the 1 isometry classes of irreducible [6,1,6]_3 codes are:

code no       1:
================
1 1 1 1 1 2
the automorphism group has order 720
and is strongly generated by the following 10 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
2 2 2 2 2 
, 
2 0 0 0 0 
0 2 0 0 0 
0 0 2 0 0 
0 0 0 0 2 
0 0 0 2 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 0 1 
0 0 0 1 0 
0 0 1 0 0 
, 
2 0 0 0 0 
0 2 0 0 0 
0 0 0 2 0 
0 0 0 0 2 
0 0 2 0 0 
, 
2 0 0 0 0 
0 2 0 0 0 
1 1 1 1 1 
0 0 0 2 0 
0 0 0 0 2 
, 
2 0 0 0 0 
0 0 0 0 2 
0 0 0 2 0 
0 0 2 0 0 
0 2 0 0 0 
, 
1 0 0 0 0 
2 2 2 2 2 
0 0 0 0 1 
0 0 0 1 0 
0 0 1 0 0 
, 
0 0 0 0 2 
0 2 0 0 0 
0 0 0 2 0 
0 0 2 0 0 
2 0 0 0 0 
, 
0 2 0 0 0 
0 0 2 0 0 
2 0 0 0 0 
0 0 0 0 2 
0 0 0 2 0 
, 
2 2 2 2 2 
0 0 0 1 0 
0 0 0 0 1 
0 0 1 0 0 
1 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6), 
(4, 5), 
(3, 5), 
(3, 5, 4), 
(3, 6), 
(2, 5)(3, 4), 
(2, 6)(3, 5), 
(1, 5)(3, 4), 
(1, 3, 2)(4, 5), 
(1, 5, 3, 4, 2, 6)
orbits: { 1, 5, 2, 6, 4, 3 }