# C27

## Blocks with defect group $C_{27}$
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(27,1,1) $kC_{27}$ 27 1 $1$ $C_{27} : C_{18}$ 1 1
M(27,1,2) $kD_{54}$ 15 2 $C_2$ $C_{18}$ 1 1
M(27,1,3) $B_0(kPSL_2(53))$ 15 2 $C_2$ 1 1
Blocks in M(27,1,2) are derived equivalent (over $\mathcal{O}$) to those in M(27,1,3).