What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?

Question

Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.

Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer correspondent of $b$.

M. Broué conjectured in the 90's that $b$ and $c$ are derived equivalent under these assumptions.

I would like to ask the following:

Questions:

1. Does there exist an up-to-date list of small groups for which this conjecture has been verified?
2. E.g., is it true for all small groups of order less than $200$, say ?
3. What is the smallest example (w.r.t. $|G|$) which is not yet verified?

Thank you very much for the help.

Answer 1

I don't know the group of smallest order for which the conjecture has not been verified.

But certainly it is known to be true for all groups of order less than 200. There are general results that deal with most small groups. For example,

• For $p$-soluble groups (and in particular soluble groups) it is known to be true, and in fact the derived equivalence is a Morita equivalence in this case. I think this was originally proved by Dade.
• For blocks with cyclic defect group it is known to be true.
• If it is true for two groups then it is true for their direct product.

For groups of order less than 200, these facts deal with all cases except for $p=2$ and $G=A_5, S_5, \operatorname{SL}(2,5)$ or $\operatorname{PSL}(2,7)$, and all of those cases have been verified.