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	[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].