CARAT Introduction / Examples / Find all Z-classes, affine classes and torsion free space groups in a given Q-class
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 12 - Find all Z-classes, affine classes and torsion free space groups in a given Q-class
Keywords: CARAT, crystallography, spacegroups, space groups, space group, spacegroup, group, mathematics, algorithms, tables, algorithm, table, examples, example, affine, affine class, Q-class, Z-class, Q class, Z class, torsion, torsion free, Bieberbach, Bieberbach group, Q_catalog, QtoZ, Extensions, Extract, Torsionfree, Presentation

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### Example 12

Find all Z-classes, affine classes and torsion free space groups in the Q-class of min.108.

### Used Programs

Extensions/Vector_systems, Extract, Presentation, Q_catalog, QtoZ, Torsionfree

### Solution

1. Call

                Q_catalog

   and enter

                s abb min.108

   to select the 5 dimensional Q-class with the CARAT name "min.108". Write the generators to the file 'min.108' by entering

                w min.108

   Enter "q" to leave the program.
2. Call

             QtoZ -D min.108

   to calculate the Z-classes in the Q-class of min.108. We get the following files:

   min.108.1.1 | min.108.1.2 | min.108.1.3 | min.108.1.4
   min.108.2.1 | min.108.2.2 | min.108.2.3 | min.108.2.4

3. Split the Z-classes into affine classes:

                for x in min.108.* ; do
                   Extensions $x > ex.$x ;
                done

   For every Z-class the programm "Extensions" calculates a presentation. Since the presentation is equal for each Z-class, it is faster to calculate a presentation with "Presentation" bevore and to start "Extensions" with the calculated presentation:

                Presentation min.108 > pres ;
                for x in min.108.* ; do
                   Extensions pres $x > ex.$x ;
                done

   Especially when there are many Z-classes, it is much faster to calcualte the presentation bevore.
4. For each of the cozycles for each Z-class write it into a file 'coz' and call

                Extract -r ex.min.108.i.j coz > min.108.i.j.k

   to construct the affine classes in the Z-class of G in standard form (i.e. the translation lattice is Zn):

   min.108.1.1.1 | min.108.1.1.2 | min.108.1.1.3 | min.108.1.1.4
   min.108.1.2.1 | min.108.1.2.2
   min.108.1.3.1 | min.108.1.3.2 | min.108.1.3.3 | min.108.1.3.4
   min.108.1.4.1 | min.108.1.4.2
   min.108.2.1.1 | min.108.2.1.2 | min.108.2.1.3 | min.108.2.1.4
   min.108.2.2.1 | min.108.2.2.2
   min.108.2.3.1 | min.108.2.3.2 | min.108.2.3.3 | min.108.2.3.4
   min.108.2.4.1 | min.108.2.4.2

5. Call

                for x in min.108.*.*.* ; do
                   echo $x ;
                   Torsionfree $x ;
                   echo ;
                done

   We get

       min.108.1.1.1
       The group in min.108.1.1.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.1.2
       The group in min.108.1.1.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.1.3
       The group in min.108.1.1.3 is torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.1.4
       The group in min.108.1.1.4 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.2.1
       The group in min.108.1.2.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.2.2
       The group in min.108.1.2.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.3.1
       The group in min.108.1.3.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.3.2
       The group in min.108.1.3.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.3.3
       The group in min.108.1.3.3 is torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.3.4
       The group in min.108.1.3.4 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.4.1
       The group in min.108.1.4.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.1.4.2
       The group in min.108.1.4.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.1.1
       The group in min.108.2.1.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.1.2
       The group in min.108.2.1.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.1.3
       The group in min.108.2.1.3 is torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.1.4
       The group in min.108.2.1.4 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.2.1
       The group in min.108.2.2.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.2.2
       The group in min.108.2.2.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.3.1
       The group in min.108.2.3.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.3.2
       The group in min.108.2.3.2 is torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.3.3
       The group in min.108.2.3.3 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.3.4
       The group in min.108.2.3.4 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.4.1
       The group in min.108.2.4.1 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

       min.108.2.4.2
       The group in min.108.2.4.2 is not torsion free
       The order of the point group is 6, and it has 3 conjugacy classes

   We see that there are 4 torsion free space groups. None has trivial center.

See also for Example 11!

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last change: 13.09.2000