Automorphism Group
The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2n n!. As a matrix group it is given by the set of all n×n signed permutation matrices. This group is isomorphic to the semidirect product
where the symmetric group Sn acts on (Z2)n by permutation (this is a classic example of a wreath product).
For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three dimensional cubic lattice, we get the group of the cube, or octahedral group, of order 48.
Read more about this topic: Integer Lattice
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