the 6 isometry classes of irreducible [6,3,4]_9 codes are: code no 1: ================ 1 1 1 2 0 0 3 2 1 0 2 0 2 3 1 0 0 2 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 2 0 0 5 5 5 8 5 7 , 0 , 2 2 2 0 7 0 2 3 1 , 0 ) acting on the columns of the generator matrix as follows (in order): (2, 4)(3, 5), (1, 4)(3, 6) orbits: { 1, 4, 2 }, { 3, 5, 6 } code no 2: ================ 1 1 1 2 0 0 3 2 1 0 2 0 6 3 1 0 0 2 the automorphism group has order 48 and is strongly generated by the following 4 elements: ( 8 0 0 0 3 0 1 3 6 , 1 , 8 0 0 4 4 4 0 0 8 , 1 , 6 0 0 4 1 2 2 2 2 , 1 , 2 4 8 2 1 4 0 7 0 , 0 ) acting on the columns of the generator matrix as follows (in order): (3, 5), (2, 4), (2, 5)(3, 4), (1, 4, 5)(2, 3, 6) orbits: { 1, 5, 3, 2, 4, 6 } code no 3: ================ 1 1 1 2 0 0 3 2 1 0 2 0 6 4 1 0 0 2 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 7 2 6 5 5 5 0 8 0 , 0 , 0 3 0 7 8 4 7 1 8 , 1 ) acting on the columns of the generator matrix as follows (in order): (1, 5, 6)(2, 3, 4), (1, 4, 6, 3, 5, 2) orbits: { 1, 6, 2, 5, 4, 3 } code no 4: ================ 1 1 1 2 0 0 3 2 1 0 2 0 7 4 1 0 0 2 the automorphism group has order 8 and is strongly generated by the following 2 elements: ( 6 0 0 0 0 7 0 5 0 , 1 , 8 5 7 0 0 8 0 8 0 , 0 ) acting on the columns of the generator matrix as follows (in order): (2, 3)(4, 5), (1, 5)(2, 3)(4, 6) orbits: { 1, 5, 4, 6 }, { 2, 3 } code no 5: ================ 1 1 1 2 0 0 3 2 1 0 2 0 4 8 1 0 0 2 the automorphism group has order 8 and is strongly generated by the following 2 elements: ( 8 0 0 4 4 4 0 0 8 , 1 , 2 6 3 0 2 0 3 6 2 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 4), (1, 6, 3, 5) orbits: { 1, 5, 3, 6 }, { 2, 4 } code no 6: ================ 1 1 1 2 0 0 4 3 1 0 2 0 5 4 1 0 0 2 the automorphism group has order 120 and is strongly generated by the following 7 elements: ( 4 0 0 0 4 0 3 7 8 , 0 , 6 0 0 0 8 0 1 1 1 , 1 , 8 0 0 0 0 3 5 5 5 , 1 , 6 0 0 4 1 3 2 2 2 , 0 , 0 4 0 8 8 8 4 0 0 , 1 , 1 5 8 7 0 0 0 3 0 , 1 , 2 6 5 0 0 6 1 1 1 , 0 ) acting on the columns of the generator matrix as follows (in order): (3, 6)(4, 5), (3, 5, 6, 4), (2, 5, 4, 3), (2, 6)(3, 4), (1, 3, 4, 2), (1, 2, 3, 5), (1, 5, 6)(2, 4, 3) orbits: { 1, 2, 5, 6, 3, 4 }