the 8 isometry classes of irreducible [23,13,6]_2 codes are:

code no       1:
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1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0
1 1 1 1 0 0 1 0 1 1 0 0 0 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 1 0 1 1 0 1 1 1 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 1 0 0 0 0 
0 1 1 0 1 0 1 0 1 0 
0 0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 0 
0 1 1 1 0 1 1 1 1 0 
0 0 0 0 0 0 1 0 0 0 
0 1 1 1 0 1 0 0 0 1 
, 
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 
1 1 0 1 1 0 1 1 1 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 1 0 0 0 0 0 
1 1 1 0 1 1 0 1 1 0 
0 1 1 0 1 0 1 0 1 0 
1 1 0 0 1 0 0 1 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(1, 18)(4, 6)(5, 16)(7, 9)(8, 19)(10, 20)(11, 13)(12, 14)(15, 17)(22, 23), 
(1, 6, 18, 4)(5, 7, 16, 9)(8, 15, 19, 17)(10, 23, 20, 22)(11, 12, 13, 14)
orbits: { 1, 18, 4, 6 }, { 2 }, { 3 }, { 5, 16, 9, 7 }, { 8, 19, 17, 15 }, { 10, 20, 22, 23 }, { 11, 13, 14, 12 }, { 21 }

code no       2:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 1
and is strongly generated by the following 0 elements:
(
)
acting on the columns of the generator matrix as follows (in order):
orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 }

code no       3:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 1
and is strongly generated by the following 0 elements:
(
)
acting on the columns of the generator matrix as follows (in order):
orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 }

code no       4:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 1
and is strongly generated by the following 0 elements:
(
)
acting on the columns of the generator matrix as follows (in order):
orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 }

code no       5:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 1
and is strongly generated by the following 0 elements:
(
)
acting on the columns of the generator matrix as follows (in order):
orbits: { 1 }, { 2 }, { 3 }, { 4 }, { 5 }, { 6 }, { 7 }, { 8 }, { 9 }, { 10 }, { 11 }, { 12 }, { 13 }, { 14 }, { 15 }, { 16 }, { 17 }, { 18 }, { 19 }, { 20 }, { 21 }, { 22 }, { 23 }

code no       6:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 0 0 1 1 1 1 1 1 0 0 0 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:
(
1 1 0 1 0 1 0 1 0 0 
0 1 1 0 1 0 1 0 1 0 
0 0 0 0 0 0 0 1 0 0 
1 0 1 1 0 1 0 0 1 0 
0 0 0 0 1 0 0 0 0 0 
1 0 1 0 1 0 1 1 0 0 
1 1 1 1 1 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 0 
1 1 0 1 1 0 1 1 1 0 
0 0 0 0 0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(1, 13)(2, 16)(3, 8)(4, 15)(6, 14)(7, 11)(9, 18)(12, 17)(20, 23)(21, 22)
orbits: { 1, 13 }, { 2, 16 }, { 3, 8 }, { 4, 15 }, { 5 }, { 6, 14 }, { 7, 11 }, { 9, 18 }, { 10 }, { 12, 17 }, { 19 }, { 20, 23 }, { 21, 22 }

code no       7:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 4
and is strongly generated by the following 1 elements:
(
0 0 0 0 0 1 0 0 0 0 
0 1 0 0 0 0 0 0 0 0 
0 1 0 1 1 0 1 0 0 1 
1 1 1 1 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 0 1 0 
0 1 1 0 1 0 1 0 1 0 
1 0 1 1 0 1 0 0 1 0 
1 0 1 0 1 0 1 1 0 0 
1 1 1 1 0 0 1 0 1 1 
)
acting on the columns of the generator matrix as follows (in order):
(1, 14, 9, 6)(3, 20)(4, 19, 12, 11)(5, 13, 22, 10)(7, 21, 18, 16)(8, 17, 23, 15)
orbits: { 1, 6, 9, 14 }, { 2 }, { 3, 20 }, { 4, 11, 12, 19 }, { 5, 10, 22, 13 }, { 7, 16, 18, 21 }, { 8, 15, 23, 17 }

code no       8:
================
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0
0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0
1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 4
and is strongly generated by the following 1 elements:
(
1 1 1 1 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 1 0 0 1 0 0 1 1 1 
1 0 1 1 0 1 1 1 0 1 
1 0 1 1 1 0 1 0 1 1 
0 0 0 0 0 0 0 0 0 1 
1 1 1 0 0 1 1 0 0 0 
0 0 0 0 0 0 0 1 0 0 
1 0 0 0 0 1 0 1 1 1 
)
acting on the columns of the generator matrix as follows (in order):
(1, 21, 6, 11)(2, 12, 8, 9)(3, 23, 10, 7)(4, 17, 13, 22)(5, 15, 16, 20)
orbits: { 1, 11, 6, 21 }, { 2, 9, 8, 12 }, { 3, 7, 10, 23 }, { 4, 22, 13, 17 }, { 5, 20, 16, 15 }, { 14 }, { 18 }, { 19 }