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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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