Does this relationship between representations in different characteristics hold? Is it possible for the dimension of the smallest faithful representation of a group to be the same or larger in all finite characteristics than in any field of characteristic $0?$ I know that this happens for the Thompson sporadic group, which has a faithful representation of dimension $248,$ but can this happen for groups where the smallest faithful representation has dimension $249$ or more?
 A: An infinite family of examples are the double covers $2.A_n$ of $A_n$ for odd $n = 2k+1 \ge 9$, where the degree of a minimal faithful representation in any characteristic except $2$ is $2^{k-1}$.
I found this result in a paper
On restrictions of modular spin representations of symmetric and alternating groups,
Alexander S. Kleshchev and Pham Huu Tiep
which you can find online.
Added later: I am very confident that these examples are correct, but I have realized now that the result of Kleshchev and Tiep does not cover the case of characteristic $2$. I though initially that that was not an issue, because a group like $2.A_n$ with a nontrivial normal $2$-subgroup has no faithful irreducible representations in characteristic $2$. But I guess that that does not rule rule out a smaller dimensional faithful representation that is reducible. As far as I can tell from examples with small $n$, the smallest dimensional faithful representations in characteristic $2$ have much larger dimension, but I don't know whether that is "known" in general.
