# C9

## Blocks with defect group $C_9$
There are three $\mathcal{O}$-Morita equivalence classes, accounting for all the possible Brauer trees.
Class Representative $k(B)$ $l(B)$ Inertial quotients ${\rm Pic}_\mathcal{O}(B)$ ${\rm Pic}_k(B)$ ${\rm mf_\mathcal{O}(B)}$ ${\rm mf_k(B)}$ Notes
M(9,1,1) $kC_9$ 9 1 $1$ $C_9 : C_6$ 1 1
M(9,1,2) $kD_{18}$ 6 2 $C_2$ $C_2 \times C_3$ 1 1
M(9,1,3) $B_0(kSL_2(8))$ 6 2 $C_2$ $C_3$ 1 1
Blocks in M(9,1,2) are derived equivalent (over $\mathcal{O}$) to those in M(9,1,3)