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Classification of the subgroups of type P32(145) of group P6222(180) with index 4

For the group G = P6222 there are 1 different subgroups Hj isomorphic to H = P32 of index 4. 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
1 180 154 145 [2 2] P6222 > P3221 > P32
(
   1   0   0    0
   0   1   0    0
   0   0   1   1/3
)






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