the 8 isometry classes of irreducible [12,5,6]_3 codes are: code no 1: ================ 1 1 1 1 1 1 1 2 0 0 0 0 2 2 1 1 1 0 0 0 2 0 0 0 2 1 2 1 0 1 0 0 0 2 0 0 0 2 2 2 1 1 0 0 0 0 2 0 1 2 0 1 2 1 0 0 0 0 0 2 the automorphism group has order 72 and is strongly generated by the following 5 elements: ( 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 1 1 1 1 1 1 1 , 2 0 0 0 0 0 0 0 0 0 0 1 0 0 2 2 1 1 1 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 1 2 1 2 0 2 0 1 1 1 1 1 1 1 , 2 0 0 0 0 0 0 2 1 2 1 0 1 0 0 2 2 2 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 2 2 2 2 2 2 2 , 0 0 0 0 2 0 0 0 0 1 0 0 0 0 2 2 1 1 1 0 0 0 0 0 1 0 0 0 0 2 2 2 1 1 0 2 0 0 0 0 0 0 2 2 2 2 2 2 2 , 2 1 0 2 1 2 0 1 2 1 2 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 2 2 2 0 0 2 2 2 2 2 2 2 ) acting on the columns of the generator matrix as follows (in order): (7, 8), (2, 5)(3, 9)(6, 10)(7, 8)(11, 12), (2, 6, 5, 10)(3, 12, 9, 11)(7, 8), (1, 6, 11, 5)(2, 10, 9, 3)(7, 8), (1, 12)(2, 10)(3, 5)(6, 9)(7, 8) orbits: { 1, 5, 12, 2, 6, 11, 3, 10, 9 }, { 4 }, { 7, 8 } code no 2: ================ 1 1 1 1 1 1 1 2 0 0 0 0 2 2 1 1 1 0 0 0 2 0 0 0 2 1 2 1 0 1 0 0 0 2 0 0 0 2 2 2 1 1 0 0 0 0 2 0 2 1 0 1 2 0 1 0 0 0 0 2 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 1 0 0 0 0 0 0 2 2 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 2 1 2 1 0 1 0 2 1 0 1 2 0 1 , 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 2 0 2 0 2 2 2 2 2 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 2 1 1 1 0 0 , 0 1 1 1 2 2 0 2 2 1 1 1 0 0 2 1 2 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (2, 9)(3, 5)(6, 10)(7, 12), (2, 12, 9, 7)(3, 6, 5, 10)(4, 8), (1, 11)(2, 9)(3, 10)(5, 6) orbits: { 1, 11 }, { 2, 9, 7, 12 }, { 3, 5, 10, 6 }, { 4, 8 } code no 3: ================ 1 1 1 1 1 1 1 2 0 0 0 0 2 2 1 1 1 0 0 0 2 0 0 0 2 1 2 1 0 1 0 0 0 2 0 0 1 2 2 1 0 0 1 0 0 0 2 0 2 1 0 2 2 1 1 0 0 0 0 2 the automorphism group has order 12 and is strongly generated by the following 3 elements: ( 2 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 2 2 1 2 1 0 1 0 2 1 1 2 0 0 2 2 1 0 2 2 1 1 0 0 2 0 0 0 0 , 2 2 1 1 1 0 0 2 1 2 1 0 1 0 1 2 2 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 , 0 0 0 0 2 0 0 0 1 0 0 0 0 0 1 1 2 2 2 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 2 2 2 2 2 2 2 1 2 2 1 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (2, 8)(3, 7)(4, 10)(5, 11)(6, 12), (1, 9)(2, 10)(3, 11)(4, 8)(5, 7), (1, 5)(3, 9)(6, 8)(7, 11)(10, 12) orbits: { 1, 9, 5, 3, 11, 7 }, { 2, 8, 10, 4, 6, 12 } code no 4: ================ 1 1 1 1 1 1 1 2 0 0 0 0 2 2 1 1 1 0 0 0 2 0 0 0 2 1 2 1 0 1 0 0 0 2 0 0 2 1 0 1 2 0 1 0 0 0 2 0 2 0 1 1 0 2 1 0 0 0 0 2 the automorphism group has order 72 and is strongly generated by the following 5 elements: ( 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 2 2 2 0 0 2 1 2 1 0 1 0 , 1 0 0 0 0 0 0 0 1 0 0 0 0 0 2 2 1 1 1 0 0 2 2 2 2 2 2 2 0 0 0 0 0 1 0 2 1 2 1 0 1 0 2 0 1 1 0 2 1 , 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 1 1 2 2 2 0 0 0 0 0 0 0 0 1 2 1 2 1 0 1 0 , 2 0 0 0 0 0 0 1 2 0 2 1 0 2 0 0 0 0 0 1 0 1 1 1 1 1 1 1 2 1 2 1 0 1 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 , 1 0 0 0 0 0 0 2 0 1 1 0 2 1 0 0 2 0 0 0 0 2 2 2 2 2 2 2 0 2 0 0 0 0 0 1 2 1 2 0 2 0 0 0 0 0 1 0 0 ) acting on the columns of the generator matrix as follows (in order): (3, 10, 7, 5)(6, 12, 11, 9), (3, 12, 7, 9)(4, 8)(5, 11, 10, 6), (2, 3, 9, 5)(6, 11, 10, 7), (2, 7, 9, 11)(3, 10, 5, 6)(4, 8), (2, 5, 7, 12)(4, 8)(6, 11, 9, 10) orbits: { 1 }, { 2, 5, 11, 12, 7, 6, 9, 10, 3 }, { 4, 8 } code no 5: ================ 1 1 1 1 1 0 0 2 0 0 0 0 2 2 1 1 0 1 0 0 2 0 0 0 2 1 2 0 1 1 0 0 0 2 0 0 1 2 0 2 1 1 0 0 0 0 2 0 2 1 2 1 0 0 1 0 0 0 0 2 the automorphism group has order 1440 and is strongly generated by the following 5 elements: ( 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 2 1 2 1 0 0 1 , 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 2 2 2 2 2 0 0 1 1 2 2 0 2 0 0 0 0 0 0 0 1 , 2 0 0 0 0 0 0 0 2 0 0 0 0 0 1 2 1 0 2 2 0 2 1 0 1 2 2 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 , 1 0 0 0 0 0 0 1 2 1 0 2 2 0 1 1 2 2 0 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 , 1 1 1 1 1 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 2 2 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 ) acting on the columns of the generator matrix as follows (in order): (7, 12), (5, 8)(6, 9)(10, 11), (3, 10)(4, 11)(8, 9), (2, 10)(3, 9)(4, 5)(6, 8), (1, 6, 11, 8)(3, 9, 5, 4) orbits: { 1, 8, 5, 9, 6, 11, 4, 3, 10, 2 }, { 7, 12 } code no 6: ================ 1 1 1 1 1 0 0 2 0 0 0 0 2 2 1 1 0 1 0 0 2 0 0 0 2 1 2 0 1 1 0 0 0 2 0 0 1 2 0 2 1 1 0 0 0 0 2 0 1 0 2 1 2 1 1 0 0 0 0 2 the automorphism group has order 1440 and is strongly generated by the following 5 elements: ( 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 1 2 1 2 2 , 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 , 2 0 0 0 0 0 0 0 2 0 0 0 0 0 1 2 0 2 1 1 0 2 1 2 0 1 1 0 1 1 1 1 1 0 0 2 2 1 1 0 1 0 0 0 0 0 0 0 1 , 2 0 0 0 0 0 0 0 0 2 0 0 0 0 2 1 2 0 1 1 0 0 0 0 0 0 2 0 2 2 2 2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 , 0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 ) acting on the columns of the generator matrix as follows (in order): (7, 12), (3, 5, 4, 6)(8, 11, 9, 10), (3, 10, 4, 11)(5, 9, 6, 8), (2, 9, 10, 3)(4, 8, 5, 6), (1, 6, 4, 3)(2, 10, 9, 8) orbits: { 1, 3, 6, 11, 10, 4, 9, 5, 8, 2 }, { 7, 12 } code no 7: ================ 1 1 1 1 1 0 0 2 0 0 0 0 2 2 1 1 0 1 0 0 2 0 0 0 2 1 2 0 1 1 0 0 0 2 0 0 2 1 2 1 0 0 1 0 0 0 2 0 2 2 1 0 1 0 1 0 0 0 0 2 the automorphism group has order 432 and is strongly generated by the following 6 elements: ( 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 , 2 0 0 0 0 0 0 0 2 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 1 1 2 2 0 2 0 1 1 2 0 2 0 2 , 1 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 2 1 2 0 1 1 0 2 1 2 1 0 0 1 , 2 0 0 0 0 0 0 2 1 2 0 1 1 0 2 2 1 1 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 2 , 1 0 0 0 0 0 0 1 1 2 2 0 2 0 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 2 1 2 1 0 0 1 0 2 0 0 0 0 0 , 1 2 1 0 2 2 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 1 1 2 2 0 2 0 0 1 0 0 0 0 0 1 2 1 2 0 0 2 ) acting on the columns of the generator matrix as follows (in order): (4, 5)(6, 7)(9, 12)(10, 11), (3, 8)(4, 5)(6, 9)(7, 12), (2, 8)(4, 5)(6, 10)(7, 11), (2, 10)(3, 9)(4, 5)(6, 8), (2, 7, 9)(3, 11, 6)(8, 12, 10), (1, 8, 9, 5, 3, 10)(2, 6, 4)(7, 11) orbits: { 1, 10, 11, 6, 2, 12, 3, 7, 9, 8, 4, 5 } code no 8: ================ 1 1 1 1 1 0 0 2 0 0 0 0 2 2 1 1 0 1 0 0 2 0 0 0 2 1 2 0 1 1 0 0 0 2 0 0 2 1 2 1 0 0 1 0 0 0 2 0 0 1 1 2 2 1 1 0 0 0 0 2 the automorphism group has order 72 and is strongly generated by the following 3 elements: ( 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 2 2 0 2 0 2 2 2 2 2 0 0 2 1 2 1 0 0 1 , 1 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 2 1 2 0 1 1 0 2 1 2 1 0 0 1 , 2 1 2 0 1 1 0 0 0 1 0 0 0 0 1 1 2 2 0 2 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 ) acting on the columns of the generator matrix as follows (in order): (3, 4)(5, 9)(6, 8)(7, 11), (2, 8)(4, 5)(6, 10)(7, 11), (1, 8, 4, 10)(2, 6, 9, 3) orbits: { 1, 10, 6, 4, 8, 2, 3, 5, 9 }, { 7, 11 }, { 12 }