the 2 isometry classes of irreducible [6,2,3]_4 codes are:

code no       1:
================
1 1 1 1 1 0
1 1 0 0 0 1
the automorphism group has order 24
and is strongly generated by the following 4 elements:
(
3 0 0 0 
0 3 0 0 
0 0 3 0 
0 0 0 3 
, 1
, 
2 0 0 0 
0 2 0 0 
0 0 2 0 
2 2 2 2 
, 1
, 
2 0 0 0 
0 2 0 0 
2 2 2 2 
0 0 2 0 
, 1
, 
0 3 0 0 
3 0 0 0 
0 0 3 0 
3 3 3 3 
, 0
)
acting on the columns of the generator matrix as follows (in order):
id, 
(4, 5), 
(3, 4, 5), 
(1, 2)(4, 5)
orbits: { 1, 2 }, { 3, 5, 4 }, { 6 }

code no       2:
================
1 1 1 1 1 0
2 1 0 0 0 1
the automorphism group has order 36
and is strongly generated by the following 6 elements:
(
3 0 0 0 
0 3 0 0 
0 0 3 0 
3 3 3 3 
, 0
, 
1 0 0 0 
0 1 0 0 
0 0 0 1 
0 0 1 0 
, 0
, 
2 0 0 0 
0 2 0 0 
2 2 2 2 
0 0 0 2 
, 0
, 
1 0 0 0 
2 3 0 0 
3 3 3 3 
0 0 3 0 
, 1
, 
0 1 0 0 
1 0 0 0 
0 0 0 1 
0 0 1 0 
, 1
, 
1 2 0 0 
0 3 0 0 
0 0 1 0 
0 0 0 1 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(4, 5), 
(3, 4), 
(3, 5), 
(2, 6)(3, 4, 5), 
(1, 2)(3, 4), 
(1, 6)
orbits: { 1, 2, 6 }, { 3, 4, 5 }