Lattice and Chains of Subgroups

Bilbao Crystallographic Server SUBGROUPGRAPH Help

Lattice and Chains of Maximal Subgroups

Online Help

Each group-subgroup pair of space groups can be represented as a chain of maximal subgroups that relate the two groups in the pair. The program SUBGROUPGRAPH permits to obtain:

All of the possible chains of maximal subgroups that relate two space groups with unspecified index - Lattice of maximal subgroups.
The chains of maximal subgroups that relate two space groups with specified index - Chains of maximal subgroups
The classification of the different subgroups of the same type for a given space group with specified index - Classes of subgroups

Lattice of Maximal Subgroups

To obtain the lattice of maximal subgroups relating two space groups you should give the numbers of these two group, as given in the International Tables for Crystallography, Vol. A or you can select them. No value for the index should be given. To obtain the lattice, click on [Construct the lattice].

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 > Z₁ > .. > 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].

Select Groups

All of the programs need as an input the number of one or two space groups as given in International Tables for Crystallography, Vol. A. If you do not know these numbers, you can select them from the Table of Space Group Symbols.

Chains of Maximal Subgroups

To obtain the chains of maximal subgroups that relate a group G with its subgroup H with specified index, you should give (or select from the table with group symbols) the group and the subgroup numbers as given in the International Tables for Crystallography, Vol. A. Also, an index of the subgroup H in G must be given. To obtain the lattice for the specified index, click on [Construct the lattice].

The resultant table contains all of the chains G > Z₁ > .. > 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 "Print this table".

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].

Classes of Subgroups

Once you have obtained all of the chains that relate the group and the subgroup with a specified index, you can go further and see how the different subgroups of the same type as H are distributed into classes of conjugate groups. To do that, click on [Classify the subgroups].

If you want to compare the result form the classification with the one obtained using the normalizer procedure mark the checkbox Check with normalizer procedure.

The different classes of subgroups are given as tables, one table for each class, which contain:

the chain from which the subgroup has been obtained,
the transformation matrix, that corresponds to that chain
and a link to other chains (if some) that give the same subgroup.

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 Check with normalizer procedure, for each one of the classes there is a button [Check class] which can be used to see the comparison of the obtained result with the one given by the normalizer procedure.

Normalizer Procedure

There is a possibility to compare the result obtained for the classes of subgroups with one given by the procedure based on the normalizers of the space group described by E. Koch (Koch E., (1984) Acta Cryst. A 40, 593-600).

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.

Graphical Representation

The graphical representation of the lattice or the chains is a graph which starts with the group G and ends with the subgroup H. The intermediate vertices correspond to the groups Z_i that appear between G and H. A group is connected with itself (loop edge) if it has a subgroup of the same type.

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.

Lattice of Maximal Subgroups

If a lattice is represented than there is one vertex for each one of the groups Z_i and for G and H. The index of H in G for each one of the possible paths to relate them is obtained by multiplying the indices on each step in the chain.

Chains of Maximal Subgroups

When the index of the subgroup in the group is specified, the resultant lattice is smaller - only the chains of maximal subgroups that correspond to the given index are shown.

Classes of Subgroups

The graph representing the classification if the different subgroups contains not only the types of the subgroups but also all of the different subgroups of the same type.

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.

Set of Transformations

For each one of the chains there is a set of transformation matrices that can be obtained following the chain.

If the chain is G > Z₁ > .. > 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) = (P₁, p₁) (P₂, p₂) ...(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.

Equivalent Subgroups

It is possible that different chains of maximal subgroups give the same subgroup. If you click on the button given in the column Equivalent of the table with the subgroups in a given class, you can see the list with all of the chains that will give the same subgroup. These chains a represented as a table which contains the chain and the transformation matrix obtained following this chain. Also, if you click on the button given in the column Transform with you can obtain the general positions of the subgroup in the basis of the supergroup, transformed with the current matrix.

More about the program

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