# C2xC2xC2

## Blocks with defect group $C_2 \times C_2 \times C_2$

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

Class Representative # lifts / $\mathcal{O}$ $k(B)$ $l(B)$ Inertial quotients ${\rm Pic}_\mathcal{O}(B)$ ${\rm Pic}_k(B)$ ${\rm mf_\mathcal{O}(B)}$ ${\rm mf_k(B)}$ Notes
M(8,5,1) $k(C_2 \times C_2 \times C_2)$ 1 8 1 $1$ $(C_2 \times C_2 \times C_2):GL_3(2)$ 1 1
M(8,5,2) $B_0(k(A_5 \times C_2))$ 1 8 3 $C_3$ $C_2 \times C_2$ 1 1
M(8,5,3) $k(A_4 \times C_2)$ 1 8 3 $C_3$ $S_3 \times C_2$ 1 1
M(8,5,4) $k((C_2 \times C_2 \times C_2):C_7)$ 1 8 7 $C_7$ $C_7:C_3$ 1 1
M(8,5,5) $B_0(kSL_2(8))$ 1 8 7 $C_7$ $C_3$ 1 1
M(8,5,6) $k((C_2 \times C_2 \times C_2):(C_7:C_3))$ 1 8 5 $C_7:C_3$ $C_3$ 1 1
M(8,5,7) $B_0(kJ_1)$ 1 8 5 $C_7:C_3$ $1$ 1 1
M(8,5,8) $B_0(k{\rm Aut}(SL_2(8)))$ 1 8 5 $C_7:C_3$ $C_3$ 1 1

M(8,5,2) and M(8,5,3) are derived equivalent over $\mathcal{O}$. 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 $\mathcal{O}$. See [Ro95].

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