the 3 isometry classes of irreducible [7,3,3]_2 codes are: code no 1: ================ 1 1 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 , 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 , 1 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 ) acting on the columns of the generator matrix as follows (in order): (4, 5), (2, 6)(3, 7), (2, 3)(4, 5)(6, 7) orbits: { 1 }, { 2, 6, 3, 7 }, { 4, 5 } code no 2: ================ 1 1 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 1 the automorphism group has order 8 and is strongly generated by the following 3 elements: ( 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 , 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 1 , 0 0 0 1 1 1 1 1 0 0 1 0 0 1 0 0 ) acting on the columns of the generator matrix as follows (in order): (4, 5), (1, 2)(4, 5), (1, 5, 2, 4)(6, 7) orbits: { 1, 2, 4, 5 }, { 3 }, { 6, 7 } code no 3: ================ 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 the automorphism group has order 48 and is strongly generated by the following 6 elements: ( 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 , 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 , 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 , 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 , 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 , 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 ) acting on the columns of the generator matrix as follows (in order): (4, 7), (3, 6), (3, 4)(6, 7), (2, 5), (2, 3)(5, 6), (2, 4, 5, 7) orbits: { 1 }, { 2, 5, 3, 7, 6, 4 }