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Blocks with defect group [math]D_{32}[/math]

These are examples of tame blocks and were first classified over [math]k[/math] by Erdmann (see [Er87] ). The classification with respect to [math]\mathcal{O}[/math] is still unknown. Note that the class [math]D(3 {\cal B})_1[/math] is only realised for defect groups [math]D_8[/math] (see [Er87, Proposition 7.5.1]).

Class Representative # lifts / [math]\mathcal{O}[/math] [math]k(B)[/math] [math]l(B)[/math] Inertial quotients [math]{\rm Pic}_\mathcal{O}(B)[/math] [math]{\rm Pic}_k(B)[/math] [math]{\rm mf_\mathcal{O}(B)}[/math] [math]{\rm mf_k(B)}[/math] Notes
M(32,18,1) [math]k(D_{32})[/math] 1 11 1 [math]1[/math] 1
M(32,18,2) [math]B_0(kPGL_2(17))[/math]  ? 11 2 [math]1[/math] 1 [math]D(2 {\cal A})[/math]
M(32,18,3) [math]B_0(kPGL_2(47))[/math]  ? 11 2 [math]1[/math] 1 [math]D(2 {\cal B})[/math]
M(32,18,4) [math]B_0(kPSL_2(97))[/math]  ? 11 3 [math]1[/math] 1 [math]D(3 {\cal A})_1[/math]
M(32,18,5) [math]B_0(kPSL_2(31))[/math]  ? 11 3 [math]1[/math] 1 [math]D(3 {\cal K})[/math]

M(32,18,2) and M(32,18,3) are derived equivalent over [math]k[/math] by [Ho97] .

M(32,18,4) and M(32,18,5) are derived equivalent over [math]k[/math] by [Li94b] .