A Universe of Sorts

Let p = (0 3 4)(1 2). Let G =
. What is the stabilizer of k=0?
purify(p) = e so we would imagine we would have H = e.
But actually, consider orbit(k). We have 0 <-> id, 3 <-> p, 4 <-> p^2.
If I now consider p * orbit(k) then I get p, p^2, p^3, where purify(p) = id, purify(p^2) = id, purify(p^3) = p^3.
Thus we find the nontrivial generator p^3.

§ Schrier Sims --- Why Purify Generators Times Coset