MNA(3,1):C2

Blocks with defect group [math]MNA(3,1):C_2=\langle a,b,c,d | a^2=b^2=c^2=d^4=1, ab=ba, ac=ca, dad^{-1}=abc, dbd^{-1}=bc=cb, cd=dc \rangle[/math]

Sambale has used GAP to show that there are no non-trivial saturated fusion systems on [math]MNA(3,1):C_2[/math] (see [Sa14]). See also [vdW91, pp. 382]. Hence every block with this defect group is nilpotent.

There is a unique [math]\mathcal{O}[/math]-Morita equivalence class.

Class	Representative	# lifts / [math]\mathcal{O}[/math]	[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(32,4,1)	[math]k(MNA(3,1):C_2)[/math]	1	11 (8,2,1)	1	[math]1[/math]			1	1