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

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

code no       2:
================
1 1 1 1 0 0
1 1 0 0 1 0
1 0 1 0 0 1
the automorphism group has order 48
and is strongly generated by the following 6 elements:
(
1 0 0 
0 1 0 
0 0 1 
, 1
, 
2 0 0 
0 0 2 
0 2 0 
, 0
, 
3 0 0 
3 0 3 
3 3 0 
, 1
, 
0 0 3 
3 3 3 
3 0 0 
, 0
, 
2 2 2 
0 0 2 
0 2 0 
, 0
, 
1 0 1 
1 1 1 
1 0 0 
, 1
)
acting on the columns of the generator matrix as follows (in order):
id, 
(2, 3)(5, 6), 
(2, 6)(3, 5), 
(1, 3)(2, 4), 
(1, 4)(2, 3), 
(1, 3, 6)(2, 5, 4)
orbits: { 1, 3, 4, 6, 2, 5 }

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

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

code no       5:
================
1 1 1 1 0 0
2 1 0 0 1 0
3 1 0 0 0 1
the automorphism group has order 48
and is strongly generated by the following 7 elements:
(
2 0 0 
0 2 0 
0 0 2 
, 1
, 
2 0 0 
0 2 0 
2 2 2 
, 0
, 
2 0 0 
3 1 0 
1 1 1 
, 1
, 
1 0 0 
2 3 0 
0 0 3 
, 0
, 
0 2 0 
2 0 0 
2 2 2 
, 1
, 
3 1 0 
0 2 0 
3 3 3 
, 1
, 
3 1 0 
2 0 0 
1 1 1 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(5, 6), 
(3, 4), 
(2, 5)(3, 4), 
(2, 5, 6), 
(1, 2)(3, 4), 
(1, 5)(3, 4), 
(1, 2, 6)(3, 4)
orbits: { 1, 2, 5, 6 }, { 3, 4 }

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