# C5

## Blocks with defect group $C_5$

These are blocks with cyclic defect groups and so they are described by Brauer trees.

There are six $\mathcal{O}$-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(5,1,1) $kC_5$ 5 1 $1$ $C_5:C_4$ 1 1
M(5,1,2) $kD_{10}$ 4 2 $C_2$ $C_2 \times C_2$ 1 1
M(5,1,3) $B_0(kA_5)$ 4 2 $C_2$ $C_2$ 1 1
M(5,1,4) $k(C_5:C_4)$ 5 4 $C_4$ $C_4$ 1 1
M(5,1,5) $B_0(kA_7)$ 5 4 $C_4$ $C_2$ 1 1
M(5,1,6) $B_{15}(k(6.A_7))$ 5 4 $C_4$ $1$ 1 1