the 9 isometry classes of irreducible [13,6,6]_3 codes are:

code no       1:
================
1 1 1 1 1 1 1 2 0 0 0 0 0
2 2 1 1 1 0 0 0 2 0 0 0 0
2 1 2 1 0 1 0 0 0 2 0 0 0
0 2 2 2 1 1 0 0 0 0 2 0 0
1 2 0 1 2 1 0 0 0 0 0 2 0
2 0 1 2 2 1 0 0 0 0 0 0 2
the automorphism group has order 720
and is strongly generated by the following 7 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 2 0 0 0 0 0 
0 2 2 2 1 1 0 
0 0 0 2 0 0 0 
2 0 1 2 2 1 0 
1 1 2 2 2 0 0 
1 1 1 1 1 1 1 
, 
1 0 0 0 0 0 0 
0 1 0 0 0 0 0 
2 0 1 2 2 1 0 
0 0 0 2 0 0 0 
0 0 2 0 0 0 0 
2 1 2 1 0 1 0 
0 0 0 0 0 0 2 
, 
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 
, 
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 
, 
1 0 2 1 1 2 0 
0 1 0 0 0 0 0 
1 2 0 1 2 1 0 
0 0 0 1 0 0 0 
0 0 0 0 1 0 0 
2 1 2 1 0 1 0 
0 0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(7, 8), 
(3, 11)(5, 13)(6, 9)(7, 8)(10, 12), 
(3, 5, 11, 13)(6, 12, 9, 10), 
(2, 5)(3, 9)(6, 10)(7, 8)(11, 12), 
(2, 6, 5, 10)(3, 12, 9, 11)(7, 8), 
(1, 12)(2, 10)(3, 5)(6, 9)(7, 8), 
(1, 13)(3, 12)(6, 10)(9, 11)
orbits: { 1, 12, 13, 10, 6, 11, 3, 5, 9, 2 }, { 4 }, { 7, 8 }

code no       2:
================
1 1 1 1 1 1 1 2 0 0 0 0 0
2 2 1 1 1 0 0 0 2 0 0 0 0
2 1 2 1 0 1 0 0 0 2 0 0 0
0 2 2 2 1 1 0 0 0 0 2 0 0
1 2 0 1 2 1 0 0 0 0 0 2 0
2 1 0 1 2 0 1 0 0 0 0 0 2
the automorphism group has order 36
and is strongly generated by the following 4 elements:
(
1 0 0 0 0 0 0 
0 0 0 0 2 0 0 
1 1 2 2 2 0 0 
0 0 0 1 0 0 0 
0 2 0 0 0 0 0 
2 1 2 1 0 1 0 
0 0 0 0 0 0 1 
, 
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 
, 
2 1 2 1 0 1 0 
1 2 0 1 2 1 0 
0 1 1 1 2 2 0 
0 0 0 1 0 0 0 
0 0 1 0 0 0 0 
1 0 0 0 0 0 0 
0 0 0 0 0 0 1 
, 
1 2 0 1 2 1 0 
2 1 2 1 0 1 0 
0 0 0 0 2 0 0 
0 0 0 2 0 0 0 
0 0 2 0 0 0 0 
2 2 1 1 1 0 0 
0 0 0 0 0 0 2 
)
acting on the columns of the generator matrix as follows (in order):
(2, 5)(3, 9)(6, 10)(11, 12), 
(2, 9)(3, 5)(6, 10)(7, 13), 
(1, 6, 10)(2, 9, 12)(3, 5, 11), 
(1, 12)(2, 10)(3, 5)(6, 9)
orbits: { 1, 10, 12, 6, 2, 11, 9, 5, 3 }, { 4 }, { 7, 13 }, { 8 }

code no       3:
================
1 1 1 1 1 1 1 2 0 0 0 0 0
2 2 1 1 1 0 0 0 2 0 0 0 0
2 1 2 1 0 1 0 0 0 2 0 0 0
0 2 2 2 1 1 0 0 0 0 2 0 0
2 1 0 1 2 0 1 0 0 0 0 2 0
2 0 1 1 0 2 1 0 0 0 0 0 2
the automorphism group has order 72
and is strongly generated by the following 4 elements:
(
1 0 0 0 0 0 0 
0 2 0 0 0 0 0 
0 0 0 0 0 2 0 
2 2 2 2 2 2 2 
2 0 1 1 0 2 1 
0 0 0 0 0 0 1 
1 2 0 2 1 0 2 
, 
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 
, 
1 0 0 0 0 0 0 
2 0 1 1 0 2 1 
0 0 0 0 2 0 0 
2 2 2 2 2 2 2 
1 1 2 2 2 0 0 
0 0 0 0 0 2 0 
0 0 1 0 0 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):
(3, 12, 7, 6)(4, 8)(5, 9, 10, 13), 
(2, 12, 9, 7)(3, 6, 5, 10)(4, 8), 
(2, 10, 12, 13)(3, 7, 9, 5)(4, 8), 
(1, 11)(2, 9)(3, 10)(5, 6)
orbits: { 1, 11 }, { 2, 7, 13, 9, 12, 3, 10, 5, 6 }, { 4, 8 }

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

code no       5:
================
1 1 1 1 1 1 1 2 0 0 0 0 0
2 2 1 1 1 0 0 0 2 0 0 0 0
2 1 2 1 0 1 0 0 0 2 0 0 0
2 1 0 1 2 0 1 0 0 0 2 0 0
2 0 1 1 0 2 1 0 0 0 0 2 0
1 0 0 2 2 2 1 0 0 0 0 0 2
the automorphism group has order 720
and is strongly generated by the following 6 elements:
(
2 0 0 0 0 0 0 
0 1 0 0 0 0 0 
0 0 1 0 0 0 0 
0 0 0 2 0 0 0 
2 1 2 1 0 1 0 
0 0 0 0 2 0 0 
1 0 2 2 0 1 2 
, 
2 0 0 0 0 0 0 
0 1 0 0 0 0 0 
0 0 2 0 0 0 0 
1 1 1 1 1 1 1 
1 0 0 2 2 2 1 
2 1 0 1 2 0 1 
0 0 0 0 1 0 0 
, 
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 
, 
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 
, 
2 0 0 0 0 0 0 
2 0 0 1 1 1 2 
0 0 0 0 0 1 0 
0 0 0 2 0 0 0 
1 2 0 2 1 0 2 
0 0 0 0 0 0 1 
0 0 2 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6, 9, 10)(7, 11, 13, 12), 
(4, 8)(5, 7, 9, 13)(6, 12, 10, 11), 
(3, 10, 7, 5)(6, 12, 11, 9), 
(3, 12, 7, 9)(4, 8)(5, 11, 10, 6), 
(2, 7, 9, 11)(3, 10, 5, 6)(4, 8), 
(2, 9, 13)(3, 7, 6)(5, 10, 11)
orbits: { 1 }, { 2, 11, 13, 7, 10, 12, 5, 9, 3, 6 }, { 4, 8 }

code no       6:
================
1 1 1 1 1 0 0 2 0 0 0 0 0
2 2 1 1 0 1 0 0 2 0 0 0 0
2 1 2 0 1 1 0 0 0 2 0 0 0
1 2 0 2 1 1 0 0 0 0 2 0 0
1 0 2 1 2 1 0 0 0 0 0 2 0
0 1 1 2 2 1 1 0 0 0 0 0 2
the automorphism group has order 15840
and is strongly generated by the following 6 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 
0 2 2 1 1 2 2 
, 
1 0 0 0 0 0 0 
0 1 0 0 0 0 0 
0 0 2 0 0 0 0 
0 0 0 0 0 2 0 
2 2 1 1 0 1 0 
2 0 1 2 1 2 0 
0 0 0 0 0 0 2 
, 
1 0 0 0 0 0 0 
0 2 0 0 0 0 0 
0 0 2 0 0 0 0 
0 0 0 0 1 0 0 
1 0 2 1 2 1 0 
1 2 1 0 2 2 0 
0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 
0 1 0 0 0 0 0 
1 1 2 2 0 2 0 
2 1 0 1 2 2 0 
2 0 1 2 1 2 0 
0 0 0 0 2 0 0 
0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 
0 0 1 0 0 0 0 
1 2 1 0 2 2 0 
0 0 0 0 0 1 0 
1 1 1 1 1 0 0 
0 0 0 0 1 0 0 
0 0 0 0 0 0 1 
, 
2 0 1 2 1 2 0 
0 0 2 0 0 0 0 
2 1 2 0 1 1 0 
1 0 0 0 0 0 0 
0 1 0 0 0 0 0 
2 2 1 1 0 1 0 
0 0 0 0 0 0 2 
)
acting on the columns of the generator matrix as follows (in order):
(7, 13), 
(4, 8, 12, 6)(5, 10, 11, 9), 
(4, 11, 12, 5)(6, 9, 8, 10), 
(3, 10, 9)(4, 8, 11)(5, 6, 12), 
(2, 9, 10, 3)(4, 8, 5, 6), 
(1, 4, 9, 6, 12)(2, 5, 11, 10, 3)
orbits: { 1, 12, 8, 11, 6, 4, 9, 10, 5, 2, 3 }, { 7, 13 }

code no       7:
================
1 1 1 1 1 0 0 2 0 0 0 0 0
2 2 1 1 0 1 0 0 2 0 0 0 0
2 1 2 0 1 1 0 0 0 2 0 0 0
1 2 0 2 1 1 0 0 0 0 2 0 0
2 1 2 1 0 0 1 0 0 0 0 2 0
0 1 1 2 2 1 1 0 0 0 0 0 2
the automorphism group has order 720
and is strongly generated by the following 4 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 
1 1 1 1 1 0 0 
2 2 1 1 0 1 0 
2 1 2 1 0 0 1 
, 
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 
, 
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):
(5, 8)(6, 9)(7, 12)(10, 11), 
(3, 10, 4, 11)(5, 9, 6, 8), 
(2, 10)(3, 9)(4, 5)(6, 8), 
(1, 6, 11, 8)(3, 9, 5, 4)
orbits: { 1, 8, 5, 6, 11, 4, 9, 10, 3, 2 }, { 7, 12 }, { 13 }

code no       8:
================
1 1 1 1 1 0 0 2 0 0 0 0 0
2 2 1 1 0 1 0 0 2 0 0 0 0
2 1 2 0 1 1 0 0 0 2 0 0 0
2 1 2 1 0 0 1 0 0 0 2 0 0
2 2 1 0 1 0 1 0 0 0 0 2 0
1 1 1 0 0 1 1 0 0 0 0 0 2
the automorphism group has order 5616
and is strongly generated by the following 8 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 
0 0 2 0 0 0 0 
0 0 0 0 0 0 2 
0 0 0 0 0 2 0 
0 0 0 0 2 0 0 
0 0 0 2 0 0 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 
, 
2 0 0 0 0 0 0 
0 2 0 0 0 0 0 
2 2 1 0 1 0 1 
0 0 0 0 2 0 0 
2 2 2 0 0 2 2 
0 0 0 0 0 0 2 
2 2 2 2 2 0 0 
, 
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 
, 
2 0 0 0 0 0 0 
1 1 1 1 1 0 0 
0 0 2 0 0 0 0 
0 0 0 0 2 0 0 
0 0 0 2 0 0 0 
1 2 1 0 2 2 0 
1 2 1 2 0 0 2 
, 
1 0 0 0 0 0 0 
0 0 0 0 2 0 0 
1 2 1 2 0 0 2 
2 2 1 1 0 1 0 
2 2 1 0 1 0 1 
0 0 0 0 0 0 1 
1 1 1 1 1 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), 
(4, 7)(5, 6)(8, 13)(9, 12), 
(3, 8)(4, 5)(6, 9)(7, 12), 
(3, 9, 12)(4, 13, 5)(6, 8, 7), 
(2, 10)(3, 9)(4, 5)(6, 8), 
(2, 8)(4, 5)(6, 10)(7, 11), 
(2, 12, 5)(3, 13, 11)(4, 10, 9)(6, 8, 7), 
(1, 8, 9, 5, 3, 10)(2, 6, 4)(7, 11)
orbits: { 1, 10, 11, 2, 6, 4, 3, 7, 13, 8, 5, 9, 12 }

code no       9:
================
1 1 1 1 1 0 0 2 0 0 0 0 0
2 2 1 1 0 1 0 0 2 0 0 0 0
2 1 2 0 1 1 0 0 0 2 0 0 0
2 1 2 1 0 0 1 0 0 0 2 0 0
2 2 1 0 1 0 1 0 0 0 0 2 0
0 1 1 2 2 1 1 0 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 
, 
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 
, 
2 0 0 0 0 0 0 
1 1 1 1 1 0 0 
0 0 2 0 0 0 0 
0 0 0 0 2 0 0 
0 0 0 2 0 0 0 
1 2 1 0 2 2 0 
1 2 1 2 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, 10)(3, 9)(4, 5)(6, 8), 
(2, 8)(4, 5)(6, 10)(7, 11), 
(2, 7, 9)(3, 11, 6)(8, 12, 10), 
(1, 8, 9, 5, 3, 10)(2, 6, 4)(7, 11)
orbits: { 1, 10, 11, 2, 6, 12, 3, 7, 8, 9, 4, 5 }, { 13 }