the 1 isometry classes of irreducible [13,3,7]_2 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0 0 1 0 0
1 1 1 0 0 0 1 1 1 0 0 1 0
1 0 0 1 1 0 1 1 0 1 0 0 1
the automorphism group has order 1536
and is strongly generated by the following 8 elements:
(
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 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 0 0 0 
0 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 1 0 
1 0 0 1 1 0 1 1 0 1 
, 
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 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 0 0 0 
0 0 0 0 0 0 0 1 0 0 
1 1 1 0 0 0 1 1 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 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 0 0 0 0 
0 0 0 0 0 0 0 1 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 
, 
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 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 0 
1 1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 1 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 
, 
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 
1 1 1 0 0 0 1 1 1 0 
0 0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 1 0 0 0 
1 1 1 1 1 1 0 0 0 0 
0 0 0 0 0 1 0 0 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 
0 1 0 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 0 
0 0 0 0 0 0 1 0 0 0 
1 1 1 0 0 0 1 1 1 0 
0 0 0 0 1 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 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 0 
1 1 1 1 1 1 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 1 1 0 0 0 1 1 1 0 
0 0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 0 
1 1 1 0 0 0 1 1 1 0 
0 0 0 0 0 1 0 0 0 0 
1 1 1 1 1 1 0 0 0 0 
1 0 0 1 1 0 1 1 0 1 
0 1 0 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 0 
0 0 0 1 0 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(10, 13), 
(9, 12), 
(7, 8), 
(6, 11)(7, 8), 
(4, 12)(5, 9)(6, 8, 11, 7), 
(4, 8)(5, 7)(6, 9, 11, 12), 
(2, 3)(4, 6, 5, 11)(7, 12)(8, 9), 
(2, 7, 12, 3, 8, 9)(4, 10, 11, 5, 13, 6)
orbits: { 1 }, { 2, 3, 9, 12, 5, 6, 8, 4, 11, 7, 13, 10 }