Difference between revisions of "MNA(3,1):C2"
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== Blocks with defect group <math>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</math> == | == Blocks with defect group <math>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</math> == | ||
| − | Sambale has used GAP to show that there are no non-trivial saturated fusion systems on <math>MNA(3,1):C_2</math> (see [[References#S|[Sa14]]]). See also [[References#V|[ | + | Sambale has used GAP to show that there are no non-trivial saturated fusion systems on <math>MNA(3,1):C_2</math> (see [[References#S|[Sa14]]]). See also [[References#V|[vdW91, pp. 382]]]. 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. | ||
Latest revision as of 16:02, 12 April 2019
Blocks with defect group [math]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[/math]
Sambale has used GAP to show that there are no non-trivial saturated fusion systems on [math]MNA(3,1):C_2[/math] (see [Sa14]). See also [vdW91, pp. 382]. 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(32,4,1) | [math]k(MNA(3,1):C_2)[/math] | 1 | 11 (8,2,1) | 1 | [math]1[/math] | 1 | 1 |
