the 6 isometry classes of irreducible [13,4,6]_2 codes are:

code no       1:
================
1 1 1 1 1 1 1 1 1 1 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0
1 1 0 1 0 1 0 1 0 0 0 0 1
the automorphism group has order 48
and is strongly generated by the following 5 elements:
(
1 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 
0 0 0 1 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 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
1 1 1 1 1 1 1 1 1 
, 
1 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 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 1 0 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 0 0 0 1 
, 
1 0 0 0 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 0 0 0 0 
0 0 0 0 0 0 0 1 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 0 0 0 
0 0 0 0 0 0 0 0 1 
, 
0 1 0 0 0 0 0 0 0 
1 0 0 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 0 0 0 1 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 1 1 1 1 1 1 1 
, 
0 0 0 0 0 0 0 0 1 
1 1 1 1 1 1 1 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 
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 
1 0 0 0 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(9, 10), 
(4, 6)(5, 7)(11, 12), 
(3, 6)(5, 8)(11, 13), 
(1, 2)(4, 6)(5, 7)(9, 10)(11, 12), 
(1, 9)(2, 10)(3, 8)(4, 7)(5, 6)
orbits: { 1, 2, 9, 10 }, { 3, 6, 8, 4, 5, 7 }, { 11, 12, 13 }

code no       2:
================
1 1 1 1 1 1 1 1 1 1 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0
1 0 0 1 1 1 0 1 0 0 0 0 1
the automorphism group has order 96
and is strongly generated by the following 6 elements:
(
1 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 
0 0 0 1 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 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
1 1 1 1 1 1 1 1 1 
, 
1 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 
0 0 0 0 1 0 0 0 0 
0 0 0 1 0 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 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 1 0 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 0 0 0 1 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 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 0 1 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 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 1 
, 
0 0 0 0 0 0 0 1 0 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 0 0 0 1 
0 0 0 0 1 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
1 0 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
, 
1 1 1 0 0 1 1 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 1 0 0 0 
1 1 1 1 1 0 0 0 0 
1 0 0 1 1 1 0 1 0 
0 1 0 0 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(9, 10), 
(4, 5), 
(2, 3), 
(2, 4, 3, 5)(7, 8)(12, 13), 
(1, 8)(2, 10)(3, 9)(4, 5)(11, 12), 
(1, 13, 8, 11, 7, 12)(2, 9, 4)(3, 10, 5)
orbits: { 1, 8, 12, 7, 13, 11 }, { 2, 3, 5, 10, 4, 9 }, { 6 }

code no       3:
================
1 1 1 1 1 0 0 0 0 1 0 0 0
1 1 1 0 0 1 1 0 0 0 1 0 0
1 1 0 1 0 1 0 1 0 0 0 1 0
1 0 1 1 0 1 0 0 1 0 0 0 1
the automorphism group has order 384
and is strongly generated by the following 8 elements:
(
1 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 
0 0 0 1 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 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
1 0 1 1 0 1 0 0 1 
, 
1 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 
0 0 0 1 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 0 0 0 1 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 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 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 1 1 0 0 1 1 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 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 
0 0 0 1 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 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 
0 0 0 0 0 1 0 0 0 
1 1 1 0 0 1 1 0 0 
0 0 0 1 0 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 0 0 0 1 
, 
1 0 0 0 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 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 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 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 0 1 
0 1 0 0 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 0 1 0 1 0 1 0 
)
acting on the columns of the generator matrix as follows (in order):
(9, 13), 
(8, 12), 
(7, 11)(8, 12), 
(5, 10)(8, 12), 
(4, 6)(5, 7, 10, 11), 
(3, 6)(5, 12, 10, 8), 
(3, 6, 4)(5, 7, 8, 10, 11, 12), 
(2, 6, 4, 3)(5, 11, 12, 9)(7, 8, 13, 10)
orbits: { 1 }, { 2, 3, 6, 4 }, { 5, 10, 11, 8, 12, 9, 7, 13 }

code no       4:
================
1 1 1 1 1 0 0 0 0 1 0 0 0
1 1 1 0 0 1 1 0 0 0 1 0 0
1 1 0 1 0 1 0 1 0 0 0 1 0
1 0 0 0 1 0 1 1 1 0 0 0 1
the automorphism group has order 96
and is strongly generated by the following 5 elements:
(
1 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 
0 0 0 1 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 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
1 0 0 0 1 0 1 1 1 
, 
1 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 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 1 0 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 0 0 0 1 
, 
1 0 0 0 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 1 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 1 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 0 
1 1 1 0 0 1 1 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 1 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 1 0 0 0 
0 0 1 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 1 1 1 1 0 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(9, 13), 
(4, 6)(5, 7)(10, 11), 
(3, 4, 6)(5, 8, 7)(10, 12, 11), 
(1, 7)(2, 11)(3, 6), 
(1, 10, 7, 2, 5, 11)(3, 4, 6)(8, 12)
orbits: { 1, 7, 11, 5, 8, 10, 12, 2 }, { 3, 6, 4 }, { 9, 13 }

code no       5:
================
1 1 1 1 1 0 0 0 0 1 0 0 0
1 1 1 0 0 1 1 0 0 0 1 0 0
1 1 0 1 0 1 0 1 0 0 0 1 0
1 0 1 0 1 0 1 1 1 0 0 0 1
the automorphism group has order 96
and is strongly generated by the following 7 elements:
(
1 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 
0 0 0 1 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 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
1 0 1 0 1 0 1 1 1 
, 
1 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 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 1 0 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 0 0 0 1 
, 
1 0 0 0 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 0 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 0 0 0 1 
, 
0 0 0 0 0 0 1 0 0 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 1 0 1 0 1 0 1 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 
1 0 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(9, 13), 
(4, 6)(5, 7)(10, 11), 
(3, 6)(5, 12)(8, 10), 
(3, 6, 4)(5, 7, 12)(8, 10, 11), 
(1, 7)(2, 11)(3, 6)(5, 10)(8, 12), 
(1, 11)(2, 7)(3, 6), 
(1, 5, 11, 12)(2, 10, 7, 8)(3, 6)
orbits: { 1, 7, 11, 12, 5, 2, 10, 8 }, { 3, 6, 4 }, { 9, 13 }

code no       6:
================
1 1 1 1 1 0 0 0 0 1 0 0 0
1 1 0 0 0 1 1 1 0 0 1 0 0
0 0 1 1 0 1 1 0 1 0 0 1 0
1 0 1 0 1 1 0 1 1 0 0 0 1
the automorphism group has order 192
and is strongly generated by the following 4 elements:
(
1 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 
0 0 0 1 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 0 0 0 1 1 1 0 
0 0 1 1 0 1 1 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 0 0 0 1 1 1 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 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 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 0 0 0 1 1 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 1 0 
1 1 0 0 0 1 1 1 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 1 0 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 10)(8, 11)(9, 12), 
(3, 8)(4, 11)(5, 6)(7, 10), 
(3, 5)(4, 10)(6, 8)(7, 11), 
(1, 7, 11, 2, 6, 8)(3, 12, 10)(4, 9, 5)
orbits: { 1, 8, 11, 3, 6, 4, 7, 5, 10, 2, 9, 12 }, { 13 }