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Let $G\le S_n$ be a transitive subgroup with trivial centralizer in $S_n$.

Can we deduce any nontrivial lower bounds on the order of $G$?

I'd also be interested in asymptotic results as $n\rightarrow\infty$.

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    That depends on $c$. For $c=1$,all you can say is that $|G| \ge n-1$, and for $c=n!$ you have $|G|=1$. For $c=(n/2)!$ (with $n$ even) you get something like $n/2-1 \le |G| \le (n/2)!$, which is a large range. Did you have any particular values of $c$ in mind? – 2017-01-09
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    @DerekHolt Okay I've edited the question. So, my interest comes from this - When $M$ is a finite nonabelian simple group, $F_2$ is a free group of rank 2, and $\varphi : F_2\twoheadrightarrow M$ is a surjection, then I've checked computationally (for the smallest 23 nonab finite simple groups) that the action of $Aut(F_2)$ on the $Aut(F_2)$-orbit of $\varphi$ has precisely $Out(M)$ as its centralizer. In particular, $Out(M)$ is always rather small. I'd like to show that the permutation image of $Aut(F_2)$ on the orbit of $\varphi$ is always "pretty large" – 2017-01-09
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    @DerekHolt In particular, if $e$ is the exponent of $M$, then I'd like to show that the permutation image always has size at least $e^3$ (computationally this is true, by massive margins). I suppose I should write this as a separate question. – 2017-01-09

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I think the best possible general bound for your edited question is $|G| \ge 2n$. For $n$ odd, the dihedral group of order $2n$ has trivial centralizer.

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    How do you see that the centralizer of $D_{2n}$ is trivial? It certainly has trivial center, but I assume you're embedding $D_{2n}$ inside $S_n$? If $n$ is even you can consider the action of $D_{2n}$ on vertices, but if $n$ is odd this doesn't work, right? – 2017-01-09
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    The order of the centralizer in $S_n$ of a transitive subgroup $G$ is equal to the number of fixed points of the stabilizer $G_a$ of a point $a$. This is $1$ for $D_{2n}$ when $n$ is odd, but $2$ for $n$ even. – 2017-01-09
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    I agree with the first sentence, and perhaps this is a stupid question - but how are you having $D_{2n}$ act on a set of $n$ elements when $n$ is odd? – 2017-01-09
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    There must be some misunderstanding here! The dihedral group of order $2n$ is often defined as the group of rotations and reflections of an $n$-sided regular polygon, so it is acting on the $n$ vertices. – 2017-01-09
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    But if $n$ is odd, the reflections don't send a vertex to another of the $n$ vertices! – 2017-01-09
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    Yes they do. For example $D_{10}$ is generated by the rotation $(1,2,3,4,5)$ and the reflection $(2,5)(3,4)$. – 2017-01-09
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    AH! Okay you're right. Thanks! – 2017-01-09