the 13 isometry classes of irreducible [13,10,3]_4 codes are:

code no       1:
================
1 1 1 1 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 0 0 0
2 1 0 0 0 1 0 0 0 0 0 0 0
3 1 0 0 0 0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 1 0 0 0 0 0
2 0 1 0 0 0 0 0 1 0 0 0 0
3 0 1 0 0 0 0 0 0 1 0 0 0
0 1 1 0 0 0 0 0 0 0 1 0 0
2 1 1 0 0 0 0 0 0 0 0 1 0
3 1 1 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 576
and is strongly generated by the following 8 elements:
(
1 0 0 
0 1 0 
0 0 1 
, 1
, 
3 0 0 
0 1 0 
0 0 1 
, 1
, 
1 0 0 
0 1 0 
1 0 1 
, 1
, 
3 0 0 
0 2 0 
1 0 2 
, 0
, 
1 0 0 
0 1 0 
0 1 1 
, 1
, 
3 0 0 
1 0 3 
1 3 0 
, 0
, 
3 0 0 
2 3 0 
2 0 3 
, 0
, 
1 0 0 
3 1 1 
2 0 1 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(6, 7)(9, 10)(12, 13), 
(4, 12)(5, 6)(8, 9), 
(3, 8)(4, 11)(6, 7), 
(3, 9, 10)(5, 7, 6)(11, 12, 13), 
(3, 11)(4, 8)(6, 7)(9, 13)(10, 12), 
(2, 9)(3, 6)(5, 10)(7, 8), 
(2, 7)(3, 10)(5, 6)(8, 9), 
(2, 4, 6, 13)(3, 9)(5, 11, 7, 12)(8, 10)
orbits: { 1 }, { 2, 9, 7, 13, 10, 8, 3, 6, 5, 11, 12, 4 }

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

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

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

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

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

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

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

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

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

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

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

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