Let $S$ be a finite simple group. All representations below are over the complex numbers. Let

- $d_0(S)$ be the smallest dimension of a faithful representation of $S$,
- $d_1(S)$ be the smallest dimension of a faithful representation of some central extension of $S$, and
- $d_2(S)$ be the smallest dimension of a faithful representation of some finite group admitting $S$ as quotient (or, equivalently, of some finite group admitting $S$ as Jordan-Hölder factor).

In a sense, $d_2(S)$ can be thought of the **smallest dimension in which $S$ occurs linearly**. On the other hand, $d_0$ and $d_1$ are well-documented, but, as far as I know, not $d_2$.

Trivially $d_0\ge d_1\ge d_2$, and the left-hand inequality can be strict: $d_0(\mathrm{Alt}_5)=3>2=d_1(\mathrm{Alt}_5)$.

Does there exist $S$ with $d_1(S)>d_2(S)$?

I'd be interested as well by any information about $d_2$, including values for small non-abelian simple groups, or some particular families.

Note: in a naive attempt to show $d_1=d_2$, we could wonder if any finite group having $S$ as quotient admits a subgroup isomorphic to some central extension of $S$; a counterexample is pointed out by Derek Holt here.