the 17 isometry classes of irreducible [12,9,3]_4 codes are: code no 1: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 2 1 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 48 and is strongly generated by the following 4 elements: ( 3 0 0 0 1 0 0 0 1 , 1 , 1 0 0 0 1 0 1 0 1 , 1 , 3 0 0 1 0 3 1 3 0 , 0 , 3 0 0 2 3 0 2 0 3 , 0 ) acting on the columns of the generator matrix as follows (in order): (4, 12)(5, 6)(8, 9), (3, 8)(4, 11)(6, 7), (2, 9)(3, 6)(5, 10)(7, 8), (2, 7)(3, 10)(5, 6)(8, 9) orbits: { 1 }, { 2, 9, 7, 8, 6, 3, 5, 10 }, { 4, 12, 11 } code no 2: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 1 0 0 2 1 0 3 0 1 , 0 , 1 0 0 2 0 1 3 1 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 6)(3, 10)(4, 11)(5, 7)(8, 9), (2, 10)(3, 6)(4, 11)(5, 9)(7, 8) orbits: { 1 }, { 2, 6, 10, 3 }, { 4, 11 }, { 5, 7, 9, 8 }, { 12 } code no 3: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 0 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 12 and is strongly generated by the following 4 elements: ( 2 0 0 0 2 0 0 0 2 , 1 , 3 0 0 0 0 3 0 3 0 , 1 , 2 0 0 1 0 3 3 1 0 , 0 , 3 0 0 2 1 0 1 0 2 , 0 ) acting on the columns of the generator matrix as follows (in order): (6, 7)(9, 10)(11, 12), (2, 3)(5, 8)(6, 10)(7, 9), (2, 9)(3, 7)(4, 11)(5, 8)(6, 10), (2, 7, 6)(3, 9, 10)(4, 11, 12) orbits: { 1 }, { 2, 3, 9, 6, 7, 10 }, { 4, 11, 12 }, { 5, 8 } code no 4: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 2 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 108 and is strongly generated by the following 6 elements: ( 1 0 0 0 3 0 0 0 2 , 1 , 3 0 0 0 2 0 3 0 3 , 1 , 3 0 0 0 2 0 1 0 2 , 1 , 2 0 0 2 0 3 3 3 0 , 1 , 3 0 0 2 2 0 3 0 2 , 0 , 3 1 0 3 2 1 0 3 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 12)(5, 7)(8, 9), (3, 8)(4, 11)(5, 7), (3, 9)(5, 7)(11, 12), (2, 9)(3, 5)(6, 10)(7, 8), (2, 7, 5)(3, 8, 9), (1, 10, 6)(2, 3, 4, 7, 8, 12)(5, 9, 11) orbits: { 1, 6, 10 }, { 2, 9, 5, 12, 8, 3, 7, 11, 4 } code no 5: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 2 1 1 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 2 and is strongly generated by the following 1 elements: ( 2 0 0 0 3 0 0 3 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (3, 10)(4, 9)(5, 6)(8, 11) orbits: { 1 }, { 2 }, { 3, 10 }, { 4, 9 }, { 5, 6 }, { 7 }, { 8, 11 }, { 12 } code no 6: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 2 1 1 0 0 0 0 0 0 0 1 0 3 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 2 0 0 0 2 0 2 2 2 , 1 , 1 0 0 0 2 0 2 0 2 , 0 ) acting on the columns of the generator matrix as follows (in order): (3, 4)(6, 7)(8, 10)(9, 11), (3, 9, 8)(4, 10, 11)(5, 6, 7) orbits: { 1 }, { 2 }, { 3, 4, 8, 11, 10, 9 }, { 5, 7, 6 }, { 12 } code no 7: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 2 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 12 and is strongly generated by the following 4 elements: ( 3 0 0 0 3 0 0 0 1 , 1 , 2 0 0 2 2 0 0 0 2 , 0 , 0 2 0 2 0 0 0 0 2 , 0 , 2 2 0 0 2 0 0 0 2 , 0 ) acting on the columns of the generator matrix as follows (in order): (4, 12)(6, 7)(8, 9)(10, 11), (2, 5)(4, 10)(6, 7)(11, 12), (1, 2)(6, 7)(8, 10)(9, 11), (1, 5)(4, 8)(6, 7)(9, 12) orbits: { 1, 2, 5 }, { 3 }, { 4, 12, 10, 8, 11, 9 }, { 6, 7 } code no 8: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 3 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 2 0 0 1 3 0 3 0 1 , 1 , 2 2 0 0 2 0 0 2 2 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 7)(3, 9)(4, 11)(10, 12), (1, 5)(3, 10)(9, 12) orbits: { 1, 5 }, { 2, 7 }, { 3, 9, 10, 12 }, { 4, 11 }, { 6 }, { 8 } code no 9: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 2 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 3 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 0 2 0 2 0 0 0 0 2 , 0 , 3 1 0 1 3 0 1 1 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (1, 2)(6, 7)(8, 10)(9, 11), (1, 7, 2, 6)(3, 12)(8, 11, 10, 9) orbits: { 1, 2, 6, 7 }, { 3, 12 }, { 4 }, { 5 }, { 8, 10, 9, 11 } code no 10: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 2 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 1 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 3 0 0 0 1 1 0 0 1 , 1 , 0 2 2 0 3 0 2 3 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 9)(4, 6)(5, 10)(7, 8), (1, 9)(3, 6)(5, 11)(7, 12) orbits: { 1, 9, 2 }, { 3, 6, 4 }, { 5, 10, 11 }, { 7, 8, 12 } code no 11: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 2 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 1 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 4 and is strongly generated by the following 2 elements: ( 3 0 0 0 0 1 0 1 0 , 1 , 2 3 2 0 1 0 0 0 2 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 3)(4, 10)(5, 8)(6, 7), (1, 12)(4, 5)(6, 7)(8, 10)(9, 11) orbits: { 1, 12 }, { 2, 3 }, { 4, 10, 5, 8 }, { 6, 7 }, { 9, 11 } code no 12: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 2 1 1 0 0 0 0 0 0 1 0 0 1 2 1 0 0 0 0 0 0 0 1 0 2 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 12 and is strongly generated by the following 3 elements: ( 2 0 0 0 3 0 0 0 3 , 1 , 3 0 0 0 1 1 0 0 1 , 1 , 3 0 1 0 2 0 0 0 2 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 10)(5, 6)(7, 8)(11, 12), (2, 9)(4, 6)(5, 10)(7, 8), (1, 8)(4, 11)(5, 12)(6, 10) orbits: { 1, 8, 7 }, { 2, 9 }, { 3 }, { 4, 10, 6, 11, 5, 12 } code no 13: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 1 2 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 6 and is strongly generated by the following 2 elements: ( 2 0 0 1 1 1 2 1 0 , 0 , 1 0 0 1 1 0 3 1 1 , 1 ) acting on the columns of the generator matrix as follows (in order): (2, 7, 4)(3, 9, 6)(5, 8, 10), (2, 5)(3, 9)(4, 8)(7, 10)(11, 12) orbits: { 1 }, { 2, 4, 5, 7, 8, 10 }, { 3, 6, 9 }, { 11, 12 } code no 14: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 0 0 0 1 0 0 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 3 0 0 0 1 0 0 0 1 , 1 , 3 0 0 0 0 1 0 1 0 , 1 , 1 2 2 0 3 2 0 2 3 , 0 ) acting on the columns of the generator matrix as follows (in order): (4, 9)(5, 6)(7, 8)(11, 12), (2, 3)(4, 9)(5, 8)(6, 7), (1, 10)(2, 11)(3, 12)(4, 9)(6, 8) orbits: { 1, 10 }, { 2, 3, 11, 12 }, { 4, 9 }, { 5, 6, 8, 7 } code no 15: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 0 0 0 1 0 0 2 2 1 0 0 0 0 0 0 0 1 0 1 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 432 and is strongly generated by the following 7 elements: ( 3 0 0 0 1 0 0 0 1 , 1 , 3 0 0 0 1 0 2 1 1 , 1 , 2 0 0 2 1 0 2 0 1 , 0 , 3 0 0 2 1 1 1 1 0 , 0 , 2 2 1 1 2 2 0 0 1 , 0 , 1 2 1 2 3 0 2 2 2 , 1 , 3 2 2 2 0 0 2 2 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (4, 9)(5, 6)(7, 8)(11, 12), (3, 10)(4, 7)(5, 6)(8, 9), (2, 5, 6)(3, 7, 8)(4, 10, 9), (2, 8, 9)(3, 4, 5)(6, 7, 10), (1, 11)(2, 10)(4, 8)(5, 7), (1, 11, 12)(2, 8, 10, 4, 3, 6)(5, 9), (1, 2, 11, 3, 12, 10)(4, 6, 8)(5, 9) orbits: { 1, 11, 12, 10, 2, 3, 4, 7, 8, 6, 9, 5 } code no 16: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 2 0 1 0 0 0 0 1 0 0 0 0 2 1 1 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 2 3 1 0 0 0 0 0 0 0 1 0 3 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 48 and is strongly generated by the following 5 elements: ( 2 0 0 2 0 1 3 3 0 , 0 , 1 0 0 0 0 2 0 2 0 , 1 , 1 0 3 0 3 0 1 0 1 , 0 , 3 2 1 0 3 0 0 1 2 , 1 , 1 1 2 1 0 1 2 2 2 , 0 ) acting on the columns of the generator matrix as follows (in order): (2, 8)(3, 5)(4, 12)(6, 7)(9, 11), (2, 3)(4, 9)(5, 8)(6, 7)(11, 12), (1, 8)(3, 7)(4, 10)(5, 9)(6, 11), (1, 11)(3, 10)(4, 7)(5, 9)(6, 8), (1, 9, 6, 12)(2, 10, 5, 7)(3, 8, 11, 4) orbits: { 1, 8, 11, 12, 2, 5, 6, 3, 9, 4, 7, 10 } code no 17: ================ 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 2 0 1 0 0 0 1 0 0 0 0 0 3 1 1 0 0 0 0 1 0 0 0 0 1 2 1 0 0 0 0 0 1 0 0 0 3 2 1 0 0 0 0 0 0 1 0 0 0 3 1 0 0 0 0 0 0 0 1 0 2 3 1 0 0 0 0 0 0 0 0 1 the automorphism group has order 576 and is strongly generated by the following 7 elements: ( 1 0 0 0 2 0 0 0 3 , 1 , 3 0 0 0 1 0 1 3 2 , 1 , 1 0 0 0 2 0 3 3 3 , 1 , 1 0 0 0 1 0 1 2 1 , 0 , 3 0 0 1 2 0 3 2 2 , 1 , 3 2 1 3 3 3 0 0 2 , 0 , 3 1 2 1 2 1 3 3 3 , 0 ) acting on the columns of the generator matrix as follows (in order): (4, 10)(5, 6)(8, 9), (3, 12)(5, 6)(7, 11), (3, 10, 12, 4)(5, 6)(7, 9, 11, 8), (3, 9)(4, 11)(7, 10)(8, 12), (2, 6)(3, 8)(4, 7), (1, 10)(2, 4)(5, 11)(6, 7), (1, 4, 3, 6, 7, 12)(2, 11, 8, 5, 10, 9) orbits: { 1, 10, 12, 4, 3, 7, 5, 8, 11, 2, 9, 6 }