CARAT Introduction / Examples / Calculate minimal/maximal klassengleich/translationengleich super-/sub-groups of a given space group up to conjugation under the affine normalizer of this group.

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### Example 14

Let R be the group generated by the matrices below. Calculate minimal/maximal klassengleich/translationengleich super-/sub-groups of R up to conjugation under its affine normalizer.

    3       % generator
     0 1 0
     1 0 0
     0 0 1
    3       % generator
     -1  0 0
      0 -1 0
      0  0 1
    3       % generator
     1 0 1
     0 1 0
     0 0 1
    3       % generator
     1 0 0
     0 1 1
     0 0 1

### Used Programs

KSubgroups, KSupergroups, TSubgroups, TSupergroups

### Solution

1. Write the matrices in a file 'R' and add

         #g4

   as top row.
2. Call

          KSubgroups -a -t -n R 2

   to get the maximal klassengleich subgroups of R with index a power of 2 up to conjugation under the affine normalizer of R:

       L1 with (0,0) yields L2

       Lengthes of the orbits:
       1 2 1

       #g4 % 1-th maximal klassengleich subgroup of R
       3       % generator
        0 1 0
        1 0 0
        0 0 1
       3       % generator
        -1  0 0
         0 -1 0
         0  0 1
       3       % generator
        1 0 1
        0 1 1
        0 0 1
       3       % generator
        1 0 0
        0 1 2
        0 0 1
       % order of the group unknown
       #g4 % 2-th maximal klassengleich subgroup of R
       3       % generator
        0 1 1
        1 0 0
        0 0 1
       3       % generator
        -1  0 1
         0 -1 0
         0  0 1
       3       % generator
        1 0 1
        0 1 1
        0 0 1
       3       % generator
        1 0 0
        0 1 2
        0 0 1
       % order of the group unknown
       #g4 % 3-th maximal klassengleich subgroup of R
       3       % generator
        0 1 1
        1 0 0
        0 0 1
       3       % generator
        -1  0 2
         0 -1 0
         0  0 1
       3       % generator
        1 0 1
        0 1 1
        0 0 1
       3       % generator
        1 0 0
        0 1 2
        0 0 1
       % order of the group unknown

3. Call

          KSupergroups -a -t -n R 2

   to get the minimal klassengleich supergroups of R with index a power of 2 up to conjugation under the affine normalizer of R:

       L1 with (0,0) yields L2

       Lengthes of the orbits:
       1

       #g4
       3       % generator
        0 1 0
        1 0 0
        0 0 1
       3       % generator
        -1  0 0
         0 -1 0
         0  0 1
       3       /2              % generator
        2 0 1
        0 2 1
        0 0 2
       3       % generator
        1 0 0
        0 1 1
        0 0 1
       % order of the group unknown

4. Call

          TSubgroups -a -t R

   to get the maximal translationengleich subgroups of R up to conjugation under the affine normalizer of R (for a generating system you have to add generators of the translation normal subgroup):

       #g1 % orbitlength: 2, pointgroup order: 2
       3       % generator
        0 1 0
        1 0 0
        0 0 1
       % order of the group unknown
       #g1 % orbitlength: 1, pointgroup order: 2
       3       % generator
        -1  0 0
         0 -1 0
         0  0 1
       % order of the group unknown

5. Call

          TSupergroups -t R

   to get the minimal translationengleich supergroups of R up to conjugation under the affine normalizer of R (for a generating system you have to add generators of the translation normal subgroup):

       #g2 % 1-th supergroup of R
       3       % generator
        1  0 0
        0 -1 0
        0  0 1
       3       % generator
        0 -1 0
        1  0 0
        0  0 1
       % order of the group unknown
       #g2 % 2-th supergroup of R
       3       /2              % generator
        2  0 -1
        0 -2  3
        0  0  2
       3       /2              % generator
        0 -2 1
        2  0 1
        0  0 2
       % order of the group unknown
       #g2 % 3-th supergroup of R
       3       % generator
         0 -1 0
        -1  0 0
         0  0 1
       3       % generator
         1 1 0
        -1 0 0
         0 0 1
       % order of the group unknown

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