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Do you know a group that has a $\frac{3}{2}$-transitive subgroup and it is not $\frac{3}{2}$-transitive itself?

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I found this group using GAP.


$$ \begin{align*} a &= (29)(36)(48)(57)\\ b &= (136)(247)(589) \\ c &= (192)(354)(687). \end{align*}$$

And let $H=\langle a,b,c\rangle$. This is a $\frac{3}{2}$-transitive subgroup of $S_9$ (it is a Frobenius group, with kernel $\langle b,c\rangle$). It is contained in a larger group $G$, generated by $H$ and the two elements

$$ \begin{align*} d &= (129) \\ e &= (12)(45)(78). \end{align*}$$

This group, just like $H$, is transitive. But the point stabilizer of $1$ only permutes $2$ and $9$ amongst themselves, while transitively moving the other $6$ points amongst themselves. In other words, it is not $\frac{3}{2}$-transitive.

To clarify my search, I simply looked for transitive, but not primitive, subgroups of a symmetric group (I chose $S_9$ because neither $9$ nor $9-1$ is prime). I then looked in those groups for subgroups that were Frobenius (as all $\frac{3}{2}$-transitive groups are either Frobenius or primitive).

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Magic $9$. $S_9$ was chosen cleverly. +1 – Babak S. Dec 20 '12 at 5:32

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