the 3 isometry classes of irreducible [8,2,4]_3 codes are: code no 1: ================ 1 1 1 1 1 1 2 0 1 1 1 0 0 0 0 2 the automorphism group has order 144 and is strongly generated by the following 5 elements: ( 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 1 1 1 1 1 1 , 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 2 0 , 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 2 2 2 2 2 2 0 0 0 0 0 1 0 0 0 0 1 0 , 2 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 2 0 , 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 2 2 2 2 2 2 0 0 0 0 1 0 0 0 0 1 0 0 ) acting on the columns of the generator matrix as follows (in order): (6, 7), (5, 6), (4, 7)(5, 6), (2, 3)(4, 5, 6), (1, 2, 3)(4, 6, 7) orbits: { 1, 3, 2 }, { 4, 7, 6, 5 }, { 8 } code no 2: ================ 1 1 1 1 1 1 2 0 2 1 1 0 0 0 0 2 the automorphism group has order 96 and is strongly generated by the following 5 elements: ( 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 1 1 1 1 1 1 , 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 1 1 1 1 1 1 0 0 0 0 0 2 , 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 2 , 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 2 2 2 2 2 2 , 1 2 2 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 2 2 2 2 2 2 0 0 0 1 0 0 ) acting on the columns of the generator matrix as follows (in order): (6, 7), (5, 7), (4, 5), (2, 3)(4, 5, 7, 6), (1, 8)(2, 3)(4, 6, 7, 5) orbits: { 1, 8 }, { 2, 3 }, { 4, 5, 6, 7 } code no 3: ================ 1 1 1 0 0 0 2 0 1 1 0 1 1 1 0 2 the automorphism group has order 192 and is strongly generated by the following 6 elements: ( 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 1 0 2 2 0 2 2 2 , 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 0 1 0 0 0 0 1 0 , 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 , 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 2 2 0 2 2 2 0 0 0 0 0 1 0 0 0 1 0 0 , 2 0 0 0 0 0 0 2 0 0 0 0 1 1 1 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 2 , 0 0 1 0 0 0 2 2 2 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 2 0 ) acting on the columns of the generator matrix as follows (in order): (6, 8), (5, 6), (4, 6), (4, 6, 5, 8), (3, 7)(4, 5), (1, 7, 2, 3)(5, 6) orbits: { 1, 3, 7, 2 }, { 4, 6, 8, 5 }