Classification of Morita equivalences for blocks with a given defect group
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. Generic classifications for classes of p-groups can be found here.
See this page for a description of the labelling conventions.
Blocks of defect zero
[math]|D|=1[/math]
|
[math]|D|[/math]
|
SmallGroup
|
Isotype
|
Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes
|
Complete (w.r.t.)?
|
Derived equiv classes (w.r.t)?
|
References
|
Notes
|
1 |
1 |
[math]1[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
|
|
Blocks for [math] p=2 [/math]
The following takes as its starting point Table 13.1 of Sambale's book [Sa14].
[math]|D|=32[/math]
|
[math]|D|[/math]
|
SmallGroup
|
Isotype
|
Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes
|
Complete (w.r.t.)?
|
Derived equiv classes (w.r.t)?
|
References
|
Notes
|
32 |
1 |
[math]C_{32}[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
|
|
32 |
2 |
[math]MNA(2,2)[/math] |
2(2) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[EKS12] |
|
32 |
3 |
[math]C_8 \times C_4[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
|
|
32 |
4 |
[math]C_8:C_4[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[CG12], [Sa12b] |
|
32 |
5 |
[math]MNA(3,1)[/math] |
|
|
|
|
|
32 |
6 |
[math]MNA(2,1):C_2[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
7 |
[math]M_{16}:C_2[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
8 |
[math]2.MNA(2,1)[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
9 |
[math]D_8:C_4[/math] |
|
|
|
Invariants known by [Sa14,10.23] |
|
32 |
10 |
[math]Q_8:C_4[/math] |
|
|
|
Invariants known by [Sa14,10.25] |
|
32 |
11 |
[math]C_4 \wr C_2[/math] |
|
|
|
Invariants known by [Ku80] |
|
32 |
12 |
[math]C_4:C_8[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[CG12], [Sa12b] |
|
32 |
13 |
[math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[CG12], [Sa12b] |
|
32 |
14 |
[math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[CG12], [Sa12b] |
|
32 |
15 |
SmallGroup(32,15) |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[CG12], [Sa12b] |
|
32 |
16 |
[math]C_{16} \times C_2[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
|
|
32 |
17 |
[math]M_{32}[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[CG12], [Sa12b] |
|
32 |
18 |
[math]D_{32}[/math] |
|
|
|
|
|
32 |
19 |
[math]SD_{32}[/math] |
|
|
|
|
|
32 |
20 |
[math]Q_{32}[/math] |
|
|
|
|
|
32 |
21 |
[math]C_4 \times C_4 \times C_2[/math] |
2(2) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[EKKS14]
|
32 |
22 |
[math]MNA(2,1) \times C_2[/math] |
|
|
|
Invariants known by [Sa14,10.25] |
|
32 |
23 |
[math](C_4:C_4) \times C_2[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
24 |
SmallGroup(32,24) |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
25 |
[math]D_8 \times C_4[/math] |
|
|
|
Invariants known by [Sa14,9.7] |
|
32 |
26 |
[math]Q_8 \times C_4[/math] |
|
|
|
Invariants known by [Sa14,9.28] |
|
32 |
27 |
SmallGroup(32,27) |
|
|
|
|
|
32 |
28 |
SmallGroup(32,28) |
|
|
|
Invariants known by [Sa14,13.11] |
|
32 |
29 |
SmallGroup(32,29) |
|
|
|
Invariants known by [Sa14,13.11] |
|
32 |
30 |
SmallGroup(32,30) |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
31 |
SmallGroup(32,31) |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
32 |
SmallGroup(32,32) |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
32 |
33 |
SmallGroup(32,33) |
|
|
|
Invariants partly known by [Sa14,13.12] |
|
32 |
34 |
SmallGroup(32,34) |
|
|
|
Invariants partly known by [Sa14,13.12] |
|
32 |
35 |
[math]C_4:Q_8[/math] |
1(1) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
[Sa14] |
|
Blocks for [math]p=3[/math]
Blocks for [math]p=5[/math]
[math]5 \leq |D| \leq 25[/math]
|
[math]|D|[/math]
|
SmallGroup
|
Isotype
|
Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes
|
Complete (w.r.t.)?
|
Derived equiv classes (w.r.t)?
|
References
|
Notes
|
5 |
1 |
[math]C_5[/math] |
6(6) |
[math]\mathcal{O}[/math] |
[math]\mathcal{O}[/math] |
|
|
25 |
1 |
[math]C_{25}[/math] |
6(6) |
No |
[math]\mathcal{O}[/math] |
|
Max 12 classes
|
25 |
2 |
[math]C_5 \times C_5[/math] |
|
|
|
|
|
Blocks for [math]p\geq 7[/math]