Bilbao Crystallographic Server SUBGROUPGRAPH Help |
Let G be the group and H the subgroup. The lattice of maximal subgroups that relate the groups G and H is represented as a table. The first row contains the group G and its maximal subgroups, given by their numbers in the International Tables for Crystallography, Vol. A, and the corresponding indices, given in brackets. The last row contains the same information about the subgroup H. The rest of the rows in the table contain the group number and symbol and a list with the maximal subgroups and their indices for all of the maximal subgroups that appear between G and H. If a chain relating G and H is represented as G > Z_{1} > .. > Z_{i} > ... Z_{n} > H then each row of the table starts with the number and the Hermann-Mauguin symbol of a group Z_{k} and contains the list with the groups Z_{j} < Z_{k} that can appear in the lattice relating G and H.
You can see a graphical representation of the lattice using the button [Draw the lattice].
The resultant table contains all of the chains G > Z_{1} > .. > Z_{i} > ... Z_{n} > H that relate the group G with the subgroup H with the given index, represented using the groups numbers and the Hermann-Mauguin symbols, and a link transformation that shows all of the transformation matrices that relate the basis of the group with that of the subgroup, obtained for the current chain.
If you want to print only the table with the chains, follow the link
In this case the lattice is a part from the bigger lattice that should be obtained if the index is not specified. You can see a graphical representation of the lattice using the button [Draw the lattice].
If you want to compare the result form the classification with the one obtained
using the normalizer procedure mark the checkbox
The different classes of subgroups are given as tables, one table for each class, which contain:
To see the general positions of the subgroup with resect to the basis of the group G, use the button in the column Transform with of the table.
All of the chains that will give the same group can be seen using the button in the column Equivalent.
If you have marked the checkbox
The page with the comparison contains the class/es obtain using the normalizer procedure and the correspondence between these classes and those that have been obtained before.
NOTE, that not always all of the classes of subgroups are obtained using the normalizer procedure.
NOTE, that if the index is large then it is possible that the graph results very complicated and difficult to use. If the graph is very big and can not be seen with the browser you can use the PostSript form and see it with a program for reading PostSript files.
As a part of the label for the vertices corresponding to the subgroup H is given the number of the class the current subgroup belongs to.
If the chain is G > Z_{1} > .. > Z_{i} > ... Z_{n} > H and (P_{i}, p_{i}) is the transformation matrix that relates the group Z_{i-1} with its maximal subgroup Z_{i}, then the matrix that relates the basis of G with that of H for this chain is obtained using (P,p) = (P_{1}, p_{1}) (P_{2}, p_{2}) ...(P_{n+1}, p_{n+1}), where (P_{n+1}, p_{n+1}) is the matrix corresponding to Z_{n} > H.
The set of transformations contains all of the matrices that can be obtained for a given chain.
If you have called the program SUBGROUPGRAPH from other program (for example form WYCKSPLIT) than for each one of the matrices there is a link to that program, so you can continue using it with the data obtained from SUBGROUPGRAPH.
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