TITLE: Sporadic subgroup of E7
QUESTION [19 upvotes]: The dimensions of some representations of the Janko group J1 coinside with dimensions of smallest representations of the Lie algebra of type E7 (56, 133). It seems to be natural that there is a subgroup of the Lie group of type E7 isomorphic to J1. Is it true?

REPLY [7 votes]: The Frobenius–Schur indicators of both $56$-dimensional representations are $+1$, meaning that in both cases the representation supports an invariant symmetric bilinear form. Since these are irreps, they do not support any other invariant bilinear forms. On the other hand, the $56$-dimensional irrep of $\mathrm{E}_7$ supports an invariant antisymmetric bilinear form. So $\mathrm{J}_1 \not \subset \mathrm{E}_7$.
According to the ATLAS, the finite group of Lie type $\mathrm{E}_7(2)$ has a $132$-dimensional "adjoint" representation over $\mathbf{F}_2$. The ATLAS does not list such a representation for $\mathrm{J}_1$, but I don't know how to confirm that it does not exist, so I cannot rule out an inclusion $\mathrm{J}_1 \overset?\subset \mathrm{E}_7(2)$.
I remark that of the three $133$-dimensional representations of $\mathrm{J}_1$, the first supports one invariant antisymmetric 3-form, which could in principle be a Lie bracket, and the other two support five each.