# MNA(3,1):C2

## Blocks with defect group $MNA(3,1):C_2=\langle a,b,c,d | a^2=b^2=c^2=d^4=1, ab=ba, ac=ca, dad^{-1}=abc, dbd^{-1}=bc=cb, cd=dc \rangle$
Sambale has used GAP to show that there are no non-trivial saturated fusion systems on $MNA(3,1):C_2$ (see [Sa14]). See also [vdW91, pp. 382]. Hence every block with this defect group is nilpotent.
There is a unique $\mathcal{O}$-Morita equivalence class.
Class Representative # lifts / $\mathcal{O}$ $k(B)$ $l(B)$ Inertial quotients ${\rm Pic}_\mathcal{O}(B)$ ${\rm Pic}_k(B)$ ${\rm mf_\mathcal{O}(B)}$ ${\rm mf_k(B)}$ Notes
M(32,4,1) $k(MNA(3,1):C_2)$ 1 11 (8,2,1) 1 $1$ 1 1