the 2 isometry classes of irreducible [28,22,4]_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 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 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
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 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 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
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 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
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 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 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 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 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
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
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
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 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 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 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
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 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 0
1 1 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 258048
and is strongly generated by the following 13 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 0 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 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 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 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 0 0 0 1 
1 1 0 0 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 
0 0 1 0 0 0 
1 0 0 1 1 0 
1 0 0 0 1 1 
0 0 0 0 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 
, 
1 1 1 0 1 1 
0 1 0 0 1 1 
1 0 0 0 1 1 
1 0 0 0 0 0 
0 1 1 0 0 1 
0 1 0 1 1 0 
)
acting on the columns of the generator matrix as follows (in order):
(6, 18)(19, 20)(21, 22)(23, 24)(25, 26)(27, 28), 
(6, 20)(18, 19)(21, 24)(22, 23)(25, 28)(26, 27), 
(5, 17)(6, 24)(11, 16)(12, 15)(13, 14)(18, 23)(19, 22)(20, 21), 
(5, 15)(6, 22)(11, 14)(12, 17)(13, 16)(18, 21)(19, 24)(20, 23), 
(5, 22)(6, 15)(11, 21)(12, 24)(13, 23)(14, 18)(16, 20)(17, 19), 
(5, 18)(6, 11)(12, 20)(13, 19)(14, 22)(15, 21)(16, 24)(17, 23), 
(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), 
(4, 21, 14)(5, 6, 25)(8, 22, 15)(9, 23, 16)(10, 24, 17)(11, 18, 26)(12, 19, 27)(13, 20, 28), 
(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), 
(1, 4, 28)(2, 10, 26)(3, 8, 25)(5, 14, 20)(6, 12, 15)(7, 9, 27)(11, 17, 18)(13, 16, 19)(21, 22, 23)
orbits: { 1, 15, 28, 12, 5, 6, 21, 11, 18, 22, 17, 8, 27, 25, 20, 4, 24, 14, 3, 16, 13, 23, 26, 2, 19, 10, 9, 7 }

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 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 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
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 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 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
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 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
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 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 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 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 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
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
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
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 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 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 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
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 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 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 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 
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 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 0 0 1 0 1 
1 0 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 
1 1 1 0 0 0 
1 1 0 1 0 0 
0 1 0 1 0 1 
1 0 0 1 1 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 
1 1 0 0 0 1 
1 1 0 0 1 0 
, 
1 0 0 0 0 0 
1 1 1 0 0 0 
0 0 1 0 0 0 
1 0 1 1 0 0 
1 1 1 1 0 1 
0 1 0 1 1 0 
, 
1 0 0 0 0 0 
1 1 0 1 0 0 
0 0 0 1 0 0 
1 0 1 1 0 0 
1 1 0 0 0 1 
0 1 0 1 1 0 
, 
0 0 0 0 0 1 
1 1 0 0 0 1 
1 0 1 0 0 1 
1 0 0 1 0 1 
1 0 0 1 1 0 
0 0 0 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 17)(6, 24)(11, 16)(12, 15)(13, 14)(18, 23)(19, 22)(20, 21), 
(5, 15)(6, 22)(11, 14)(12, 17)(13, 16)(18, 21)(19, 24)(20, 23), 
(5, 21)(6, 14)(11, 22)(12, 23)(13, 24)(15, 18)(16, 19)(17, 20), 
(5, 24)(6, 17)(11, 23)(12, 22)(13, 21)(14, 20)(15, 19)(16, 18), 
(3, 7)(4, 8)(5, 22)(6, 14)(11, 21)(12, 23)(13, 24)(15, 18)(16, 20)(17, 19)(25, 26), 
(3, 4)(5, 18)(6, 11)(7, 8)(12, 22)(13, 21)(14, 20)(15, 19)(16, 24)(17, 23)(27, 28), 
(2, 7)(4, 9)(5, 22, 12, 24)(6, 17, 19, 15)(11, 23, 13, 21)(14, 20, 16, 18)(25, 27), 
(2, 7, 8)(3, 9, 4)(5, 18)(6, 13, 21, 11, 19, 15)(12, 24, 14, 20, 16, 22)(17, 23)(25, 27, 28), 
(1, 21, 4, 6)(2, 22, 8, 18)(3, 23, 9, 19)(5, 14)(7, 24, 10, 20)(11, 15)(12, 16)(13, 17)
orbits: { 1, 6, 24, 22, 14, 17, 11, 15, 4, 19, 13, 5, 16, 12, 7, 2, 20, 18, 23, 21, 8, 3, 9, 10 }, { 25, 26, 27, 28 }