CARAT Introduction / Examples / Find all Z-classes, affine classes and torsion free space groups in a given Q-class

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

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.

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

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.

For each of the cocycles 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

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.