# C7

## Blocks with defect group $C_7$

These are blocks with cyclic defect groups and so they are described by Brauer trees. There are candidate Brauer trees with no known block realising them.

All $k$-Morita equivalence classes lift to $\mathcal{O}$-classes.

CLASSIFICATION INCOMPLETE
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(7,1,1) $kC_7$ 1 7 1 $1$ $C_7:C_6$ 1 1
M(7,1,2) $kD_{14}$ 1 5 2 $C_2$ $C_2 \times C_3$ 1 1
M(7,1,3) $B_0(kPSL_2(13))$ 1 5 2 $C_2$ $C_3$ 1 1
M(7,1,4) $k(C_7:C_3)$ 1 5 3 $C_3$ $C_3 \times C_2$ 1 1
M(7,1,5) $B_9(k(3.A_7))$ 1 5 3 $C_3$ $C_2$ 1 1
M(7,1,6) $B_0(kA_7)$ 1 5 3 $C_3$ $C_2$ 1 1
M(7,1,7) $B_{15}(k(6.A_7))$ 1 5 3 $C_3$ $C_2$ 1 1
M(7,1,8) $k(C_7:C_6)$ 1 7 6 $C_6$ $C_6$ 1 1
M(7,1,9) $B_0(k(S_7))$ 1 7 6 $C_6$ $C_2$ 1 1
M(7,1,10) $B_0(kPSL_3(29^3).3)$ 1 7 6 $C_6$ $C_3$ 1 1
M(7,1,11) $B_{16}(k(2.J_2))$ 1 7 6 $C_6$  ? 1 1
M(7,1,12) $B_{29}(k(2.Ru))$ 1 7 6 $C_6$ $1$ 1 1
M(7,1,13) $B_0(k({}^2G_2(27)))$ 1 7 6 $C_6$ $1$ 1 1
M(7,1,14) $B_0(k({}^2G_2(243)))$[1] 1 7 6 $C_6$ $1$ 1 1
7 6 $C_6$ 1 1
7 6 $C_6$ 1 1
7 6 $C_6$ 1 1
7 6 $C_6$ 1 1
7 6 $C_6$ 1 1
7 6 $C_6$ 1 1
7 6 $C_6$ 1 1

To do:

• Picard group for M(7,1,11) trivial or $C_2$?

## Notes

1. Theorem 3.8 of [Du14] gives existence for $q=3^{2m+1}$ such that $(3^{2m+1}-3^{m+1}+1)_7=7$