the 2 isometry classes of irreducible [12,2,7]_2 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0 0 1 0
1 1 0 0 0 0 1 1 1 1 0 1
the automorphism group has order 57600
and is strongly generated by the following 15 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 1 0 0 0 0 1 1 1 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 
0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 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 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 0 0 0 0 1 1 1 1 
0 0 0 0 0 0 0 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 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 
, 
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 0 0 1 
0 0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1 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 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 
0 0 0 0 0 1 0 0 0 0 
1 1 0 0 0 0 1 1 1 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 1 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 
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 0 1 
0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 
0 0 0 1 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 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 
0 0 0 0 0 0 0 1 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 0 0 1 
0 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 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 1 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 0 0 
0 0 0 1 0 0 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 
, 
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 
1 1 1 1 1 1 0 0 0 0 
0 0 0 1 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 
0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 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 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 1 0 0 0 0 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 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 
1 1 0 0 0 0 1 1 1 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 
1 1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 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 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 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 1 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 
)
acting on the columns of the generator matrix as follows (in order):
(10, 12), 
(9, 10), 
(9, 10, 12), 
(8, 9), 
(8, 10), 
(7, 8), 
(7, 10, 12)(8, 9), 
(6, 11)(7, 8)(9, 10), 
(5, 11)(8, 10, 9), 
(5, 6)(8, 9, 10), 
(4, 6)(8, 9), 
(4, 6, 11, 5)(7, 8)(9, 10), 
(3, 6, 5, 4)(7, 9, 8), 
(3, 8, 11, 7, 6, 9, 4, 10, 5, 12), 
(1, 2)(7, 8, 9)
orbits: { 1, 2 }, { 3, 4, 12, 6, 5, 9, 10, 11, 7, 8 }

code no       2:
================
1 1 1 1 1 1 0 0 0 0 1 0
1 1 1 0 0 0 1 1 1 1 0 1
the automorphism group has order 17280
and is strongly generated by the following 11 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 1 1 0 0 0 1 1 1 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 
0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 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 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 1 
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 1 0 0 0 
0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 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 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 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 0 0 0 0 1 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 
1 1 1 0 0 0 1 1 1 1 
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 1 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 
1 1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 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 
, 
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 
1 1 1 1 1 1 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 0 0 1 
0 0 0 0 0 0 0 1 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 0 0 1 0 0 0 0 
0 0 0 1 0 0 0 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 1 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 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 0 0 1 
0 0 0 0 0 0 0 1 0 0 
, 
0 0 1 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 1 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 0 0 0 0 1 
0 0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(10, 12), 
(9, 10), 
(9, 12), 
(8, 9, 10), 
(7, 9)(8, 10), 
(7, 10, 9, 12), 
(6, 11)(7, 8, 9), 
(5, 11)(8, 10, 9), 
(4, 6, 5)(7, 10, 9, 8), 
(2, 3)(8, 10, 9), 
(1, 2, 3)(4, 5, 6)(8, 10)
orbits: { 1, 3, 2 }, { 4, 5, 6, 11 }, { 7, 9, 12, 8, 10 }