the 6 isometry classes of irreducible [14,5,6]_2 codes are:

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

code no       2:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0
1 0 1 1 0 1 0 0 1 0 0 0 0 1
the automorphism group has order 48
and is strongly generated by the following 4 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 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 
)
acting on the columns of the generator matrix as follows (in order):
(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)
orbits: { 1, 10 }, { 2, 4, 9, 6, 3, 5, 7, 8 }, { 11, 12, 13, 14 }

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

code no       4:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0
1 0 0 1 1 1 0 1 0 0 0 0 1 0
0 1 0 1 0 1 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 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 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 
1 1 1 1 1 1 1 1 1 
, 
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 0 1 0 
0 0 0 0 0 0 1 0 0 
1 1 1 1 1 1 1 1 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 0 1 0 0 
1 0 0 1 1 1 0 1 0 
0 1 0 1 0 1 1 1 0 
0 0 0 1 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 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 
1 1 1 0 0 1 1 0 0 
1 0 0 1 1 1 0 1 0 
1 1 1 1 1 1 1 1 1 
, 
0 1 0 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 0 0 0 1 0 
1 1 1 1 1 0 0 0 0 
1 0 0 1 1 1 0 1 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 
)
acting on the columns of the generator matrix as follows (in order):
(9, 10), 
(2, 4)(3, 5)(7, 8)(9, 10)(12, 13), 
(2, 8)(3, 12)(4, 7)(5, 13)(6, 14), 
(1, 11)(2, 5)(3, 4)(7, 12)(8, 13)(9, 10), 
(1, 7, 12, 11, 5, 2)(3, 13, 6, 8, 4, 14)
orbits: { 1, 11, 2, 12, 4, 8, 5, 13, 3, 7, 6, 14 }, { 9, 10 }

code no       5:
================
1 1 1 1 1 0 0 0 0 1 0 0 0 0
1 1 1 0 0 1 1 0 0 0 1 0 0 0
1 1 0 1 0 1 0 1 0 0 0 1 0 0
1 0 1 0 1 0 1 1 0 0 0 0 1 0
1 0 0 1 1 1 1 1 1 0 0 0 0 1
the automorphism group has order 1152
and is strongly generated by the following 7 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 
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 0 0 0 0 1 0 
1 0 0 1 1 1 1 1 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 1 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 1 0 0 0 0 0 0 
1 0 1 0 1 0 1 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 
1 1 1 1 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 
1 1 0 1 0 1 0 1 0 
0 0 0 0 1 0 0 0 0 
1 0 1 0 1 0 1 1 0 
0 1 0 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
, 
0 0 1 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 0 0 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 0 
1 1 1 1 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 1 0 0 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(9, 14), 
(4, 6)(5, 7)(10, 11), 
(3, 6, 4)(5, 7, 12)(8, 10, 11), 
(2, 7)(4, 13)(8, 10), 
(2, 6, 7, 13, 5, 4)(3, 12)(8, 11, 10), 
(1, 5, 3)(2, 13, 10, 7, 4, 8)(6, 11, 12), 
(1, 7)(2, 11)(3, 6)(5, 10)(8, 12)
orbits: { 1, 3, 7, 4, 12, 5, 6, 2, 10, 13, 11, 8 }, { 9, 14 }

code no       6:
================
1 1 1 1 1 0 0 0 0 1 0 0 0 0
1 1 0 0 0 1 1 1 0 0 1 0 0 0
0 0 1 1 0 1 1 0 1 0 0 1 0 0
1 0 1 0 1 1 0 1 1 0 0 0 1 0
0 1 0 1 1 0 1 1 1 0 0 0 0 1
the automorphism group has order 2688
and is strongly generated by the following 9 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 
0 0 0 0 1 0 0 0 0 
0 0 1 1 0 1 1 0 1 
0 0 0 0 0 0 0 0 1 
0 1 0 1 1 0 1 1 1 
0 0 0 0 0 0 1 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 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 1 0 1 1 0 1 1 1 
1 0 1 0 1 1 0 1 1 
0 0 1 1 0 1 1 0 1 
1 1 0 0 0 1 1 1 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 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
1 1 0 0 0 1 1 1 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 1 0 0 
0 1 0 1 1 0 1 1 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 1 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 0 0 0 1 1 1 0 
0 0 1 1 0 1 1 0 1 
, 
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 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 0 0 0 1 1 1 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 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 0 0 0 1 1 1 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 1 0 0 0 0 0 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 0 0 0 1 
0 0 1 1 0 1 1 0 1 
1 0 1 0 1 1 0 1 1 
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 0 0 0 0 1 0 0 0 
, 
0 0 0 0 0 0 0 1 0 
1 1 0 0 0 1 1 1 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 0 1 
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 1 0 0 0 0 0 
, 
0 1 0 1 1 0 1 1 1 
1 0 1 0 1 1 0 1 1 
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 
1 1 1 1 1 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 
)
acting on the columns of the generator matrix as follows (in order):
(6, 12)(7, 9)(8, 14)(11, 13), 
(6, 14)(7, 13)(8, 12)(9, 11), 
(6, 11)(7, 8)(9, 14)(12, 13), 
(5, 10)(8, 11)(9, 12), 
(3, 5)(4, 10)(6, 8)(7, 11), 
(3, 8)(4, 11)(5, 6)(7, 10), 
(3, 6, 9)(4, 7, 12)(5, 8, 13)(10, 11, 14), 
(1, 7, 11, 2, 6, 8)(3, 12, 10)(4, 9, 5), 
(1, 9, 14)(2, 12, 13)(3, 4)(5, 7, 11, 10, 6, 8)
orbits: { 1, 8, 14, 12, 7, 11, 6, 3, 5, 9, 13, 2, 10, 4 }