the order of A is 24*6 = 144.
a subgroup of order 72 must be normal. why? why is it that ANY subgroup of index 2 is ALWAYS normal? (think of how many cosets there are, and then compare left and right cosets).
a sylow 3-subgroup is of order 9. sylow theory tells us that the number of sylow 3-subgroups must be 1,4,or 16. since sylow subgroups are all conjugate, the only way a sylow subgroup can be normal is if it is the only sylow subgroup (for that prime divisor).
since <((1 2 3),e),(e,(1 2 3))> is one such subgroup, and <((2 3 4),e),(e,(1 2 3))> another, we can conclude that the sylow 3-subgroups are not normal.
a sylow 2-subgroup is of order 16. again we have either 1,3 or 9 of them, and a sylow 2-subgroup will be normal only if there is only one. again <((1 2 3 4),e),(e,(1 2))> is one such subgroup, and <((1 2 3 4), e), (e,(2 3))> is another, so the sylow 2-subgroups are not normal.