# C25

## Blocks with defect group $C_{25}$

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

There are 12 possible Brauer trees. It is not known which give rise to $\mathcal{O}$-Morita equivalence classes.

CLASSIFICATION INCOMPLETE
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(25,1,1) $kC_{25}$ 25 1 $1$ $C_{25} : C_{20}$ 1 1
M(25,1,2) $kD_{50}$ 14 2 $C_2$ $C_2 \times C_{10}$ 1 1
M(25,1,3) $B_0(kPSL_2(49))$ 14 2 $C_2$ $C_{10}$ 1 1
M(25,1,4) $k(C_{25}:C_4)$ 10 4 $C_4$ $C_4 \times C_5$ 1 1
M(25,1,5) $B_0(kPGL_2(49))$ 10 4 $C_4$ $C_2 \times C_5$ 1 1
M(25,1,6) $B_0(kSz(32))$ 10 4 $C_4$ $C_5$ 1 1

Blocks in $M(25,1,2)$ are derived equivalent (over $\mathcal{O}$) to those in $M(25,1,3)$.

Blocks in $M(25,1,4)$ are derived equivalent (over $\mathcal{O}$) to those in $M(25,1,5)$.