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).