the 6 isometry classes of irreducible [13,7,5]_3 codes are:

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

code no       2:
================
1 1 1 1 1 1 2 0 0 0 0 0 0
2 1 1 1 0 0 0 2 0 0 0 0 0
1 2 1 0 1 0 0 0 2 0 0 0 0
2 2 0 1 1 0 0 0 0 2 0 0 0
0 1 1 2 1 0 0 0 0 0 2 0 0
2 1 0 2 0 1 0 0 0 0 0 2 0
1 0 0 1 2 1 0 0 0 0 0 0 2
the automorphism group has order 16
and is strongly generated by the following 3 elements:
(
2 0 0 0 0 0 
0 0 0 0 1 0 
0 0 2 0 0 0 
2 2 0 1 1 0 
0 1 0 0 0 0 
1 2 0 1 0 2 
, 
2 1 2 0 2 0 
0 0 0 0 2 0 
0 0 1 0 0 0 
2 1 1 1 0 0 
0 2 0 0 0 0 
0 0 0 0 0 1 
, 
0 0 0 1 0 0 
0 2 2 1 2 0 
0 0 1 0 0 0 
0 0 0 0 1 0 
1 1 0 2 2 0 
2 0 0 2 1 2 
)
acting on the columns of the generator matrix as follows (in order):
(2, 5)(4, 10)(6, 12)(8, 11), 
(1, 9)(2, 5)(4, 8)(10, 11), 
(1, 11, 2, 8, 9, 10, 5, 4)(6, 7, 12, 13)
orbits: { 1, 9, 4, 8, 10, 5, 11, 2 }, { 3 }, { 6, 12, 13, 7 }

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

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

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

code no       6:
================
1 1 1 1 1 1 2 0 0 0 0 0 0
2 1 1 1 0 0 0 2 0 0 0 0 0
1 2 1 0 1 0 0 0 2 0 0 0 0
2 2 1 0 0 1 0 0 0 2 0 0 0
1 0 2 2 1 1 0 0 0 0 2 0 0
0 2 2 1 2 1 0 0 0 0 0 2 0
1 2 1 2 2 1 0 0 0 0 0 0 2
the automorphism group has order 78
and is strongly generated by the following 4 elements:
(
1 0 0 0 0 0 
2 1 2 1 1 2 
2 2 1 0 0 1 
0 0 0 0 2 0 
1 2 2 2 0 0 
0 1 0 0 0 0 
, 
1 0 0 0 0 0 
2 2 1 0 0 1 
0 0 0 0 0 1 
0 1 1 2 1 2 
1 2 1 0 1 0 
0 0 1 0 0 0 
, 
0 0 0 1 0 0 
2 2 2 2 2 2 
0 1 1 2 1 2 
1 0 0 0 0 0 
1 1 2 0 0 2 
2 1 1 1 0 0 
, 
0 0 0 0 2 0 
2 1 1 1 0 0 
0 0 0 1 0 0 
2 2 1 0 0 1 
0 2 2 1 2 1 
2 1 2 0 2 0 
)
acting on the columns of the generator matrix as follows (in order):
(2, 6, 13)(3, 7, 10)(4, 8, 5)(9, 12, 11), 
(2, 10)(3, 6)(4, 12)(5, 9)(7, 13)(8, 11), 
(1, 4)(2, 7)(3, 12)(5, 10)(6, 8)(9, 11), 
(1, 12, 5)(2, 11, 8)(3, 10, 4)(6, 7, 9)
orbits: { 1, 4, 5, 12, 10, 8, 9, 3, 7, 2, 11, 6, 13 }