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

These were classified in [Ea16] using the CFSG. The Picard groups with respect to [math]\mathcal{O}[/math] are computed in [EL18c] with the exception of the principal block of [math]J_1[/math], which has been computed by Eisele.

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] [math](C_2 \times C_2 \times C_2):GL_3(2)[/math] 1 1
M(8,5,2) [math]B_0(k(A_5 \times C_2))[/math] 1 8 3 [math]C_3[/math] [math]C_2 \times C_2[/math] 1 1
M(8,5,3) [math]k(A_4 \times C_2)[/math] 1 8 3 [math]C_3[/math] [math]S_3 \times C_2[/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] [math]C_7:C_3[/math] 1 1
M(8,5,5) [math]B_0(kSL_2(8))[/math] 1 8 7 [math]C_7[/math] [math]C_3[/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] [math]C_3[/math] 1 1
M(8,5,7) [math]B_0(kJ_1)[/math] 1 8 5 [math]C_7:C_3[/math] [math]1[/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] [math]C_3[/math] 1 1

M(8,5,2) and M(8,5,3) are derived equivalent over [math]\mathcal{O}[/math]. This is a consequence of the derived equivalence between M(4,2,2) and M(4,2,3) (see [Ri96]).

M(8,5,4) and M(8,5,5) are derived equivalent over [math]\mathcal{O}[/math]. See [Ro95].

M(8,5,6), M(8,5,7) and M(8,5,8) are derived equivalent over [math]\mathcal{O}[/math]. See [Go97], [Ok97] and [CR13].