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Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ be a $G$-invariant complex irreducible character of $R(G)$ which is not extendible to $G$. What is the set of degrees of irreducible characters of $G$ lying over $\theta$.

Thanks.

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    $\begingroup$ The character degrees of ${\rm SL}(2,3^p)$ are going to be very relevant here $\endgroup$
    – Geoff Robinson
    Jul 5, 2018 at 9:14
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    $\begingroup$ Of faithful characters, that is $\endgroup$
    – Geoff Robinson
    Jul 5, 2018 at 11:45
  • $\begingroup$ @GeoffRobinson, could you please explain more? $\endgroup$
    – asad
    Jul 8, 2018 at 6:01
  • $\begingroup$ @Geoff Robinson, could you please explain more? $\endgroup$
    – asad
    Jul 8, 2018 at 7:55
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    $\begingroup$ You need to construct a central extension ${\hat G}$ of $G$ such that $\theta$ extends to a character of ${\hat G}$. This really comes from a central extension of $G/R(G)$ and that central extension will have ${\rm SL}(2,3^{p})$ as its unique component. $\endgroup$
    – Geoff Robinson
    Jul 8, 2018 at 9:53

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