the 1 isometry classes of irreducible [12,4,6]_2 codes are:

code no       1:
================
1 1 1 1 1 0 0 0 1 0 0 0
1 1 1 0 0 1 1 0 0 1 0 0
1 1 0 1 0 1 0 1 0 0 1 0
1 0 1 0 1 0 1 1 0 0 0 1
the automorphism group has order 576
and is strongly generated by the following 5 elements:
(
1 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 1 0 0 
0 0 0 0 0 0 1 0 
0 0 0 1 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 1 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 
0 0 1 0 0 0 0 0 
0 0 0 0 0 0 1 0 
0 0 0 1 0 0 0 0 
1 1 0 1 0 1 0 1 
1 1 1 1 1 0 0 0 
, 
1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 
0 1 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 
0 0 0 0 0 1 0 0 
0 0 0 0 1 0 0 0 
0 0 1 0 0 0 0 0 
1 1 1 0 0 1 1 0 
, 
0 0 0 0 0 0 0 1 
1 1 0 1 0 1 0 1 
0 0 1 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 1 0 0 
0 0 0 0 1 0 0 0 
1 0 0 0 0 0 0 0 
, 
0 0 0 0 1 0 0 0 
1 1 1 1 1 0 0 0 
0 0 0 1 0 0 0 0 
0 0 0 0 0 1 0 0 
1 1 1 0 0 1 1 0 
0 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 6)(5, 7)(9, 10), 
(3, 4, 6)(5, 11, 7)(8, 10, 9), 
(2, 3, 7, 12, 11, 4)(5, 6)(8, 9, 10), 
(1, 8)(2, 11)(5, 7)(9, 10), 
(1, 7, 9, 2, 10, 5)(3, 6, 4)(8, 11)
orbits: { 1, 8, 5, 9, 10, 11, 7, 6, 2, 12, 3, 4 }