C2xC2xC2

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Blocks with defect group [math]C_2 \times C_2 \times C_2[/math]

Each of the eight [math]k[/math]-Morita equivalence classes lifts to an unique class over [math]\mathcal{O}[/math]. The classification uses the CFSG.

Class Representative [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
[math]M(8,5,1)[/math] [math]k(C_2 \times C_2 \times C_2)[/math] 8 1 [math]1[/math] 1 1
[math]M(8,5,2)[/math] [math]k(A_4 \times C_2)[/math] 8 3 [math]C_3[/math] 1 1
[math]M(8,5,3)[/math] [math]B_0(k(A_5 \times C_2))[/math] 8 3 [math]C_3[/math] 1 1
[math]M(8,5,4)[/math] [math]k((C_2 \times C_2 \times C_2):C_7)[/math] 8 7 [math]C_7[/math] 1 1
[math]M(8,5,5)[/math] [math]B_0(kSL_2(8))[/math] 8 7 [math]C_7[/math] 1 1
[math]M(8,5,6)[/math] [math]k((C_2 \times C_2 \times C_2):(C_7:C_3))[/math] 8 5 [math]C_7:C_3[/math] 1 1
[math]M(8,5,7)[/math] [math]B_0(kJ_1)[/math] 8 5 [math]C_7:C_3[/math] 1 1
[math]M(8,5,8)[/math] [math]B_0(k{\rm Aut}(SL_2(8)))[/math] 8 5 [math]C_7:C_3[/math] 1 1

[math]M(8,5,2)[/math] and [math]M(8,5,3)[/math] are derived equivalent over [math]\mathcal{O}[/math].

[math]M(8,5,4)[/math] and [math]M(8,5,5)[/math] are derived equivalent over [math]\mathcal{O}[/math].

[math]M(8,5,6)[/math], [math]M(8,5,7)[/math] and [math]M(8,5,8)[/math] are derived equivalent over [math]\mathcal{O}[/math].