D32

Blocks with defect group $D_{32}$

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. Note that the class $D(3 {\cal B})_1$ is only realised for defect groups $D_8$ (see [Er87, Proposition 7.5.1]).

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(32,18,1) $k(D_{32})$ 1 11 1 $1$ 1
M(32,18,2) $B_0(kPGL_2(17))$  ? 11 2 $1$ 1 $D(2 {\cal A})$
M(32,18,3) $B_0(kPGL_2(47))$  ? 11 2 $1$ 1 $D(2 {\cal B})$
M(32,18,4) $B_0(kPSL_2(97))$  ? 11 3 $1$ 1 $D(3 {\cal A})_1$
M(32,18,5) $B_0(kPSL_2(31))$  ? 11 3 $1$ 1 $D(3 {\cal K})$

M(32,18,2) and M(32,18,3) are derived equivalent over $k$ by [Ho97] .

M(32,18,4) and M(32,18,5) are derived equivalent over $k$ by [Li94b] .