the 3 isometry classes of irreducible [15,6,8]_4 codes are:

code no       1:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
3 2 2 2 1 1 1 0 0 0 1 0 0 0 0
2 3 2 1 2 1 0 1 0 0 0 1 0 0 0
2 2 3 1 1 2 0 0 1 0 0 0 1 0 0
3 2 2 1 0 0 2 1 1 0 0 0 0 1 0
2 3 2 0 1 0 1 2 1 0 0 0 0 0 1
the automorphism group has order 24
and is strongly generated by the following 3 elements:
(
2 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 3 
2 1 1 1 3 3 3 0 0 
0 0 0 2 0 0 0 0 0 
2 2 3 1 1 2 0 0 1 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 3 0 0 0 
0 0 0 0 0 0 0 2 0 
0 1 0 0 0 0 0 0 0 
, 0
, 
2 1 2 3 2 3 0 3 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 2 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 3 0 0 0 
0 2 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 2 
3 0 0 0 0 0 0 0 0 
3 1 1 1 2 2 2 0 0 
, 1
, 
0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 2 
3 1 3 2 3 2 0 2 0 
0 0 0 0 1 0 0 0 0 
1 1 2 3 3 1 0 0 3 
0 0 0 0 0 0 0 3 0 
0 0 0 0 0 2 0 0 0 
0 0 0 0 0 0 1 0 0 
3 0 0 0 0 0 0 0 0 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(2, 9)(3, 11)(5, 13)(6, 7)(14, 15), 
(1, 8, 4, 12)(2, 6, 5, 3)(7, 13, 11, 9), 
(1, 9, 2)(3, 11, 12)(4, 13, 5)(6, 7, 8)
orbits: { 1, 12, 2, 4, 11, 9, 3, 8, 5, 13, 7, 6 }, { 10 }, { 14, 15 }

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

code no       3:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
3 2 2 2 1 1 1 0 0 0 1 0 0 0 0
2 3 2 1 2 1 0 1 0 0 0 1 0 0 0
2 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
1 3 0 0 1 2 3 3 1 0 0 0 0 0 1
the automorphism group has order 72
and is strongly generated by the following 4 elements:
(
2 0 0 0 0 0 0 0 0 
0 2 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 2 0 0 0 0 0 
0 0 0 0 2 0 0 0 0 
3 1 3 2 3 2 0 2 0 
0 0 0 0 0 0 0 0 1 
0 0 3 0 0 0 0 0 0 
0 0 0 0 0 0 3 0 0 
, 0
, 
3 0 0 0 0 0 0 0 0 
1 1 2 3 3 1 0 0 3 
0 0 0 0 0 0 0 3 0 
0 0 0 3 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 
2 3 2 1 2 1 0 1 0 
0 0 0 0 0 0 3 0 0 
0 0 3 0 0 0 0 0 0 
0 0 0 0 2 0 0 0 0 
, 0
, 
0 0 0 0 0 0 3 0 0 
0 0 0 0 0 0 0 3 0 
0 0 0 0 0 0 0 0 3 
2 3 3 3 1 1 1 0 0 
3 2 3 1 3 1 0 1 0 
3 3 2 1 1 3 0 0 1 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 1 0 0 0 
, 1
, 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 0 3 
1 2 1 3 1 3 0 3 0 
0 0 3 0 0 0 0 0 0 
2 2 3 1 1 2 0 0 1 
0 0 0 0 0 0 0 1 0 
0 1 0 0 0 0 0 0 0 
0 0 0 3 0 0 0 0 0 
1 3 3 3 2 2 2 0 0 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(3, 8)(6, 12)(7, 9)(10, 15)(11, 13), 
(2, 13)(3, 8)(5, 9)(6, 12)(10, 14), 
(1, 11, 4, 7)(2, 12, 5, 8)(3, 13, 6, 9), 
(1, 12, 3, 4, 8, 6)(2, 7, 13, 5, 11, 9)
orbits: { 1, 7, 6, 9, 4, 2, 12, 13, 8, 5, 11, 3 }, { 10, 15, 14 }