the 1 isometry classes of irreducible [4,1,4]_27 codes are:

code no       1:
================
1 1 1 2
the automorphism group has order 72
and is strongly generated by the following 6 elements:
(
3 0 0 
0 3 0 
0 0 3 
, 2
, 
25 0 0 
0 25 0 
14 14 14 
, 0
, 
7 0 0 
0 0 7 
0 7 0 
, 2
, 
15 0 0 
21 21 21 
0 15 0 
, 2
, 
0 0 15 
15 0 0 
0 15 0 
, 1
, 
1 1 1 
2 0 0 
0 2 0 
, 2
)
acting on the columns of the generator matrix as follows (in order):
id, 
(3, 4), 
(2, 3), 
(2, 3, 4), 
(1, 2, 3), 
(1, 2, 3, 4)
orbits: { 1, 3, 4, 2 }