C2xC2xC2
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 | # 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(8,5,1) | [math]k(C_2 \times C_2 \times C_2)[/math] | 1 | 8 | 1 | [math]1[/math] | 1 | 1 | |||
| M(8,5,2) | [math]B_0(k(A_5 \times C_2))[/math] | 1 | 8 | 3 | [math]C_3[/math] | 1 | 1 | |||
| M(8,5,3) | [math]k(A_4 \times C_2)[/math] | 1 | 8 | 3 | [math]C_3[/math] | 1 | 1 | |||
| M(8,5,4) | [math]k((C_2 \times C_2 \times C_2):C_7)[/math] | 1 | 8 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(8,5,5) | [math]B_0(kSL_2(8))[/math] | 1 | 8 | 7 | [math]C_7[/math] | 1 | 1 | |||
| M(8,5,6) | [math]k((C_2 \times C_2 \times C_2):(C_7:C_3))[/math] | 1 | 8 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(8,5,7) | [math]B_0(kJ_1)[/math] | 1 | 8 | 5 | [math]C_7:C_3[/math] | 1 | 1 | |||
| M(8,5,8) | [math]B_0(k{\rm Aut}(SL_2(8)))[/math] | 1 | 8 | 5 | [math]C_7:C_3[/math] | 1 | 1 |
M(8,5,2) and M(8,5,3) are derived equivalent over [math]\mathcal{O}[/math].
M(8,5,4) and M(8,5,5) are derived equivalent over [math]\mathcal{O}[/math].
M(8,5,6), M(8,5,7) and M(8,5,8) are derived equivalent over [math]\mathcal{O}[/math].
