# Difference between revisions of "Q8"

## Blocks with defect group $Q_8$
These are examples of tame blocks and were first classified over $k$ by Erdmann (see [Er87] ). The Morita equivalence classes lift to unique Morita equivalence classes over $\mathcal{O}$ by [HKL07], [Ei16] and the theory of nilpotent blocks.
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(8,4,1) $kQ_8$ 1 5 1 $1$ $S_4$ 1 1
M(8,4,2) $B_0(kSL_2(5))$ 1 7 3 $C_3$ $C_2$ 1 $Q(3 {\cal A})_2$
M(8,4,3) $kSL_2(3)$ 1 7 3 $C_3$ $S_3$ 1 1 $Q(3 {\cal K})$
M(8,4,2) and M(8,4,3) are derived equivalent over $k$ by [Ho97] .