Embedding of finite groups into general linear group

Question

It's clear that for any field $\mathbb{F}$ any finite group $G$ can be embedded into $GL_{n}(\mathbb{F})$ for some $n$.

My question is about one modification of this result. Let's fix positive integer $N$. Is it true that for any finite group $G$ there exist a field $\mathbb{F}$ such that $G$ is embeddable into $GL_N(\mathbb{F})$?

UPD: For $N=1$ it's false. Let's consider $N>1$.

Answer 1

Let $p$ be a prime that is not the characteristic of the field. Consider elementary abelian groups $C_p \times \cdots \times C_p$.

Now $x^p = 1$ has no repeated roots, so any of these elementary abelian groups in $GL_n$ will be simultaenously diagonalizable. But for fixed $n$, there are only finitely many diagonal matrices $A \in GL_n$ satisfying $A^p = 1$. Thus if the number of factors is large enough, then $C_p \times \cdots C_p$ does not embed into $GL_n$

Answer 2

The symmetric group $S_n$ has no faithful representation of degree less than $n-2$ over any field, so the answer to the question is no for every $N$.