the 4 isometry classes of irreducible [26,20,4]_2 codes are:

code no       1:
================
1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 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 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 1 1 0 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
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 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 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
1 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
0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 18432
and is strongly generated by the following 9 elements:
(
1 0 0 0 0 0 
0 1 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 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 0 0 1 0 0 
0 1 0 1 0 1 
0 1 0 1 1 0 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 0 0 1 0 0 
1 1 1 1 0 1 
1 1 1 1 1 0 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 0 0 1 0 0 
0 1 0 1 1 0 
0 1 0 1 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 0 0 1 0 0 
0 1 1 0 1 0 
0 1 1 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 1 1 1 0 0 
0 1 0 1 0 1 
0 1 0 1 1 0 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
1 1 0 1 0 0 
0 1 0 1 0 1 
0 1 0 1 1 0 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 1 1 1 0 0 
1 1 0 1 0 0 
0 1 0 1 1 0 
0 1 0 1 0 1 
, 
0 0 1 1 1 0 
1 1 1 1 1 0 
1 0 0 1 1 0 
0 0 0 0 1 0 
0 1 0 0 0 0 
1 1 1 1 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(6, 18)(19, 20)(21, 22)(23, 24)(25, 26), 
(5, 22)(6, 15)(11, 21)(12, 24)(13, 23)(14, 18)(16, 20)(17, 19), 
(5, 24)(6, 17)(11, 23)(12, 22)(13, 21)(14, 20)(15, 19)(16, 18), 
(5, 15)(6, 22)(11, 14)(12, 17)(13, 16)(18, 21)(19, 24)(20, 23), 
(5, 13)(6, 20)(11, 12)(14, 17)(15, 16)(18, 19)(21, 24)(22, 23), 
(4, 10)(5, 23, 13, 22)(6, 16, 20, 15)(8, 9)(11, 24, 12, 21)(14, 18, 17, 19), 
(4, 8)(5, 21, 11, 22)(6, 14, 18, 15)(9, 10)(12, 23, 13, 24)(16, 20, 17, 19), 
(3, 9, 7, 10)(4, 8)(5, 14, 11, 15)(6, 21, 18, 22)(16, 17)(23, 24), 
(1, 11, 8, 16)(2, 5, 4, 17)(3, 13, 10, 14)(6, 22, 19, 24)(7, 12, 9, 15)(18, 21, 20, 23)
orbits: { 1, 16, 20, 18, 13, 15, 6, 19, 17, 8, 14, 23, 21, 5, 3, 11, 9, 22, 24, 12, 4, 10, 2, 7 }, { 25, 26 }

code no       2:
================
1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 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 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 1 1 0 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
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 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 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
1 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
0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 768
and is strongly generated by the following 7 elements:
(
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
1 1 0 1 0 0 
1 1 0 0 1 0 
1 1 0 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
1 0 1 1 0 0 
1 0 1 0 1 0 
1 0 1 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
1 1 0 0 1 0 
1 1 0 1 0 0 
1 1 0 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
1 1 1 0 0 0 
0 0 0 1 0 0 
0 0 0 0 1 0 
1 1 0 0 0 1 
, 
1 0 0 0 0 0 
0 0 1 0 0 0 
1 1 1 0 0 0 
1 0 1 0 1 0 
1 0 1 1 0 0 
0 0 0 0 0 1 
, 
1 0 1 0 1 0 
0 1 1 0 1 0 
0 0 0 0 1 0 
0 0 1 1 1 0 
0 0 1 0 0 0 
0 0 0 0 0 1 
, 
1 0 0 1 1 0 
0 1 0 1 1 0 
0 0 1 1 1 0 
0 0 0 0 1 0 
0 0 0 1 0 0 
0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 8)(5, 11)(6, 18)(9, 10)(12, 13)(19, 20), 
(4, 9)(5, 12)(6, 19)(8, 10)(11, 13)(18, 20), 
(4, 11)(5, 8)(6, 18)(9, 13)(10, 12)(19, 20)(21, 24)(22, 25)(23, 26), 
(3, 7)(6, 18)(9, 10)(12, 13)(16, 17)(21, 22)(24, 25), 
(2, 7, 3)(4, 11, 9, 5, 8, 12)(10, 13)(15, 17, 16)(18, 20, 19)(21, 25, 23, 24, 22, 26), 
(1, 12)(2, 13)(3, 5)(4, 16)(7, 11)(8, 17)(9, 14)(10, 15), 
(1, 14)(2, 15)(3, 16)(4, 5)(7, 17)(8, 11)(9, 12)(10, 13)
orbits: { 1, 12, 14, 13, 5, 10, 8, 9, 11, 2, 3, 4, 15, 17, 7, 16 }, { 6, 18, 19, 20 }, { 21, 24, 22, 26, 25, 23 }

code no       3:
================
1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 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 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 1 1 0 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
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 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 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
1 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
0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 11520
and is strongly generated by the following 8 elements:
(
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 0 0 1 0 0 
1 1 0 0 1 0 
1 1 0 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
1 0 1 1 0 0 
1 1 0 0 1 0 
0 1 1 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 1 0 0 0 
0 1 1 1 0 0 
0 1 1 0 1 0 
0 1 1 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
1 0 1 1 0 0 
0 0 1 0 0 0 
0 0 0 0 1 0 
1 0 1 0 0 1 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
1 1 0 1 0 0 
0 1 1 1 0 0 
0 1 1 0 1 0 
1 0 0 1 0 1 
, 
1 0 0 0 0 0 
0 0 1 0 0 0 
0 1 0 0 0 0 
1 1 0 0 1 0 
1 0 1 1 0 0 
0 1 1 0 0 1 
, 
1 0 0 0 0 0 
1 0 1 0 1 0 
1 1 0 1 0 0 
0 1 0 1 1 0 
0 0 1 1 1 0 
0 1 1 1 1 1 
, 
0 1 0 1 1 0 
1 0 0 1 1 0 
1 1 1 1 1 0 
0 1 1 0 1 0 
0 0 0 1 0 0 
0 1 1 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(5, 11)(6, 18)(12, 13)(14, 15)(16, 17)(19, 20)(21, 22), 
(4, 9)(5, 11)(6, 20)(8, 10)(12, 13)(14, 17)(15, 16)(18, 19)(21, 22)(23, 24), 
(4, 10)(5, 13)(6, 20)(8, 9)(11, 12)(18, 19)(25, 26), 
(3, 4, 9)(6, 21, 19)(7, 8, 10)(12, 14, 16)(13, 15, 17)(18, 22, 20)(23, 25, 24), 
(3, 9, 8)(4, 7, 10)(5, 17, 13)(6, 20, 21)(11, 16, 12)(18, 19, 22)(23, 25, 26), 
(2, 3)(4, 11)(5, 9)(6, 20)(8, 13)(10, 12)(14, 17)(21, 24)(22, 23), 
(2, 13, 12)(3, 10, 8)(4, 11, 15)(5, 9, 16)(6, 20, 26)(7, 14, 17)(18, 23, 22)(19, 21, 24), 
(1, 16, 7, 15)(2, 17, 3, 14)(4, 5, 10, 13)(6, 20)(8, 11, 9, 12)(18, 19)(21, 22)(23, 24)
orbits: { 1, 15, 14, 16, 13, 11, 7, 17, 12, 3, 9, 5, 8, 2, 10, 4 }, { 6, 18, 20, 19, 21, 26, 22, 24, 25, 23 }

code no       4:
================
1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 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 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 1 1 0 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
1 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
1 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 0 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 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 720
and is strongly generated by the following 7 elements:
(
1 0 0 0 0 0 
0 1 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 1 0 
, 
1 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 1 0 0 
, 
1 0 0 0 0 0 
0 1 0 0 0 0 
0 0 0 0 0 1 
0 0 1 0 0 0 
0 0 0 0 1 0 
0 0 0 1 0 0 
, 
1 0 0 0 0 0 
0 0 1 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 1 
, 
1 0 0 0 0 0 
0 0 0 1 0 0 
0 0 1 0 0 0 
0 0 0 0 0 1 
0 0 0 0 1 0 
0 1 0 0 0 0 
, 
0 1 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 1 0 
0 0 0 1 0 0 
, 
0 0 0 0 0 1 
0 0 1 0 0 0 
0 1 0 0 0 0 
0 0 0 1 0 0 
1 0 0 0 0 0 
0 0 0 0 1 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6)(11, 17)(12, 18)(13, 19)(14, 20)(15, 21)(16, 22), 
(4, 6, 5)(8, 17, 11)(9, 18, 12)(10, 19, 13)(14, 20, 23)(15, 21, 24)(16, 22, 25), 
(3, 4, 6)(7, 8, 17)(9, 20, 18)(10, 21, 19)(12, 14, 23)(13, 15, 24)(16, 26, 25), 
(2, 3)(8, 9)(11, 12)(15, 16)(17, 18)(21, 22)(24, 25), 
(2, 6, 4)(7, 18, 9)(8, 17, 20)(10, 19, 22)(11, 23, 14)(13, 25, 16)(15, 24, 26), 
(1, 3, 2)(4, 6)(8, 18, 10, 17, 9, 19)(11, 12, 13)(14, 25, 15, 23, 16, 24)(20, 22, 21), 
(1, 5, 6)(2, 3)(7, 13, 19)(8, 16, 21, 9, 15, 22)(11, 25, 17, 12, 24, 18)(14, 26, 20)
orbits: { 1, 2, 6, 3, 4, 5 }, { 7, 17, 9, 19, 11, 8, 18, 10, 25, 12, 21, 13, 14, 20, 22, 24, 26, 23, 15, 16 }