the 2 isometry classes of irreducible [6,2,3]_4 codes are: code no 1: ================ 1 1 1 1 1 0 1 1 0 0 0 1 the automorphism group has order 24 and is strongly generated by the following 4 elements: ( 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3 , 1 , 2 0 0 0 0 2 0 0 0 0 2 0 2 2 2 2 , 1 , 2 0 0 0 0 2 0 0 2 2 2 2 0 0 2 0 , 1 , 0 3 0 0 3 0 0 0 0 0 3 0 3 3 3 3 , 0 ) acting on the columns of the generator matrix as follows (in order): id, (4, 5), (3, 4, 5), (1, 2)(4, 5) orbits: { 1, 2 }, { 3, 5, 4 }, { 6 } code no 2: ================ 1 1 1 1 1 0 2 1 0 0 0 1 the automorphism group has order 36 and is strongly generated by the following 6 elements: ( 3 0 0 0 0 3 0 0 0 0 3 0 3 3 3 3 , 0 , 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 , 0 , 2 0 0 0 0 2 0 0 2 2 2 2 0 0 0 2 , 0 , 1 0 0 0 2 3 0 0 3 3 3 3 0 0 3 0 , 1 , 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 , 1 , 1 2 0 0 0 3 0 0 0 0 1 0 0 0 0 1 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 5), (3, 4), (3, 5), (2, 6)(3, 4, 5), (1, 2)(3, 4), (1, 6) orbits: { 1, 2, 6 }, { 3, 4, 5 }