the 3 isometry classes of irreducible [9,3,6]_4 codes are:

code no       1:
================
1 1 1 1 1 1 1 0 0
3 2 2 1 1 0 0 1 0
3 2 1 2 0 1 0 0 1
the automorphism group has order 108
and is strongly generated by the following 5 elements:
(
2 0 0 0 0 0 
0 2 0 0 0 0 
0 0 0 2 0 0 
0 0 2 0 0 0 
0 0 0 0 0 2 
0 0 0 0 2 0 
, 0
, 
3 0 0 0 0 0 
0 0 0 0 0 3 
3 3 3 3 3 3 
0 0 3 0 0 0 
0 3 0 0 0 0 
0 0 0 0 3 0 
, 0
, 
1 0 0 0 0 0 
0 0 0 1 0 0 
0 0 0 0 1 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
1 1 1 1 1 1 
, 1
, 
2 3 3 1 1 0 
2 2 2 2 2 2 
0 0 0 0 0 2 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 0 0 3 0 0 
, 1
, 
0 0 0 0 1 0 
2 2 2 2 2 2 
1 3 2 3 0 2 
1 0 0 0 0 0 
0 0 0 2 0 0 
0 0 1 0 0 0 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(3, 4)(5, 6)(8, 9), 
(2, 5, 6)(3, 4, 7), 
(2, 4)(3, 5)(6, 7), 
(1, 8)(2, 4, 6, 3, 5, 7), 
(1, 4, 5)(2, 8, 7)(3, 6, 9)
orbits: { 1, 8, 5, 9, 2, 6, 3, 4, 7 }

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

code no       3:
================
1 1 1 1 1 0 1 0 0
2 2 1 1 0 1 0 1 0
3 2 2 0 3 1 0 0 1
the automorphism group has order 432
and is strongly generated by the following 5 elements:
(
3 0 0 0 0 0 
0 3 0 0 0 0 
0 0 0 0 1 0 
1 1 1 1 1 0 
0 0 0 0 0 1 
0 0 1 0 0 0 
, 0
, 
2 0 0 0 0 0 
0 2 0 0 0 0 
2 2 2 2 2 0 
0 0 0 0 2 0 
0 0 0 2 0 0 
1 1 2 2 0 2 
, 1
, 
3 0 0 0 0 0 
0 0 0 0 0 2 
1 1 2 2 0 2 
2 2 2 2 2 0 
0 3 0 0 0 0 
0 0 3 0 0 0 
, 1
, 
0 0 0 0 2 0 
2 2 2 2 2 0 
0 2 0 0 0 0 
2 0 0 0 0 0 
0 0 2 0 0 0 
1 1 3 3 0 3 
, 0
, 
2 1 1 0 2 3 
0 0 0 1 0 0 
0 0 0 0 3 0 
0 0 0 0 0 3 
3 0 0 0 0 0 
3 3 2 2 0 2 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(3, 6, 5)(4, 8, 7), 
(3, 7)(4, 5)(6, 8), 
(2, 5, 7, 4, 9, 8, 3, 6), 
(1, 4, 7, 2, 3, 5)(6, 8), 
(1, 5, 3, 9)(2, 8, 6, 4)
orbits: { 1, 5, 9, 6, 4, 2, 3, 8, 7 }