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The Wikipedia pages for $E_6$ and $E_7$ list three series of groups notated as each of $E_6(q)$, $^2E_6(q)$, and $E_7(q)$:

  • The simple form, analogous to $\operatorname{PSL}_n(q)$
  • The adjoint form, analogous to $\operatorname{PGL}_n(q)$
  • The universal form, analogous to $\operatorname{SL}_n(q)$

Is there a fourth series analogous to $\operatorname{GL}_n(q)$?

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  • $\begingroup$ But the general linear group is not semisimple, so it is not like the others. $\endgroup$
    – Ben McKay
    Oct 3 at 16:51
  • $\begingroup$ Yet it still has analogs for the classical group families in the form of the general unitary groups, general symplectic groups, and pin groups. $\endgroup$ Oct 3 at 17:49

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