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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).