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Apr 11, 2020 at 7:14 comment added Joshua Grochow Certainly doesn't answer the question, but seems at least tangentially relevant: all modular representations are a direct summand of a module that admits a finite resolution by permutation modules arxiv.org/abs/2003.04373.
May 9, 2019 at 22:06 vote accept Henri Johnston
May 9, 2019 at 15:03 answer added Alex B. timeline score: 10
Jul 28, 2016 at 20:59 comment added Mark Wildon A quick search by computer algebra found some further examples: the dihedral group of order $8$, and the split extension of $\langle x, y \rangle \cong C_3 \times C_3$ by a fixed-point-free automorphism $t$ of order $2$, acting as $x^t = x^2$, $y^t = y^2$.
Jul 28, 2016 at 13:52 history asked Henri Johnston CC BY-SA 3.0