Bilbao Crystallographic Server Supergroups Help

Bilbao Crystallographic Server contains two programs related with supergroup-group pairs:
• MINSUP which gives all of the minimal supergroups of index 2, 3, and 4 of a given group.
• SUPERGROUPS, which calculates all of the different supergroups of a given type.

## Minimal Supergroups

The information you can obtain using the program MINSUP is as follows: for a given space group the program shows a table with all of the minimal supergroups of index 2, 3, and 4.

To specify the group for which you want to see the minimal supergroups, you should give its number in the International Tables for Crystallography, Vol.A or you can select it from the table with the space group numbers and symbols, which is accessible through the link ``choose''.

The list with the minimal supergroups is represented as a table which contains:

• The number of the supergroup in the International Tables for Crystallography, Vol. A, and its Hermann-Mauguin symbol.
• The index of the group in the supergroup and the supergroup type are also given.
• A link to the list with the transformation matrices that related the basis of the supergroup with that of the subgroup.

The table described above contains only the types of the minimal supergroups. If you are interested in all of the supergroups of a given type, then you should select the type of the supergroup, marking the corresponding checkbox and click on the button [Show].

## Different Supergroups of the Same Type

For a given supergroup-group pair with specified index the program SUPERGROUPS gives all of the different supergroups of the given type. The result contains:
• The transformation matrices that relate the basis of the supergroup with that of the subgroup.
• One representative from each coset in the decomposition of the supergroup with respect to the group and a link to the full cosets in this decomposition
• If you click oh the button [Full Cosets] you can see the the coset decomposition of the supergroup with respect to the group.

NOTE, that the coset representatives are given with respect to the basis of the group.

For some group-supergroup pairs there is a parameter r,s, or/and t in the transformation matrix and in the translational part of the coset representatives. This parameter can have any value and each one of this values will give different supergroup of the same type.

## Decomposition of the supergroup with respect to the group

The result from the calculation of all of the different supergroups contains:
• the transformation matrices and
• one representative from each coset in the left cosets decomposition of the supergroup with respect to the group.

The button [Full Cosets] is provided to access the full cosets in this decomposition. The page obtained using this button contains the correspondent transformation matrix and the elements of the supergroup, given with respect to the group and distributed into left cosets.

## Supergroups of a Given Type

The program SUPERGROUPS gives the possibility to obtain all of the different supergroups of the same type even if the supergroups are not minimal. To do that you should give the supergroup and the group number as given in the International Tables for Crystallography, vol.A, and the index of the group in the supergroup. All of this data is obligatory.. If you do not know the group number you can select it from the table with the space group numbers and symbols.

When all of the data is given, click on [Find the Supergroups] to obtain all of the different supergroups of the given type.