# MNA(2,1)

## Blocks with defect group $MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle$
The defect groups are minimal nonabelian $2$-groups. The invariants $k(B)$, $l(B)$ and $k_i(B)$ for all $i$ are determined in [Sa11]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the CFSG. The automorphism group of $MNA(2,1)$ is a $2$-group, but by [Sa14,12.7] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) $\cong A_4:C_4$. By [Sa16] all non-nilpotent blocks with this defect group are isotypic.
CLASSES NOT CLASSIFIED
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(16,3,1) $k(MNA(2,1))$ 1 10 1 $1$ 1 1
M(16,3,2) $B_0(k(A_5:C_4))$  ? 10 2 $1$ 1 1
M(16,3,3) $k(A_4:C_4)$  ? 10 2 $1$ 1 1
If $B$ is not nilpotent, then $k(B)=10, k_1(B)=2, l(B)=2$.