the 2 isometry classes of irreducible [27,21,4]_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 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 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
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 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 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
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 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
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 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 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 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 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
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
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
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
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 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
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 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 0 1
the automorphism group has order 9216
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 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 
1 1 1 1 1 0 
1 1 1 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 0 1 
0 1 1 0 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 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 0 1 1 0 0 
0 0 1 1 1 0 
0 0 1 1 0 1 
, 
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 
1 1 1 0 0 0 
1 1 0 1 0 0 
1 0 0 1 0 1 
0 1 0 1 1 0 
, 
1 0 0 0 0 0 
1 1 1 0 0 0 
0 0 1 0 0 0 
0 1 1 1 0 0 
0 1 1 0 1 0 
1 1 0 0 0 1 
, 
0 1 0 1 1 0 
1 0 0 1 1 0 
1 1 1 1 1 0 
1 1 0 0 1 0 
1 0 0 0 0 0 
0 1 0 1 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(5, 15)(6, 22)(11, 14)(12, 17)(13, 16)(18, 21)(19, 24)(20, 23), 
(5, 17)(6, 24)(11, 16)(12, 15)(13, 14)(18, 23)(19, 22)(20, 21), 
(5, 20)(6, 13)(11, 19)(12, 18)(14, 24)(15, 23)(16, 22)(17, 21), 
(5, 22)(6, 15)(11, 21)(12, 24)(13, 23)(14, 18)(16, 20)(17, 19), 
(4, 9)(5, 14, 12, 16)(6, 21, 19, 23)(8, 10)(11, 15, 13, 17)(18, 22, 20, 24), 
(4, 8)(5, 21, 11, 22)(6, 14, 18, 15)(9, 10)(12, 23, 13, 24)(16, 20, 17, 19), 
(3, 7)(4, 8)(5, 21)(6, 15)(11, 22)(12, 24)(13, 23)(14, 18)(16, 19)(17, 20)(25, 26), 
(2, 7)(4, 8, 9, 10)(5, 11, 12, 13)(6, 20, 19, 18)(15, 17)(21, 23)(25, 27), 
(1, 5, 8, 15)(2, 11, 4, 14)(3, 12, 10, 17)(6, 22)(7, 13, 9, 16)(18, 21)(19, 24)(20, 23)
orbits: { 1, 15, 5, 12, 23, 6, 11, 18, 17, 8, 20, 22, 16, 21, 13, 24, 14, 3, 19, 2, 10, 4, 9, 7 }, { 25, 26, 27 }

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 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 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
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 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 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
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 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
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 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 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 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 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
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
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
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
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
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 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 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
the automorphism group has order 1920
and is strongly generated by the following 5 elements:
(
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 0 
0 0 0 0 0 1 
, 
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 0 
0 0 0 0 0 1 
, 
1 0 0 0 0 0 
0 1 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 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 
)
acting on the columns of the generator matrix as follows (in order):
(4, 5)(8, 11)(9, 12)(10, 13)(21, 24)(22, 25)(23, 26), 
(3, 7)(4, 8)(5, 11)(16, 17)(19, 20)(21, 22)(24, 25), 
(3, 5)(7, 11)(9, 14)(10, 15)(19, 24)(20, 25)(23, 27), 
(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)
orbits: { 1, 12, 9, 8, 14, 11, 4, 5, 17, 7, 16, 3, 15, 2, 10, 13 }, { 6 }, { 18, 19, 20, 24, 25, 21, 23, 22, 26, 27 }