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Dec 3, 2021 at 9:09 comment added David A. Craven For finite simple groups, rather than all finite groups, I think the minimal non-trivial projective dimension (i.e., dimension of a minimal representation of a central extension) is known explicitly. (There might be a couple of cases where it's not completely known down to the exact number, but I think for the complexes it is.) The papers you are looking for are the unique ones by Landazuri and Seitz, and a follow up by Seitz and Zalesski, There have been a few others since then, but their bounds are usually pretty good.
Dec 2, 2021 at 19:39 history became hot network question
Dec 2, 2021 at 12:19 vote accept Dustin G. Mixon
Dec 2, 2021 at 12:03 answer added Sean Eberhard timeline score: 12
Dec 2, 2021 at 11:56 comment added Sean Eberhard A natural way of writing this is to ask for the largest finite simple subgroup of the unitary group $U(n)$, and the obvious guess is $A_{n+1}$.
Dec 2, 2021 at 11:36 history asked Dustin G. Mixon CC BY-SA 4.0