C7

Blocks with defect group [math]C_7[/math]

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 [math]k[/math]-Morita equivalence classes lift to [math]\mathcal{O}[/math]-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(7,1,1)	[math]kC_7[/math]	7	1	[math]1[/math]			1	1
M(7,1,2)	[math]kD_{15}[/math]	5	2	[math]C_2[/math]			1	1
M(7,1,3)	[math]B_0(kPSL_2(13))[/math]	5	2	[math]C_2[/math]			1	1
M(7,1,4)	[math]k(C_7:C_3)[/math]	5	3	[math]C_3[/math]			1	1
M(7,1,5)	[math]B_9(k(3.A_7))[/math]	5	3	[math]C_3[/math]			1	1
M(7,1,6)	[math]B_0(kA_7)[/math]	5	3	[math]C_3[/math]			1	1
M(7,1,7)	[math]B_{15}(k(6.A_7))[/math]	5	3	[math]C_3[/math]			1	1
M(7,1,8)	[math]k(C_7:C_6)[/math]	7	6	[math]C_6[/math]			1	1
M(7,1,9)	[math]B_0(k(S_7))[/math]	7	6	[math]C_6[/math]			1	1
M(7,1,10)	[math]B_0(kPSL_3(29^3).3)[/math]	7	6	[math]C_6[/math]			1	1
M(7,1,11)	[math]B_{16}(k(2.J_2))[/math]	7	6	[math]C_6[/math]			1	1
M(7,1,12)	[math]B_{29}(k(2.Ru))[/math]	7	6	[math]C_6[/math]			1	1
M(7,1,13)	[math]B_0(k({}^2G_2(27)))[/math]	7	6	[math]C_6[/math]			1	1
M(7,1,14)		7	6	[math]C_6[/math]			1	1
		7	6	[math]C_6[/math]			1	1
		7	6	[math]C_6[/math]			1	1
		7	6	[math]C_6[/math]			1	1
		7	6	[math]C_6[/math]			1	1
		7	6	[math]C_6[/math]			1	1

To do:

• The tree labelling the potential class M(7,1,14) is believed to be realised, but an example has not been confirmed.