Integer Lattice - Automorphism Group

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 S_n 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.