4 4] Fm-3m > Fm-3 > F23 > P23 > R3
(
   1   0   1   1/2
   1  -1  -1   1/2
   0   1  -1    0
)
10 225 202 196 195 146 [2 2 4 4] Fm-3m > Fm-3 > F23 > P23 > R3
(
   1   0   1   1/4
  -1   1   1   1/4
   0  -1   1   1/4
)
11 225 202 196 195 146 [2 2 4 4] Fm-3m > Fm-3 > F23 > P23 > R3
(
  -1   0  -1   3/4
  -1   1   1   3/4
   0   1  -1   1/4
)
12 225 202 200 195 146 [2 4 2 4] Fm-3m > Fm-3 > Pm-3 > P23 > R3
(
   1   0   1    0
   1  -1  -1   1/2
   0   1  -1   1/2
)
13 225 202 200 195 146 [2 4 2 4] Fm-3m > Fm-3 > Pm-3 > P23 > R3
(
  -1   0  -1    0
   1  -1  -1   1/2
   0  -1   1   1/2
)
14 225 216 215 195 146 [2 4 2 4] Fm-3m > F-43m > P-43m > P23 > R3
(
   1   0   1   1/4
   1  -1  -1   3/4
   0   1  -1   3/4
)
15 225 216 196 195 146 [2 2 4 4] Fm-3m > F-43m > F23 > P23 > R3
(
   1   0   1   3/4
   1  -1  -1   3/4
   0   1  -1   1/4
)
16 225 221 200 148 146 [4 2 4 2] Fm-3m > Pm-3m > Pm-3 > R-3 > R3
(
  -1   0  -1    0
  -1   1   1    0
   0   1  -1    0
)






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