Related: math.stackexchange.com/questions/4556720/…

Thanks for that too!

Great. This is an interesting question. I think it would be true for small groups where the smallest rep in char zero is 2 (for non-abelian groups) or 1 (for abelian ones) and so there is no room to go down any further. But for groups with large smallest reps I have no idea! Very nice question!

It depends on the field too. For example cyclic group $C_4$ has a faithful $2$-dimensional irreducible representation over $\mathbb{R}$, but no faithful $1$-dimensional rep over $\mathbb{R}$. But of course there is a faithful $1$-dimensional rep over $\mathbb{C}$. If you only care about the characteristic, I guess you can just look at irreducible representations over algebraically closed fields.

If you are interested in these things it would probably be a good idea to forget about simple groups, really... Simplicity has in all likelihood nothing to do with what you are asking, and it is getting you stuck with wierds things like the Conway group and that number 25 which is really entirely unbased in anything :-!

I just found some group somewhere that works.

Well, a group of order 4,157,776,806,543,360,000 is rarely the best candidate to understand anything...

There are some mathematical numbers, like Skewes' number, that are huge...

And one can write entire books about the number 2. Both facts are irrelevant.

24 is a very nice number for things of characteristic 0 if we look at the Leech lattice, and that is exactly what $Co_1$ is about.

Let us continue this discussion in chat.

The Thompson sporadic group has minimal dimension 248 for all fields.

What's the proof that that number isn't smaller in nonzero characteristic?

Also, 248 * 15 is 3720.

Take direct products. The irreducibles of a direct product are tensor products of irreducibles for the factors. This result holds for any field, so the minimal dimension of a direct product is the sum of the minimal dimensions, and for irreducible faithful it's the product of the minimal faithful dimensions.

Conway 1 does not have a faithful complex representation of dimension 24, by the way.

BTW, it is certainly not possible for the minimal degree to be larger in all finite characteristics than in characteristic $0$ because, for any prime $p$ that does not divide $|G|$, the representations in characteristic $p$ correspond exactly to those in characteric $0$, at least over algebraically closed fields.