the 3 isometry classes of irreducible [27,22,3]_2 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0
1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 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 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 0 0 0 1 0 0 0 0 0 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 0 0
1 1 0 1 0 0 0 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 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 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 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 0 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 1 0 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 1 0 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 0 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 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 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 0 1 0 0 0 0
1 1 1 0 1 0 0 0 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 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 0 1 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 0 0 1
the automorphism group has order 384
and is strongly generated by the following 8 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 1 1 0 1 
0 1 1 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 1 0 1 
0 0 1 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 1 0 0 1 
0 1 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 0 1 0 1 
1 0 1 1 0 
, 
1 0 0 0 0 
1 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 0 0 0 1 
, 
1 0 0 0 0 
0 1 1 0 0 
0 0 1 0 0 
1 0 0 0 1 
0 0 0 1 0 
, 
1 1 1 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 0 0 0 1 
1 0 0 1 0 
, 
0 1 1 0 0 
1 0 1 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 1 1 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 23)(5, 16)(11, 24)(12, 21)(13, 22)(14, 19)(15, 20)(17, 18), 
(4, 21)(5, 14)(11, 22)(12, 23)(13, 24)(15, 18)(16, 19)(17, 20), 
(4, 19)(5, 12)(11, 20)(13, 18)(14, 23)(15, 24)(16, 21)(17, 22), 
(4, 22)(5, 15)(11, 21)(12, 24)(13, 23)(14, 18)(16, 20)(17, 19), 
(2, 7)(5, 18)(9, 10)(12, 13)(16, 17)(21, 22)(25, 26), 
(2, 9)(4, 5, 11, 18)(6, 27)(7, 10)(12, 23, 13, 24)(14, 21, 15, 22)(16, 19, 17, 20)(25, 26), 
(1, 10)(4, 24, 16, 18)(5, 17, 23, 11)(6, 26)(7, 8)(12, 22, 14, 20)(13, 19, 15, 21), 
(1, 9)(2, 8)(5, 24)(6, 25)(11, 16)(12, 15)(20, 21)
orbits: { 1, 10, 9, 7, 2, 8 }, { 3 }, { 4, 23, 21, 19, 22, 18, 12, 14, 13, 17, 16, 11, 15, 20, 5, 24 }, { 6, 27, 26, 25 }

code no       2:
================
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0
1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 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 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 0 0 0 1 0 0 0 0 0 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 0 0
1 1 0 1 0 0 0 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 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 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 0 1 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 0 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 1 0 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 1 0 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 0 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 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 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 0 1 0 0 0 0
1 1 1 0 1 0 0 0 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 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 0 1 0
0 1 1 1 1 0 0 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 9216
and is strongly generated by the following 11 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 1 1 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 1 0 0 1 
0 1 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 1 1 0 1 
0 1 1 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 0 1 
0 0 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 1 1 0 
1 1 1 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 0 1 0 0 
1 0 1 1 0 
0 0 1 0 1 
, 
1 0 0 0 0 
1 1 0 0 0 
1 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 1 0 0 0 
0 0 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
1 0 1 0 1 
1 1 0 0 1 
1 1 1 1 1 
1 1 1 0 1 
, 
1 1 1 0 0 
0 1 0 0 0 
1 1 0 0 0 
1 0 0 0 1 
0 1 1 1 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 24)(6, 25)(18, 23)(19, 22)(20, 21)(26, 27), 
(5, 18)(6, 27)(19, 20)(21, 22)(23, 24)(25, 26), 
(4, 19)(5, 12)(11, 20)(13, 18)(14, 23)(15, 24)(16, 21)(17, 22), 
(4, 23)(5, 16)(11, 24)(12, 21)(13, 22)(14, 19)(15, 20)(17, 18), 
(4, 5)(11, 18)(12, 19)(13, 20)(14, 21)(15, 22)(16, 23)(17, 24), 
(4, 17)(5, 24)(11, 16)(12, 15)(13, 14)(18, 23)(19, 22)(20, 21), 
(3, 8)(4, 14, 11, 15)(5, 22, 18, 21)(9, 10)(12, 16, 13, 17)(19, 24, 20, 23)(25, 26), 
(2, 7)(3, 8)(12, 13)(14, 15)(19, 20)(21, 22), 
(2, 3)(7, 8)(12, 14)(13, 15)(19, 21)(20, 22), 
(2, 19, 8, 22)(3, 21, 7, 20)(4, 25, 17, 6)(5, 24)(11, 26, 16, 27)(18, 23), 
(1, 9, 10)(3, 8, 7)(4, 24, 17, 23, 11, 18)(5, 16)(6, 25, 26)(12, 22, 15, 21, 13, 20)(14, 19)
orbits: { 1, 10, 9 }, { 2, 7, 3, 22, 8, 21, 20, 19, 17, 13, 15, 5, 12, 16, 14, 18, 11, 24, 4, 23, 25, 26, 27, 6 }

code no       3:
================
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0
1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 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 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 0 0 0 1 0 0 0 0 0 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 0 0
1 1 0 1 0 0 0 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 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 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 0 1 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 1 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 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 0 1 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 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
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 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
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
1 1 0 1 1 0 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 0 1 0
1 0 1 1 1 0 0 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 2304
and is strongly generated by the following 8 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 0 1 
0 0 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 0 1 0 
0 0 0 0 1 
1 1 1 0 0 
, 
1 0 0 0 0 
0 1 0 1 1 
1 1 0 0 1 
0 0 0 0 1 
1 1 1 0 0 
, 
1 0 0 0 0 
1 0 1 1 1 
0 0 1 1 0 
0 0 0 1 0 
1 1 1 0 0 
, 
1 0 0 0 0 
1 1 0 0 1 
0 1 0 1 1 
0 0 0 0 1 
1 1 1 0 0 
, 
0 1 0 0 0 
1 0 0 0 0 
0 0 0 0 1 
0 0 1 0 0 
1 1 0 1 0 
, 
0 0 0 0 1 
1 1 0 0 1 
0 1 0 0 0 
0 1 1 0 1 
1 0 0 1 1 
, 
1 0 1 0 0 
0 0 1 1 1 
1 0 1 0 1 
0 0 1 0 1 
1 1 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(4, 5)(11, 17)(12, 18)(13, 19)(14, 20)(15, 21)(16, 22), 
(3, 19, 13)(4, 10, 5)(8, 18, 12)(9, 17, 11)(14, 20, 25)(15, 21, 24)(16, 22, 23), 
(2, 15, 24)(3, 26, 19)(4, 10, 5)(7, 14, 25)(8, 27, 18)(9, 17, 11)(16, 22, 23), 
(2, 27)(3, 14)(5, 10)(7, 26)(8, 15)(9, 17)(12, 21)(13, 20)(16, 23)(18, 24)(19, 25), 
(2, 13, 15, 26, 21, 19)(3, 24)(4, 10, 5)(7, 12, 14, 27, 20, 18)(8, 25)(9, 17, 11)(16, 22, 23), 
(1, 2)(3, 4, 19, 10, 13, 5)(6, 26)(8, 12, 18)(9, 11, 17)(14, 25, 20)(15, 23, 21, 16, 24, 22), 
(1, 10, 16, 6, 23, 5)(2, 3, 15, 26, 24, 19)(4, 22)(8, 12, 18)(9, 11, 17)(13, 21)(14, 25, 20), 
(1, 14, 10, 27, 16, 25, 6, 18, 23, 7, 5, 8)(2, 26)(3, 15, 13, 24, 19, 21)(4, 12, 22, 20)(9, 17, 11)
orbits: { 1, 2, 5, 8, 24, 27, 19, 26, 4, 10, 13, 23, 7, 12, 18, 15, 25, 21, 3, 16, 14, 6, 22, 20 }, { 9, 11, 17 }