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Let $G \leq S_n$ be $2$-transitive other than $A_n$ and $S_n$. Is it possible that there exists $N\lhd G$ with $N\neq G$, $N$ transitive and $G/N$ cyclic?

I am interested mostly in the answer when $n$ is large and also when the group $G$ is $3$-transitive.

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    $\begingroup$ $G = S_n$ and $N = A_n$? $\endgroup$ – spin Jan 19 '20 at 11:22
  • $\begingroup$ Other than that, I forgot to mention it. I will fix the question $\endgroup$ – Lior Bary-Soroker Jan 19 '20 at 12:18
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    $\begingroup$ I suppose you want $G \neq N$ as well. For all $n \geq 2$ there is a classification of $n$-transitive groups, so I guess a starting point would be looking at these lists for examples. $\endgroup$ – spin Jan 19 '20 at 12:23
  • $\begingroup$ Thanks, I fixed that too. Sorry for being sloppy. $\endgroup$ – Lior Bary-Soroker Jan 19 '20 at 13:51
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    $\begingroup$ $PGL_n(F_q)$ and $PSL_n(F_q)$ should do the trick I think, the determinant on $PGL_n(F_q)$ is a well-defined element of the group $(F_q^*) / (F_q^*)^n$, which can be non-trivial (and always non-trivial for n=2, q odd). For $n=2$ you will also get $3$-transitivity of $G$, not $N$ though. $\endgroup$ – Lev Soukhanov Jan 19 '20 at 14:56
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For $G = \operatorname{Aut}(M_{22})$ and $N = M_{22}$, with the action of $M_{22}$ on $22$ points you have $N \triangleleft G < S_{22}$. Here both $N$ and $G$ are $3$-transitive, and $G/N \cong C_2$.

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The most obvious family of examples is $AGL(1,q)$ for $q$ a prime power.

As spin said in the comments, finite $2$-transitive groups are classified. They are all almost simple or of affine type (like the example I gave). The almost simple ones are quite explicitly listed, so you would just have to go through the list. You should get plenty more examples. The classification of affine ones is a little less explicit (see Have finite doubly transitive groups been classified?)

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