the 4 isometry classes of irreducible [10,2,7]_4 codes are: code no 1: ================ 1 1 1 1 1 1 1 1 1 0 2 2 2 1 1 1 0 0 0 1 the automorphism group has order 1296 and is strongly generated by the following 7 elements: ( 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 2 2 2 2 2 2 2 2 , 0 , 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 2 0 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 0 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 3 3 3 3 3 3 3 3 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 , 1 , 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 , 0 , 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 , 1 , 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 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 2 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (8, 9), (7, 8, 9), (5, 6)(7, 8, 9), (4, 9, 6, 8)(5, 7), (2, 3)(7, 9, 8), (1, 3, 2)(4, 8, 6, 9)(5, 7), (1, 5, 2, 4, 3, 6)(7, 8, 9) orbits: { 1, 2, 6, 3, 5, 9, 8, 4, 7 }, { 10 } code no 2: ================ 1 1 1 1 1 1 1 1 1 0 3 2 2 1 1 1 0 0 0 1 the automorphism group has order 288 and is strongly generated by the following 6 elements: ( 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 3 3 3 3 3 3 3 3 , 0 , 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 , 0 , 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 2 0 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 3 3 3 3 3 3 3 3 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 , 1 , 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 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 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 2 , 0 , 3 1 1 2 2 2 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (8, 9), (7, 8), (5, 6)(7, 8, 9), (4, 8, 5, 7, 6, 9), (2, 3)(4, 6, 5)(7, 9), (1, 10)(2, 3)(5, 6)(7, 8, 9) orbits: { 1, 10 }, { 2, 3 }, { 4, 9, 5, 8, 6, 7 } code no 3: ================ 1 1 1 1 1 1 1 1 1 0 3 3 2 2 1 1 0 0 0 1 the automorphism group has order 288 and is strongly generated by the following 7 elements: ( 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 2 2 2 2 2 2 2 2 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 , 0 , 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 , 1 , 2 0 0 0 0 0 0 0 0 2 0 0 0 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 2 0 0 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 2 , 1 , 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 , 1 , 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 , 1 ) acting on the columns of the generator matrix as follows (in order): (8, 9), (7, 9, 8), (5, 6)(7, 9, 8), (3, 6)(4, 5)(7, 8), (3, 6, 4, 5)(7, 9), (1, 3)(2, 4)(5, 6), (1, 2)(3, 6, 4, 5)(7, 8) orbits: { 1, 3, 2, 6, 5, 4 }, { 7, 8, 9 }, { 10 } code no 4: ================ 1 1 1 1 1 1 0 0 1 0 2 2 1 1 0 0 1 1 0 1 the automorphism group has order 576 and is strongly generated by the following 7 elements: ( 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 1 1 3 3 0 0 3 3 , 0 , 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 3 3 3 3 3 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 , 0 , 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 3 3 3 3 3 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 , 0 , 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 2 2 0 0 2 2 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 , 1 , 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 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 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 , 1 ) acting on the columns of the generator matrix as follows (in order): (8, 10), (7, 8), (6, 9), (5, 6, 9), (5, 8, 6, 7, 9, 10), (3, 4)(5, 9, 6)(7, 8), (1, 4)(2, 3)(5, 9) orbits: { 1, 4, 3, 2 }, { 5, 9, 10, 6, 7, 8 }