the 5 isometry classes of irreducible [15,6,6]_2 codes are:

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

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

code no       3:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0
1 0 1 1 0 1 0 0 1 0 0 0 0 1 0
1 0 0 0 1 0 1 1 1 0 0 0 0 0 1
the automorphism group has order 720
and is strongly generated by the following 6 elements:
(
1 0 0 0 0 0 0 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 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 1 1 0 0 1 1 0 0 
1 1 0 1 0 1 0 1 0 
1 0 1 1 0 1 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 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 0 0 0 0 0 0 1 0 
0 0 1 0 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 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 1 0 0 
, 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 
0 0 0 0 1 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
, 
0 0 0 0 0 0 0 0 1 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 11)(7, 12)(8, 13)(9, 14)(10, 15), 
(4, 6)(5, 7)(11, 12), 
(3, 6)(5, 8)(11, 13), 
(2, 6, 3, 4)(5, 8, 7, 9)(11, 13, 12, 14), 
(1, 10)(2, 9)(3, 8)(4, 5)(6, 7)(11, 12), 
(1, 9)(2, 10)(3, 7)(4, 8)(5, 6)(12, 13)(14, 15)
orbits: { 1, 10, 9, 15, 2, 14, 7, 4, 12, 5, 8, 6, 3, 11, 13 }

code no       4:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0
1 0 1 0 1 1 0 0 1 0 0 0 0 1 0
0 1 1 0 0 1 0 1 1 0 0 0 0 0 1
the automorphism group has order 96
and is strongly generated by the following 5 elements:
(
1 0 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 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 1 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 
, 
1 0 0 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 0 
1 0 1 0 1 1 0 0 1 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 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 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
1 0 1 0 1 1 0 0 1 
1 1 0 1 0 1 0 1 0 
0 1 1 0 0 1 0 1 1 
0 0 0 0 0 0 1 0 0 
0 1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
, 
1 1 1 1 1 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
1 0 1 0 1 1 0 0 1 
1 1 0 1 0 1 0 1 0 
, 
0 0 0 0 0 0 0 1 0 
0 1 1 0 0 1 0 1 1 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
1 0 1 0 1 1 0 0 1 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 1 0 0 
0 0 1 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 0 
)
acting on the columns of the generator matrix as follows (in order):
(2, 3)(4, 5)(8, 9)(13, 14), 
(2, 13)(3, 14)(4, 8)(5, 9)(10, 12), 
(2, 8)(3, 9)(4, 14)(5, 13)(6, 15), 
(1, 11)(2, 4)(3, 5)(8, 14)(9, 13), 
(1, 3, 8)(2, 4, 15)(5, 11, 14)(6, 13, 9)
orbits: { 1, 11, 8, 5, 9, 4, 2, 14, 3, 13, 15, 6 }, { 7 }, { 10, 12 }

code no       5:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0
1 0 0 1 1 1 0 1 0 0 0 0 1 0 0
0 1 0 1 0 1 1 1 0 0 0 0 0 1 0
0 1 0 1 1 0 1 0 1 0 0 0 0 0 1
the automorphism group has order 360
and is strongly generated by the following 6 elements:
(
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
1 0 0 1 1 1 0 1 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 1 0 0 0 0 
1 1 1 1 1 1 1 1 1 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
1 0 0 1 1 1 0 1 0 
0 0 0 0 0 0 0 1 0 
1 1 1 0 0 1 1 0 0 
0 1 0 1 0 1 1 1 0 
0 1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
1 1 1 1 1 1 1 1 1 
, 
1 0 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 1 1 1 1 0 0 0 0 
0 1 0 0 0 0 0 0 0 
1 1 1 1 1 1 1 1 1 
0 1 0 1 0 1 1 1 0 
0 1 0 1 1 0 1 0 1 
1 1 1 0 0 1 1 0 0 
, 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 0 0 1 1 1 0 1 0 
1 1 1 0 0 1 1 0 0 
1 1 1 1 1 1 1 1 1 
, 
0 0 0 1 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 0 0 1 0 0 
1 1 1 1 1 0 0 0 0 
1 1 1 0 0 1 1 0 0 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 0 0 1 
, 
1 1 1 0 0 1 1 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 1 0 0 0 0 
1 0 0 1 1 1 0 1 0 
0 0 0 0 0 0 1 0 0 
0 0 1 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, 13)(4, 14)(6, 10)(7, 11)(8, 9)(12, 15), 
(2, 7)(3, 13)(4, 8)(5, 12)(6, 14)(9, 10), 
(2, 5)(4, 11)(6, 10)(7, 14)(8, 15)(9, 12), 
(1, 11)(2, 3)(4, 5)(7, 13)(8, 12)(9, 10), 
(1, 7, 4)(2, 14, 3)(5, 13, 11)(6, 8, 12), 
(1, 12)(2, 11)(3, 8)(4, 14)(6, 13)(9, 10)
orbits: { 1, 11, 4, 12, 7, 13, 2, 14, 8, 5, 15, 9, 3, 6, 10 }