the 4 isometry classes of irreducible [13,9,4]_4 codes are:

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

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

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

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