C2xC2xC2

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].