Difference between revisions of "C4:C4"
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== Blocks with defect group <math>C_4:C_4</math> == | == Blocks with defect group <math>C_4:C_4</math> == | ||
| − | <math>C_4:C_4</math> is a [[Metacyclic defect groups|metacyclic group]] of nonmaximal class, and so by [[References|[Sa12b]]] there is only one possible [[Glossary#Fusion system|fusion system]]. Hence every block with this defect group is nilpotent. | + | <math>C_4:C_4</math> is a [[Metacyclic defect groups|metacyclic group]] of nonmaximal class, and so by [[References#C|[CG12]] or [[References#S|[Sa12b]]] there is only one possible [[Glossary#Fusion system|fusion system]]. Hence every block with this defect group is nilpotent. |
There is a unique <math>\mathcal{O}</math>-Morita equivalence class. | There is a unique <math>\mathcal{O}</math>-Morita equivalence class. | ||
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| − | |[[M(16,4,1)]] || <math>k(C_4:C_4)</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 || | + | |[[M(16,4,1)]] || <math>k(C_4:C_4)</math> || 1 ||10 (8,2) ||1 ||<math>1</math> || || ||1 ||1 || |
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Latest revision as of 17:05, 28 January 2019
Blocks with defect group [math]C_4:C_4[/math]
[math]C_4:C_4[/math] is a metacyclic group of nonmaximal class, and so by [CG12 or [Sa12b] there is only one possible fusion system. 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(16,4,1) | [math]k(C_4:C_4)[/math] | 1 | 10 (8,2) | 1 | [math]1[/math] | 1 | 1 |
