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#g2 3 % generator 1 0 0 1 -1 0 0 0 -1 3 % generator -1 1 0 -1 0 0 0 0 1 2^1 * 3^1 = 6 % order of the groupWrite this into a file 'G' and call
Graph GYou will get the following output:
3 % graph for the arithmetic classes (1) 3 2 1 0 2 1 (2) 1 0 3 There are 3 Z-Classes with 2 1 2 Space Groups! (3) 1: 1 (2, 2^1) (4) 1: 1 (4, 2^2) 1: 1 (3, 3^1) 2 (6, 3^1) (5) 2: 2 (2, 2^1) 2: 2 (4, 2^2) 1: 3 (18, 3^1) 1: 4 (9, 3^1) 2: 5 (9, 3^1) 3: 3 (2, 2^1) 3: 3 (4, 2^2) 3: 4 (3, 3^1) 5 (6, 3^1) 4: 1 (1, 3^1) 5: 2 (1, 3^1) 4: 4 (2, 2^1) 4: 4 (4, 2^2) 4: 4 (3, 3^1) 5 (6, 3^1) 5: 5 (2, 2^1) 5: 5 (4, 2^2) 5 % inclusions for all spacegroups 3 1 1 1 0 (6) 0 2 0 0 1 0 0 2 1 1 1 0 0 3 1 0 1 0 0 2Comments to the output:
(1) | The number in this line is the number of Z-classes. |
(2) | This matrix yields information about the graph of the arithmetic classes. There are 3 + 2 + 1 = 6 maximal sublattices of a representative of the first Z-class which are invariant under this group. If you conjugate this representative with the sublattices, you get 3 groups in the first Z-class, 2 groups in the second Z-class and 1 group in the third Z-class. |
(3) | This line gives you the numbers of the affine classes in the various Z-classes. The affine classes are numbered in ascending order with respect to the Z-classes. So the first affine class of the third Z-class gets the number 4. The first affine class in each Z-class contains the symmorphic space groups. |
(4) | The first space group has 2 maximal k-subgroups of index 2^1 which are conjugated under the affine normalizer of the spacegroup. These subgroups are in the first affine class. |
(5) | The first space group has 3 maximal k-subgroups of index 3^1 which are conjugated under the affine normalizer of the spacegroup. These subgroups are in the first affine class. There are 2 maximal k-subgroups of index 3^1 which are conjugated under the affine normalizer of the spacegroup. The translation lattices for all these subgroups are in one orbit under the stabilizer of the cocycle of the spacegroup, so we print the orbits in one line. |
(6) | This matrix gives you the numbers of orbits under the affine normalizer of a spacegroup on the maximal k-subgroups. There are 3 + 1 + 1 + 1 orbits for a representative of the first affine class. The groups in 3 of these orbits are in the first affine class. The groups in one orbit are in the second affine class, etc. |
#g2 3 % generator -1 0 0 0 -1 0 0 0 -1 3 % generator 0 1 0 -1 -1 0 0 0 1 2^1 * 3^1 = 6 % order of the groupWrite this into a file 'G' and call
Graph GYou will get the following output:
2 % graph for the arithmetic classes 4 2 1 2 There are 2 Z-Classes with 1 1 Space Groups! 1: 1 (2, 2^1) 1: 1 (4, 2^2) 1: 1 (3, 3^1) (1) 1: 1 (3, 3^1) (2) 1: 2 (6, 3^1) 2: 1 (3, 3^1) 2: 2 (2, 2^1) 2: 2 (4, 2^2) 2 % inclusions for all spacegroups 4 1 1 2Comments to the output:
In this example, there are two orbits each with 3 maximal k-subgroups of a representative of the first affine class. They are printed in separate lines ((1) and (2)) because the translation lattices for these groups are NOT conjugated under the stabilizer of the cocycle of a representative for the first affine class.
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