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Blocks with defect group [math]D_8 * C_4[/math]

The invariants [math]k(B)[/math], [math]k_i(B)[/math] and [math]l(B)[/math] are determined in [Sa13b]. There is precisely one saturated fusion system on [math]D_8 * C_4 \cong Q_8*C_4[/math]. There is as yet no classification of blocks with these defect groups, and Donovan's conjecture is not known in any form.


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(16,13,1) [math]k(Q_8*C_4)[/math] 1 10 1 [math]1[/math] 1 1
M(16,13,2) [math]B_0(k(SL_2(5)*C_4))[/math]  ? 14 3 [math]1[/math] 1 1
M(16,13,3) [math]k(SL_2(3)*C_4)[/math]  ? 14 3 [math]1[/math] 1 1

If [math]B[/math] is not nilpotent, then [math]k_0(B)=8, k_1(B)=6, l(B)=3[/math].