# D8

## Blocks with defect group $D_8$

These are examples of tame blocks and were first classified over $k$ by Erdmann (see [Er87] ). The classification with respect to $\mathcal{O}$ is still unknown.

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(8,3,1) $kD_8$ 1 5 1 $1$ 1
M(8,3,2) $B_0(kPGL_2(5))$  ? 5 2 $1$ 1 $D(2 {\cal A})$
M(8,3,3) $kS_4$  ? 5 2 $1$ 1 $D(2 {\cal B})$
M(8,3,4) $B_0(kPSL_2(9))$  ? 5 3 $1$ 1 $D(3 {\cal A})_1$
M(8,3,5) $B_0(kA_7)$  ? 5 3 $1$ 1 $D(3 {\cal B})_1$
M(8,3,6) $B_0(kPSL_2(7))$  ? 5 3 $1$ 1 $D(3 {\cal K})$

M(8,3,2) and M(8,3,3) are derived equivalent over $k$ by [Ho97] .

M(8,3,4), M(8,3,5) and M(8,3,6) are derived equivalent over $k$ by [Li94b] .