the 3 isometry classes of irreducible [16,8,7]_4 codes are:

code no       1:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
2 2 2 1 1 1 0 0 0 1 0 0 0 0 0 0
3 2 1 2 1 0 1 0 0 0 1 0 0 0 0 0
2 3 1 1 2 0 0 1 0 0 0 1 0 0 0 0
2 2 1 0 0 2 1 1 0 0 0 0 1 0 0 0
3 2 0 1 0 1 2 1 0 0 0 0 0 1 0 0
1 3 3 2 1 3 2 1 0 0 0 0 0 0 1 0
2 3 2 3 1 2 3 1 0 0 0 0 0 0 0 1
the automorphism group has order 60
and is strongly generated by the following 4 elements:
(
2 0 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 
0 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 3 
1 3 1 3 2 1 3 2 
2 3 3 1 2 3 1 2 
3 2 1 1 3 0 0 1 
, 1
, 
2 3 1 3 1 0 1 0 
0 0 1 0 0 0 0 0 
2 3 0 1 0 1 3 1 
2 1 3 3 2 0 0 3 
0 3 0 0 0 0 0 0 
2 2 3 0 0 2 3 3 
0 0 0 2 0 0 0 0 
1 1 1 2 2 2 0 0 
, 1
, 
0 0 0 1 0 0 0 0 
0 0 0 0 0 0 1 0 
3 0 0 0 0 0 0 0 
0 0 2 0 0 0 0 0 
0 0 0 0 0 2 0 0 
2 1 1 3 2 1 3 2 
1 1 3 0 0 1 3 3 
1 1 1 3 3 3 0 0 
, 0
, 
3 3 3 2 2 2 0 0 
1 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 2 
1 2 1 2 3 1 2 3 
2 1 0 3 0 3 1 3 
2 3 1 1 2 0 0 1 
0 0 0 1 0 0 0 0 
0 0 0 0 0 3 0 0 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(2, 12, 8, 5)(3, 4)(6, 13, 11, 16)(7, 14, 10, 15), 
(1, 13, 6, 11)(2, 5, 14, 3)(4, 7, 15, 12)(8, 10), 
(1, 3, 4)(2, 13, 7)(5, 15, 6)(8, 16, 10)(11, 12, 14), 
(1, 2, 12, 6, 8, 3, 13, 14, 5, 16, 4, 7, 11, 15, 10)
orbits: { 1, 11, 4, 10, 13, 6, 14, 7, 3, 12, 16, 8, 15, 2, 5 }, { 9 }

code no       2:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
2 2 2 1 1 1 0 0 0 1 0 0 0 0 0 0
3 2 1 2 1 0 1 0 0 0 1 0 0 0 0 0
2 3 1 1 2 0 0 1 0 0 0 1 0 0 0 0
1 2 3 2 0 2 1 1 0 0 0 0 1 0 0 0
0 2 2 3 0 1 2 1 0 0 0 0 0 1 0 0
0 3 0 1 1 3 2 1 0 0 0 0 0 0 1 0
3 3 1 0 1 2 3 1 0 0 0 0 0 0 0 1
the automorphism group has order 960
and is strongly generated by the following 6 elements:
(
2 0 0 0 0 0 0 0 
0 2 0 0 0 0 0 0 
0 2 0 1 1 2 3 1 
2 2 1 0 1 3 2 1 
0 0 0 0 0 0 0 3 
0 0 2 0 0 0 0 0 
0 0 0 2 0 0 0 0 
1 1 1 1 1 1 1 1 
, 1
, 
1 0 0 0 0 0 0 0 
1 1 1 1 1 1 1 1 
1 1 1 3 3 3 0 0 
2 1 3 1 3 0 3 0 
0 0 0 0 0 0 0 1 
0 1 1 2 0 3 1 3 
3 1 2 1 0 1 3 3 
0 0 0 0 1 0 0 0 
, 0
, 
1 0 0 0 0 0 0 0 
0 2 2 1 0 3 2 3 
0 0 0 0 0 0 3 0 
0 0 0 0 0 0 0 3 
0 0 0 3 0 0 0 0 
0 2 0 0 0 0 0 0 
3 3 3 1 1 1 0 0 
2 3 1 3 1 0 1 0 
, 1
, 
1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 
1 3 2 3 0 3 1 1 
0 0 0 0 3 0 0 0 
0 1 0 3 3 1 2 3 
0 0 0 0 0 0 0 1 
0 0 0 0 0 0 1 0 
0 0 0 0 0 1 0 0 
, 1
, 
2 1 3 3 2 0 0 3 
0 0 0 3 0 0 0 0 
0 3 3 2 0 1 3 1 
3 3 3 3 3 3 3 3 
0 0 0 0 0 0 2 0 
0 0 3 0 0 0 0 0 
0 0 0 0 0 0 0 2 
3 2 1 2 0 2 3 3 
, 1
, 
3 2 1 2 0 2 3 3 
0 3 0 2 2 3 1 2 
3 2 1 1 3 0 0 1 
0 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 
0 0 0 0 0 0 3 0 
0 2 0 0 0 0 0 0 
1 1 1 1 1 1 1 1 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(3, 6, 13, 15)(4, 7, 14, 16)(5, 12, 9, 8)(10, 11), 
(2, 9)(3, 10)(4, 11)(5, 8)(6, 14)(7, 13), 
(2, 6, 9, 14)(3, 13, 10, 7)(4, 5, 11, 8)(12, 15), 
(2, 15, 5, 4)(3, 10, 16, 13)(6, 8)(9, 11, 12, 14), 
(1, 12)(2, 11, 9, 4)(3, 6, 10, 14)(5, 13, 8, 7), 
(1, 5, 14, 13)(2, 7, 6, 15)(3, 4, 11, 12)(8, 9)
orbits: { 1, 12, 13, 5, 15, 11, 6, 7, 3, 16, 14, 8, 4, 2, 10, 9 }

code no       3:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0
2 2 2 1 1 1 0 0 0 1 0 0 0 0 0 0
3 2 1 2 1 0 1 0 0 0 1 0 0 0 0 0
3 1 2 1 2 0 0 1 0 0 0 1 0 0 0 0
3 0 0 3 3 1 1 1 0 0 0 0 1 0 0 0
0 1 0 2 1 3 2 1 0 0 0 0 0 1 0 0
3 2 1 3 2 3 2 1 0 0 0 0 0 0 1 0
0 0 3 3 1 2 3 1 0 0 0 0 0 0 0 1
the automorphism group has order 960
and is strongly generated by the following 5 elements:
(
3 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 
0 0 2 2 1 3 2 1 
1 2 3 2 3 0 3 0 
2 2 2 3 3 3 0 0 
0 0 0 0 0 0 2 0 
0 0 0 0 0 2 0 0 
0 0 3 0 0 0 0 0 
, 1
, 
2 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 
0 1 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 
3 1 2 1 2 0 2 0 
2 2 2 2 2 2 2 2 
0 0 0 0 0 2 0 0 
, 1
, 
1 2 3 1 2 1 2 3 
2 3 1 3 1 0 1 0 
3 3 3 3 3 3 3 3 
2 2 2 3 3 3 0 0 
0 0 0 0 0 0 0 1 
2 1 3 1 3 0 0 1 
0 0 0 0 0 1 0 0 
0 1 0 0 0 0 0 0 
, 1
, 
3 3 3 2 2 2 0 0 
3 2 1 3 2 3 2 1 
2 1 3 1 3 0 3 0 
2 2 2 2 2 2 2 2 
0 0 0 0 0 0 0 2 
0 0 3 0 0 0 0 0 
0 3 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 
, 0
, 
0 0 2 2 3 1 2 3 
0 0 3 0 0 0 0 0 
3 2 1 2 1 0 1 0 
2 1 3 2 1 2 1 3 
0 0 0 0 1 0 0 0 
1 2 3 2 3 0 0 2 
0 0 0 0 0 0 0 3 
0 1 0 2 1 3 2 1 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(3, 8, 12, 16)(4, 13, 15, 11)(5, 14, 9, 10)(6, 7), 
(2, 4, 5, 3)(6, 8, 13, 11)(7, 10, 12, 9)(14, 15), 
(1, 15)(2, 8, 5, 11)(3, 10, 4, 9)(6, 7, 13, 12), 
(1, 13, 11, 3, 6, 15, 2, 7, 8, 5, 14, 9, 4, 12, 10), 
(1, 10, 12, 6, 13, 16)(2, 14, 8, 7, 11, 3)(4, 9, 15)
orbits: { 1, 15, 10, 16, 13, 14, 6, 9, 7, 3, 12, 4, 8, 5, 2, 11 }