# SD16

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

These are examples of tame blocks and were first classified over $k$ by Erdmann (see [Er88c], [Er90b]). Further work was carried out in [Mac], where $SD(3 {\cal H})$ was eliminated, and the block $B_0(kPSL_3(3))$ reattributed to $SD(3 {\cal D})$, so that the class $SD(3 {\cal B})_1$ is now a class of algebras with no known block-realised representative. It is not known whether there are blocks realising the class $SD(3 {\cal C})_2$.

Until these remaining cases are resolved the labelling is provisional.

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(16,8,1) $kSD_{16}$ 1 7 1 $1$ 1
M(16,8,2) $B_5(kPSU_3(5))$  ? 8 2 $1$ 1 $SD(2 {\cal A})_1$
M(16,8,3) $B_0(kM_{10})=B_0(k(A_6.2_3))$  ? 7 2 $1$ 1 $SD(2 {\cal A})_2$
M(16,8,4) $B_3(k(3.M_{10}))=B_3(k(3.A_6.2_3))$  ? 7 2 $1$ 1 $SD(2 {\cal B})_1$
M(16,8,5) $B_1(kPSL_3(11))$  ? 8 2 $1$ 1 $SD(2 {\cal B})_2$
M(16,8,6) $B_0(kPSU_3(5))$  ? 8 3 $1$ 1 $SD(3 {\cal A})_1$
M(16,8,7) $B_0(kPSL_3(3))$  ? 8 3 $1$ 1 $SD(3 {\cal D})$. See note above.
? 8 3 $1$ 1 $SD(3 {\cal B})_1$
? 8 3 $1$ 1 $SD(3 {\cal B})_2$ or $SD(3 {\cal C})_2$[1]

M(16,8,2) and M(16,8,5) are derived equivalent over $k$ by [Ho97] .

M(16,8,3) and M(16,8,4) are derived equivalent over $k$ by [Ho97] .

All Morita equivalence classes with three simple modules are derived equivalent over $k$ by [Ho97] .

## Notes

1. See discussion in [Mac]: A block of the Monster group could be in one or other of these classes.