the 1 isometry classes of irreducible [4,1,4]_4 codes are:

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