the 4 isometry classes of irreducible [10,6,3]_2 codes are:

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

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

code no       3:
================
1 1 1 1 1 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0
1 0 1 0 0 0 1 0 0 0
0 1 1 0 0 0 0 1 0 0
1 0 0 1 0 0 0 0 1 0
0 1 1 1 0 0 0 0 0 1
the automorphism group has order 64
and is strongly generated by the following 6 elements:
(
1 0 0 0 
0 1 0 0 
0 0 1 0 
1 1 1 1 
, 
1 0 0 0 
0 1 0 0 
0 0 1 0 
1 0 0 1 
, 
1 0 0 0 
1 1 0 0 
1 0 1 0 
0 0 0 1 
, 
1 0 0 0 
0 0 1 0 
0 1 0 0 
0 0 0 1 
, 
1 0 0 0 
0 1 1 1 
0 0 0 1 
1 0 1 0 
, 
0 1 1 0 
1 0 1 0 
0 0 1 0 
1 1 1 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 5)(9, 10), 
(4, 9)(5, 10), 
(2, 6)(3, 7), 
(2, 3)(6, 7), 
(2, 5, 6, 10)(3, 9, 7, 4), 
(1, 8)(2, 7)(4, 5)
orbits: { 1, 8 }, { 2, 6, 3, 10, 7, 5, 4, 9 }

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