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Blocks with defect group [math]MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle[/math]

The defect groups are minimal nonabelian [math]2[/math]-groups. The invariants [math]k(B)[/math], [math]l(B)[/math] and [math]k_i(B)[/math] for all [math]i[/math] 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 [math]MNA(2,1)[/math] is a [math]2[/math]-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) [math]\cong A_4:C_4[/math]. By [Sa16] all non-nilpotent blocks with this defect group are isotypic.

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(16,3,1) [math]k(MNA(2,1))[/math] 1 10 1 [math]1[/math] 1 1
M(16,3,2) [math]B_0(k(A_5:C_4))[/math]  ? 10 2 [math]1[/math] 1 1
M(16,3,3) [math]k(A_4:C_4)[/math]  ? 10 2 [math]1[/math] 1 1

If [math]B[/math] is not nilpotent, then [math]k(B)=10, k_1(B)=2, l(B)=2[/math].