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].
