Difference between revisions of "MNA(2,1)"
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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> == | == 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 [[References|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but there exists | + | 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 [[References|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References|[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>. |
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>''' | '''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>''' | ||
Revision as of 17:04, 24 November 2018
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].
CLASSIFICATION INCOMPLETE
| 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]k(SmallGroup(48,30))[/math] | ? | 10 | 2 | [math]1[/math] | 1 | 1 |
