Does the (simply connected compact) Lie group $E_7$ contain a finite subgroup $G \subset E_7$ such that the $56$-dimensional irrep of $E_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-dimensional adjoint representation remains simple when restricted to $G$? By "$28 \oplus \overline{28}$" I mean of course a $28$-dimensional complex irrep plus its dual.

Such a subgroup is definitely Lie primitive (meaning it doesn't fit inside a proper Lie subgroup $L \subset E_7$), since an imprimitive subgroup would split $133 = \mathfrak{l} \oplus (\dots)$. According to Griess and Ryba - Finite simple groups which projectively embed in an exceptional Lie group are classified!, of the (quasi)simple subgroups of $E_7$, only three are primitive, and none of them work. ($133$ splits over $SL_2(29)$ and $SL_2(37)$, and $56$ remains simple over $PSU_3(8)$.) But I don't know about nonsimple subgroups.

This feels like a good homework problem, but in fact came up in my research: such a subgroup would allow me to build a superconformal field theory with some nice properties.

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    $\begingroup$ I love the title of that paper. I know Bob as a very calm person, so can only imagine how much the result must have excited him for him to put that exclamation mark. $\endgroup$ – LSpice Jun 22 '19 at 2:01
  • $\begingroup$ I don't think this is homework problem. I haven't worked out all details, and I will probably write up some sort of answer. I think the difficult case might be if the $28$-dimensional irreducible representation is induced from $1$-dimensional representation of a subgroup of index $28$. $\endgroup$ – Geoff Robinson Jun 22 '19 at 17:43
  • $\begingroup$ @GeoffRobinson I take it that you think that there is no such $G$ (which also seems most likely to me). I look forward to your write up. $\endgroup$ – Theo Johnson-Freyd Jun 23 '19 at 16:17
  • $\begingroup$ I am not certain at present that there is no such $G$, though it would be my inclination to believe that. I am currently trying to prove that such a $G$ would have a quasisimple subnormal subgroup acting irreducibly on the $28$-dimensional module and on the $133$-dimensional module, which you have essentially already excluded. If the $28$-dimensional rep of $G$ is not monomial, I think I can do it. If it is monomial, there seems to be more to be done. $\endgroup$ – Geoff Robinson Jun 23 '19 at 16:46

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