the 4 isometry classes of irreducible [15,7,7]_4 codes are:

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

code no       2:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
2 2 2 1 1 1 0 0 0 1 0 0 0 0 0
3 2 1 2 1 0 1 0 0 0 1 0 0 0 0
2 3 1 1 2 0 0 1 0 0 0 1 0 0 0
1 2 3 2 0 2 1 1 0 0 0 0 1 0 0
0 2 2 3 0 1 2 1 0 0 0 0 0 1 0
0 3 0 1 1 3 2 1 0 0 0 0 0 0 1
the automorphism group has order 60
and is strongly generated by the following 4 elements:
(
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
, 
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):
(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), 
(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, 15, 11, 7, 3, 5, 14, 6, 4, 2, 10, 8, 9 }

code no       3:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
2 2 2 1 1 1 0 0 0 1 0 0 0 0 0
3 2 1 2 1 0 1 0 0 0 1 0 0 0 0
3 1 2 1 2 0 0 1 0 0 0 1 0 0 0
3 0 0 3 3 1 1 1 0 0 0 0 1 0 0
0 1 0 2 1 3 2 1 0 0 0 0 0 1 0
3 2 1 3 2 3 2 1 0 0 0 0 0 0 1
the automorphism group has order 120
and is strongly generated by the following 4 elements:
(
2 0 0 0 0 0 0 0 
0 2 0 0 0 0 0 0 
0 0 2 0 0 0 0 0 
0 0 0 2 0 0 0 0 
0 0 0 0 2 0 0 0 
3 3 3 2 2 2 0 0 
1 3 2 3 2 0 2 0 
1 2 3 2 3 0 0 2 
, 0
, 
1 0 0 0 0 0 0 0 
0 0 3 0 0 0 0 0 
0 0 0 0 3 0 0 0 
0 3 0 0 0 0 0 0 
0 0 0 3 0 0 0 0 
0 0 0 0 0 0 1 0 
2 0 0 2 2 1 1 1 
3 3 3 1 1 1 0 0 
, 1
, 
0 1 0 3 1 2 3 1 
0 0 0 0 0 0 2 0 
2 0 0 2 2 1 1 1 
0 0 0 0 0 1 0 0 
3 2 1 2 1 0 0 2 
0 0 1 0 0 0 0 0 
0 0 0 0 2 0 0 0 
2 2 2 2 2 2 2 2 
, 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
)
acting on the columns of the generator matrix as follows (in order):
(6, 10)(7, 11)(8, 12)(9, 13)(14, 15), 
(2, 4, 5, 3)(6, 12, 13, 7)(8, 9, 11, 10), 
(1, 14)(2, 12, 5, 7)(3, 6, 4, 13)(8, 10, 11, 9), 
(1, 13, 11, 3, 6, 15, 2, 7, 8, 5, 14, 9, 4, 12, 10)
orbits: { 1, 14, 10, 15, 5, 6, 11, 8, 12, 4, 7, 3, 9, 13, 2 }

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