C25

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Blocks with defect group [math]C_{25}[/math]

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 [math]\mathcal{O}[/math]-Morita equivalence classes.

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

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

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