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: $d\_0(\mathrm{Alt}\_5)=3>2=d\_1(\mathrm{Alt}\_5)$. 



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> Does there exist $S$ with $d\_1(S)>d\_2(S)$?
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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](https://math.stackexchange.com/questions/1166315/minimal-finite-groups-with-a-given-simple-factor). 



Linear occurrences of finite simple groups
Seems that $S = O\_{2n}^\pm(2)$ are examples of this for $n=5$, and probably for all $n \geq 5$. Such $S$ is a Jordan-Hölder factor of the automorphism group of the extraspecial group $2^{1+2n}\_\pm$, so $d\_2 \leq 2^n$. But the Schur multiplier is trivial, so $d\_0 = d\_1$, and the ATLAS of Conway et al. reports minimal faithful representations of dimensions $154$ for $O\_{10}^-(2)$ and $155$ for $O\_{10}^+(2)$, both larger than $2^5 = 32$.


