the 1 isometry classes of irreducible [4,1,4]_5 codes are:

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