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We have Y which is made up of the following elements Y=(1,2),(1,3),(1,4),(2,3)(2,4)(3,4), and we are considering the action of $S_4$ on Y. Let $\phi: S_4 \rightarrow Sym(Y)$ be the action homomorphism. The first part of the question is compute the $\phi(a)$ and $\phi(b)$ of the generators a= (1,2) and b=(1,2,3,4) as permutations of the set Y, which I have done. The second part of the question is:

Label the elements of Y by the numbers from {1,2,3,4,5,6} in the lexicographic order and let $\psi:S_4 \rightarrow S_6$ be the corresponding action homomorphism equivalent to $\phi$. Compute $\psi(a)$ and $\psi(b)$.

I have relabelled the elements of Y and I have rewritten it so the a and b are in the new notation, but I am not too sure how to compute the corresponding maps! Help please! I think that I may have misunderstoon the word lexicographic.

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    The elements of $Y$ are in lexicographic order as you've listed them. The pairs beginning with $1$ are first, with ties broken by looking at the second coordinate. So $(1,2)$ has label $1$, $(1,3)$ has label $2$, and so on. – 2017-02-18
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    You wrote "the action of $S_4$ on $Y$" but did not specify which particular action you are talking about. – 2017-02-19
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    @EthanBolker Normally when you talk about lexicographical orderings of permutations, you use the list of images rather than cycle notation, which is in any case not unique. So $(3,4)$ has image list $1,2,4,3$ and comes first under this ordering, and the complete ordering of $Y$ is $(3,4), (2,3), (2,4), (1,2), (1,3), (1,4)$. – 2017-02-19
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    @DerekHolt I am not too sure I understand. I have also modified the question now to include the action homomorphism on Y. I missed that out. – 2017-02-19

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