it's not.
we see that φ(ij) = φ(k) = (2 3), whereas:
φ(i)φ(j) = (1 2)(1 3) = (1 3 2) ≠ (2 3).
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alternatively, if φ WAS a homomorphism, then φ(Q) would be a subgroup of S_{3} of order 4, but 4 does not divide 6.
it's not.
we see that φ(ij) = φ(k) = (2 3), whereas:
φ(i)φ(j) = (1 2)(1 3) = (1 3 2) ≠ (2 3).
****************
alternatively, if φ WAS a homomorphism, then φ(Q) would be a subgroup of S_{3} of order 4, but 4 does not divide 6.