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Classification of the subgroups of type Cmm2(35) of group Fm-3m(225) with index 48

For the group G = Fm-3m there are 15 different subgroups Hj isomorphic to H = Cmm2 of index 48. These subgroups are distributed in 3 classes 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
1 225 139 123 099 099 035 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4mm > P4mm > Cmm2
(
  -1  -1   0    0
   0   0   1    0
  -1   1   0    0
)
2 225 139 123 123 065 035 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4/mmm > Cmmm > Cmm2
(
  -1  -1   0    0
   0   0   1    0
  -1   1   0   1/2
)
3 225 216 215 111 035 [2 4 3 2] Fm-3m > F-43m > P-43m > P-42m > Cmm2
(
   1   1   0    0
  -1   1   0    0
   0   0   1    0
)
4 225 221 123 099 035 [4 3 2 2] Fm-3m > Pm-3m > P4/mmm > P4mm > Cmm2
(
   0   0   1    0
   1   1   0    0
  -1   1   0    0
)
5 225 221 215 111 035 [4 2 3 2] Fm-3m > Pm-3m > P-43m > P-42m > Cmm2
(
   0   0   1   1/2
   1   1   0   1/2
  -1   1   0    0
)
6 225 221 123 111 035 [4 3 2 2] Fm-3m > Pm-3m > P4/mmm > P-42m > Cmm2
(
   1   1   0    0
  -1   1   0   1/2
   0   0   1   1/2
)




Class 2

Check Chain [indices] Chain with HM symbols Transformation Identical
7 225 139 123 123 111 035 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4/mmm > P-42m > Cmm2
(
   0   1   0    0
  -1   0   0    0
   0   0   2    0
)
8 225 139 121 111 111 035 [3 2 2 2 2] Fm-3m > I4/mmm > I-42m > P-42m > P-42m > Cmm2
(
   0   0   2    0
   0   1   0    0
  -1   0   0    0
)
9 225 139 123 123 111 035 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4/mmm > P-42m > Cmm2
(
  -1   0   0    0
   0   0   2   1/2
   0   1   0    0
)




Class 3

Check Chain [indices] Chain with HM symbols Transformation Identical
10 225 139 119 115 111 035 [3 2 2 2 2] Fm-3m > I4/mmm > I-4m2 > P-4m2 > P-42m > Cmm2
(
  -1   1   0   3/4
  -1  -1   0   1/4
   0   0   1   1/4
)
11 225 139 119 115 111 035 [3 2 2 2 2] Fm-3m > I4/mmm > I-4m2 > P-4m2 > P-42m > Cmm2
(
  -1  -1   0   1/4
   0   0   1   1/4
  -1   1   0   3/4
)
12 225 216 119 115 111 035 [2 3 2 2 2] Fm-3m > F-43m > I-4m2 > P-4m2 > P-42m > Cmm2
(
  -1   1   0   1/4
  -1  -1   0   1/4
   0   0   1   1/4
)
13 225 216 119 115 111 035 [2 3 2 2 2] Fm-3m > F-43m > I-4m2 > P-4m2 > P-42m > Cmm2
(
  -1  -1   0   1/4
   0   0   1   1/4
  -1   1   0   1/4
)
14 225 216 215 111 035 [2 4 3 2] Fm-3m > F-43m > P-43m > P-42m > Cmm2
(
   0   0   1   3/4
   1   1   0   1/4
  -1   1   0   3/4
)
15 225 216 215 111 035 [2 4 3 2] Fm-3m > F-43m > P-43m > P-42m > Cmm2
(
   0   0   1   1/4
   1   1   0   3/4
  -1   1   0   3/4
)






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