the 1 isometry classes of irreducible [24,20,4]_5 codes are:

code no       1:
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1 1 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 1 1 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 2 1 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 3 1 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 1 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 2 0 1 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 4 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 1 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0
4 3 1 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0
2 4 1 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0
2 0 2 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0
1 2 2 1 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0
0 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0
3 4 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0
0 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0
4 2 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0
2 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0
1 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0
1 0 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0
4 1 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
the automorphism group has order 96
and is strongly generated by the following 3 elements:
(
2 0 0 0 
0 2 0 0 
2 0 4 4 
2 0 2 1 
, 
2 0 0 0 
2 0 3 2 
0 2 1 2 
3 4 1 2 
, 
0 0 1 0 
3 2 4 3 
4 0 1 4 
3 2 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(3, 12)(4, 15)(5, 7)(6, 13)(8, 14)(9, 16)(10, 18)(11, 17)(19, 22)(20, 21), 
(2, 23)(3, 19)(4, 20)(5, 9)(6, 13)(7, 16)(8, 11)(10, 18)(12, 22)(14, 17)(15, 21), 
(1, 22, 2, 7, 24, 9, 23, 3)(4, 11, 20, 13, 15, 14, 21, 10)(5, 6, 16, 8, 12, 18, 19, 17)
orbits: { 1, 3, 12, 19, 23, 22, 8, 18, 2, 9, 14, 11, 16, 10, 5, 24, 17, 15, 4, 7, 6, 21, 13, 20 }