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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 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
  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