C27
Blocks with defect group [math]C_{27}[/math]
These are blocks with cyclic defect groups and so they are described by Brauer trees.
There are three [math]\mathcal{O}[/math]-Morita equivalence classes, accounting for all the possible Brauer trees.
| Class | Representative | [math]k(B)[/math] | [math]l(B)[/math] | Inertial quotients | [math]{\rm Pic}_\mathcal{O}(B)[/math] | [math]{\rm Pic}_k(B)[/math] | [math]{\rm mf_\mathcal{O}(B)}[/math] | [math]{\rm mf_k(B)}[/math] | Notes |
|---|---|---|---|---|---|---|---|---|---|
| M(27,1,1) | [math]kC_{27}[/math] | 27 | 1 | [math]1[/math] | [math]C_{27} : C_{18}[/math] | 1 | 1 | ||
| M(27,1,2) | [math]kD_{54}[/math] | 15 | 2 | [math]C_2[/math] | [math]C_{18}[/math] | 1 | 1 | ||
| M(27,1,3) | [math]B_0(kPSL_2(53))[/math] | 15 | 2 | [math]C_2[/math] | 1 | 1 |
Blocks in M(27,1,2) are derived equivalent (over [math]\mathcal{O}[/math]) to those in M(27,1,3).
