Example 10

   6
   1 3 1 0 0 0
   0 1 3 1 0 0
   0 0 1 3 1 0
   0 0 0 1 3 1
   1 0 0 0 1 3
   3 1 0 0 0 1

Used Programs

Solution

First Solution

1. Call

            Bravais_catalog
         

   and input 6-1 as family symbol. Write the Bravais group 6-1_1_1 onto file 'B' (i. e. calculate the bravais-groups, input filename 'B', choose "a selection of bravais-groups", and enter 1 for i and j). Now 'B' contains the Bravais group of I₆.
2. Write the above matrix M on a file 'm' and call

            Elt m
         

   to make sure that the lattice spanned by the columns of M has full rank. We get:

            6       % ELT of matrix
             1 0 0 0 0  0
             0 1 0 0 0  0
             0 0 1 0 0  0
             0 0 0 1 0  0
             0 0 0 0 8  0
             0 0 0 0 0 40
         

3. Call

            Orbit -g -S m B > mB
         

   to write the orbit and a generating set for the stabilizer on the file 'mB'. The length of the orbit runs out to be 1920, and hence the order of the stabilizer is 24. Edit the file 'mB' by omitting the orbit and leaving the generators of the stabilizer (i.e. the last 23 matrices). By calling

            Order mB
         

   one sees that the order is 24 indeed. Now 'mB' contains a set of generators of the stabilizer.

Second Solution

1. Prepare a file 'f' with the following content

            #3
   
            12 d0
            1
   
            12 d1
            1 1 1 1 1 1 0 0 0 0 0 0
   
            12x0
            0
            0 0
            0 0 0
            0 0 0 0
            0 0 0 0 0
            0 0 0 0 0 0
            1 3 1 0 0 0 0
            0 1 3 1 0 0 0 0
            0 0 1 3 1 0 0 0 0
            0 0 0 1 3 1 0 0 0 0
            1 0 0 0 1 3 0 0 0 0 0
            3 1 0 0 0 1 0 0 0 0 0 0
         

   and note that the Bravais group of the space of forms generated by these three matrices consists of the matrices diag(g,m^-1 g m) with g in the stabilizer of the lattice generated by the columns of m in the Bravais group of I₆.
2. Call

            Aut_grp f > Bf
         

   to get generators of the group described above in the file 'Bf'.
   Note the stabilizer of the lattice generated by the colums of M can be extracted from the file 'Bf' in its action on the standard lattice and also in its action on the sublattice.