the 2 isometry classes of irreducible [13,9,5]_16 codes are:

code no       1:
================
1 1 1 1 1 0 0 0 0 0 0 0 0
4 3 2 1 0 1 0 0 0 0 0 0 0
3 8 5 1 0 0 1 0 0 0 0 0 0
11 15 6 1 0 0 0 1 0 0 0 0 0
15 9 7 1 0 0 0 0 1 0 0 0 0
12 6 8 1 0 0 0 0 0 1 0 0 0
10 4 9 1 0 0 0 0 0 0 1 0 0
7 13 10 1 0 0 0 0 0 0 0 1 0
8 14 11 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:
(
3 0 0 0 
6 13 9 8 
0 0 15 0 
6 14 5 11 
, 2
, 
9 12 1 13 
8 12 9 7 
0 0 8 0 
1 0 0 0 
, 2
)
acting on the columns of the generator matrix as follows (in order):
(2, 11)(4, 6)(7, 12)(8, 13)(9, 10), 
(1, 4, 5, 6)(2, 12, 8, 10)(7, 11, 9, 13)
orbits: { 1, 6, 4, 5 }, { 2, 11, 10, 7, 9, 8, 12, 13 }, { 3 }

code no       2:
================
1 1 1 1 1 0 0 0 0 0 0 0 0
4 3 2 1 0 1 0 0 0 0 0 0 0
3 8 5 1 0 0 1 0 0 0 0 0 0
11 15 6 1 0 0 0 1 0 0 0 0 0
15 9 7 1 0 0 0 0 1 0 0 0 0
12 6 8 1 0 0 0 0 0 1 0 0 0
10 4 9 1 0 0 0 0 0 0 1 0 0
7 13 10 1 0 0 0 0 0 0 0 1 0
14 10 15 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:
(
3 0 0 0 
0 15 0 0 
0 0 1 0 
0 0 0 5 
, 2
, 
5 0 0 0 
0 7 0 0 
0 0 0 13 
0 0 4 0 
, 3
, 
3 5 15 8 
0 8 0 0 
4 4 4 4 
8 13 5 9 
, 3
, 
1 4 6 7 
0 4 0 0 
14 7 3 5 
15 7 12 2 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(5, 7)(6, 8)(10, 13)(11, 12), 
(3, 4)(5, 13, 7, 10)(6, 11, 8, 12), 
(1, 12, 4, 7)(3, 11, 9, 5)(8, 10), 
(1, 11, 13)(3, 5, 10)(4, 12, 8)(6, 9, 7)
orbits: { 1, 7, 13, 5, 4, 9, 10, 11, 3, 12, 8, 6 }, { 2 }