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What is

$$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$

$G$ transitive permutation group?

And what are the examples of finite permutation groups containing a large conjugacy class?

$\deg(G)$ resp. $\operatorname{ord}(G) $ are degree and order of $G$, $\operatorname{conj}(x)$ is the conjugacy class of $ x\in G $

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    $\begingroup$ Whenever $n$ is odd, the dihedral group of order $2n$ acts transitively and has a conjugacy class (the order two elements) of order $n$. So $1/2$ is achievable. $\endgroup$ – David E Speyer Oct 17 '18 at 15:57
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    $\begingroup$ You mean "acts transitively and faithfully on $n$ elements". $\endgroup$ – YCor Oct 17 '18 at 16:48
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    $\begingroup$ So well, clearly $|conj(x)|\le |G|/2$ for every nontrivial finite group and $x\in G$, so the term in parenthesis is $\le 1/2$ for every $G$ transitive on $n\ge 2$ elements. So David's remark settles the problem: the limsup is 1/2. $\endgroup$ – YCor Oct 17 '18 at 16:51

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