# Q16

## Blocks with defect group $Q_{16}$

These are examples of tame blocks and were first classified over $k$ by Erdmann (see [Er88a], [Er88b]) with some exceptions. It is not known which algebras in the infinite families $Q(2 {\cal A})$ and $Q(2 {\cal B})_1$ are realised by blocks, and as such Donovan's conjecture is still open for $Q_{16}$ for blocks with two simple modules. Until this is resolved the labelling is provisional.

For blocks with three simple modules the $k$-Morita equivalence classes lift to unique $\mathcal{O}$-classes by [Ei16], but otherwise the classification with respect to $\mathcal{O}$ is still unknown.

CLASSIFICATION INCOMPLETE
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(16,9,1) $kSD_{16}$ 1 7 1 $1$ 1
M(16,9,2) $B_0(k \tilde{S}_5)$[1]  ? 8 2 $1$ 1 $Q(2 {\cal A})$
M(16,9,3) $B_0(k \tilde{S}_4)$[2]  ? 8 2 $1$ 1 $Q(2 {\cal B})_1$
M(16,9,4) $B_0(kSL_2(9))$ 1 9 3 $1$ 1 $Q(3 {\cal A})_2$
M(16,9,5) $B_0(k(2.A_7))$ 1 10 3 $1$ 1 $Q(3 {\cal B})$
M(16,9,6) $B_0(kSL_2(7))$ 1 9 3 $1$ 1 $Q(3 {\cal K})$

M(16,9,2) and M(16,9,3) are derived equivalent over $k$ by [Ho97], in which it is further proved that all blocks with defect group $Q_{16}$ and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).

M(16,9,4), M(16,9,5) and M(16,9,6) are derived equivalent over $\mathcal{O}$ by [Ei16][3].

## Notes

1. This is the double cover SmallGroup(240,89)
2. This is the double cover SmallGroup(48,28)
3. This result was obtained over $k$ in [Ho97]