Difference between revisions of "Classification by p-group"
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! scope="col"| SmallGroup | ! scope="col"| SmallGroup | ||
! scope="col"| Isotype | ! scope="col"| Isotype | ||
| + | ! scope="col"| Donovan (w.r.t.)? | ||
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes | ! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes | ||
! scope="col"| Complete (w.r.t.)? | ! scope="col"| Complete (w.r.t.)? | ||
| Line 19: | Line 20: | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
|- | |- | ||
| − | | 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | | 1 || 1 || <math>1</math> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | | 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | | 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | | 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | | 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | | 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] || | + | | 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] || |
|- | |- | ||
| − | |8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | |8 || [[C8|1]] || [[C8|<math>C_8</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | |8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | |8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | |8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] | + | |8 || [[D8|3]] || [[D8|<math>D_8</math>]] || <math>\mathcal{O}</math> ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
| − | |8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || | + | |8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || |
|- | |- | ||
| − | |8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG | + | |8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG |
|} | |} | ||
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|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || | |16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || | ||
|- | |- | ||
| − | |16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math> | + | |16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>\mathcal{O}</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
| − | |16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes | + | |16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes. Donovan's conjecture known over <math>\mathcal{O}</math> for <math>l(B) \neq 2</math> by [[References#L|[La16]]] (see discussion in [[References#E|[EEKLS26]]]). |
|- | |- | ||
| − | |16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || | + | |16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || <math>\mathcal{O}</math> || 6(?) || || <math>k</math> || [[References#E|[Er88a], [Er88b], [EEKLS26]]], [[References#H|[Ho97]]] || Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. When <math>l(B)=2</math> there are between 2 and 4 Morita equivalence classes over <math>k</math>. Finite list of Morita equivalence classes over <math>\mathcal{O}</math>, with possible repetition, by [[References#E|[EEKLS26]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]] |
|- | |- | ||
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] || | |16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] || | ||
| Line 129: | Line 130: | ||
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || | |32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || | ||
|- | |- | ||
| − | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math> | + | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>\mathcal{O}</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
| − | |32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || | + | |32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || Donovan's conjecture known over <math>\mathcal{O}</math> for <math>l(B) \neq 2</math> by [[References#L|[La16]]] (see discussion in [[References#E|[EEKLS26]]]). |
|- | |- | ||
| − | | | + | |16 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || <math>\mathcal{O}</math> || || || <math>k</math> || [[References#E|[Er88a], [Er88b], [EEKLS26]]], [[References#H|[Ho97]]] || Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. When <math>l(B)=2</math> there are between 2 and 4 Morita equivalence classes over <math>k</math>. Finite list of Morita equivalence classes over <math>\mathcal{O}</math>, with possible repetition, by [[References#E|[EEKLS26]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]] |
|- | |- | ||
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] | |32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] | ||
Revision as of 16:51, 13 August 2026
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.
Contents
Blocks for [math] p=2 [/math]
| [math]1 \leq |D| \leq 8[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t.)? | 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] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 2 | 1 | [math]C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 4 | 1 | [math]C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 4 | 2 | [math]C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Er82], [Li94] | |
| 8 | 1 | [math]C_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 8 | 2 | [math]C_4 \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 8 | 3 | [math]D_8[/math] | [math]\mathcal{O}[/math] | 6(?) | [math]k[/math] | [math]k[/math] | [Er87] | See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 8 | 4 | [math]Q_8[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]k[/math] | [Er88a], [Er88b], [HKL07], [Ei16] | |
| 8 | 5 | [math]C_2 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 8(8) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Ea16] | Uses CFSG |
| [math]|D|=16[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|---|
| 16 | 1 | [math]C_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 16 | 2 | [math]C_4 \times C_4[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKKS14] | |
| 16 | 3 | MNA(2,1) | No | 3(?) | No | [Sa11] | Block invariants known | |
| 16 | 4 | [math]C_4:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 16 | 5 | [math]C_8 \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 16 | 6 | [math]M_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 16 | 7 | [math]D_{16}[/math] | [math]\mathcal{O}[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 16 | 8 | [math]SD_{16}[/math] | [math]k[/math] | 7(?) | [Er88c], [Er90b] | Two other possible classes. Donovan's conjecture known over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/math] by [La16] (see discussion in [EEKLS26]). | ||
| 16 | 9 | [math]Q_{16}[/math] | [math]\mathcal{O}[/math] | 6(?) | [math]k[/math] | [Er88a], [Er88b], [EEKLS26], [Ho97] | Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16]. When [math]l(B)=2[/math] there are between 2 and 4 Morita equivalence classes over [math]k[/math]. Finite list of Morita equivalence classes over [math]\mathcal{O}[/math], with possible repetition, by [EEKLS26]. Principal blocks classified up to source algebra equivalence in [KoLa20b] | |
| 16 | 10 | [math]C_4 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL18a] | |
| 16 | 11 | [math]D_8 \times C_2[/math] | No | [Sa12] | Block invariants known | |||
| 16 | 12 | [math]Q_8 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | No | [EL20] | Block invariants known by [Sa13] | |
| 16 | 13 | [math]D_8*C_4[/math] | No | 3(?) | No | [Sa13b] | Block invariants known | |
| 16 | 14 | [math](C_2)^4[/math] | [math]\mathcal{O}[/math] | 16(16) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Ea18] |
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [Sa14].
| [math]|D|=32[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | 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] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 32 | 2 | [math]MNA(2,2)[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKS12] | |
| 32 | 3 | [math]C_8 \times C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 32 | 4 | [math]C_8:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 5 | [math]MNA(3,1)[/math] | No | [Sa11] | Invariants known | |||
| 32 | 6 | [math]MNA(2,1):C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 7 | SmallGroup(32,7) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | [math]M_{16}:C_2[/math] |
| 32 | 8 | [math]2.MNA(2,1)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 9 | [math]D_8:C_4[/math] | No | [Sa14,10.23] | Invariants known | |||
| 32 | 10 | [math]Q_8:C_4[/math] | No | [Sa14,10.25] | Invariants known | |||
| 32 | 11 | [math]C_4 \wr C_2[/math] | No | 6(6) | No | [Ku80], [KoLaSa23] | Invariants known. Principal blocks classified up to source algebra equivalence in [KoLaSa23] | |
| 32 | 12 | [math]C_4:C_8[/math] | [math]\mathcal{O}[/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] | [math]\mathcal{O}[/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] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 15 | SmallGroup(32,15) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 16 | [math]C_{16} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 32 | 17 | [math]M_{32}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 18 | [math]D_{32}[/math] | [math]\mathcal{O}[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 32 | 19 | [math]SD_{32}[/math] | [math]k[/math] | Donovan's conjecture known over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/math] by [La16] (see discussion in [EEKLS26]). | ||||
| 16 | 20 | [math]Q_{32}[/math] | [math]\mathcal{O}[/math] | [math]k[/math] | [Er88a], [Er88b], [EEKLS26], [Ho97] | Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16]. When [math]l(B)=2[/math] there are between 2 and 4 Morita equivalence classes over [math]k[/math]. Finite list of Morita equivalence classes over [math]\mathcal{O}[/math], with possible repetition, by [EEKLS26]. Principal blocks classified up to source algebra equivalence in [KoLa20b] | ||
| 32 | 21 | [math]C_4 \times C_4 \times C_2[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKKS14] | |
| 32 | 22 | [math]MNA(2,1) \times C_2[/math] | No | [Sa14,10.25] | Invariants known | |||
| 32 | 23 | [math](C_4:C_4) \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 24 | SmallGroup(32,24) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 25 | [math]D_8 \times C_4[/math] | No | [Sa14,9.7] |
Invariants known | |||
| 32 | 26 | [math]Q_8 \times C_4[/math] | [math]\mathcal{O}[/math] | 3(3) | No | [EL20] | Invariants known by [Sa14,9.28] | |
| 32 | 27 | SmallGroup(32,27) | No | |||||
| 32 | 28 | SmallGroup(32,28) | No | [Sa14,13.11] | Invariants known | |||
| 32 | 29 | SmallGroup(32,29) | No | [Sa14,13.11] | Invariants known | |||
| 32 | 30 | SmallGroup(32,30) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 31 | SmallGroup(32,31) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 32 | SmallGroup(32,32) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 33 | SmallGroup(32,33) | No | [Sa14,13.12] | Invariants partly known | |||
| 32 | 34 | SmallGroup(32,34) | No | [Sa14,13.12] | Invariants partly known | |||
| 32 | 35 | [math]C_4:Q_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 36 | [math]C_8 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL18a] | |
| 32 | 37 | [math]M_{16} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 38 | [math]D_8*C_8[/math] | No | [Sa14,9.18] | Invariants known | |||
| 32 | 39 | [math]D_{16} \times C_2[/math] | No | [Sa14,9.7] | Invariants known | |||
| 32 | 40 | [math]SD_{16} \times C_2[/math] | No | [Sa14,9.37] | Invariants known | |||
| 32 | 41 | [math]Q_{16} \times C_2[/math] | No | [Sa14,9.28] | Invariants known | |||
| 32 | 42 | [math]D_{16}*C_4[/math] | No | [Sa14,9.18] | Invariants known | |||
| 32 | 43 | SmallGroup(32,43) | No | |||||
| 32 | 44 | SmallGroup(32,44) | No | |||||
| 32 | 45 | [math]C_4 \times C_2 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 8(8) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL23] | |
| 32 | 46 | [math]D_8 \times C_2 \times C_2[/math] | No | |||||
| 32 | 47 | [math]Q_8 \times C_2 \times C_2[/math] | No | |||||
| 32 | 48 | [math](D_8*C_4) \times C_2[/math] | No | |||||
| 32 | 49 | [math]D_8*D_8[/math] | No | [Sa13c] | Invariants partly known | |||
| 32 | 50 | [math]D_8*Q_8[/math] | No | [Sa13c] | Invariants partly known | |||
| 32 | 51 | [math](C_2)^5[/math] | [math]\mathcal{O}[/math] | 34 (34) | [math]\mathcal{O}[/math] | [Ar19] | Derived eq. classes determined for 30 of the 34 Morita eq. classes. |
Blocks for [math]p=3[/math]
| [math]1 \leq |D| \leq 27[/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] | ||
| 3 | 1 | [math]C_3[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 9 | 1 | [math]C_9[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 9 | 2 | [math]C_3 \times C_3[/math] | |||||
| 27 | 1 | [math]C_{27}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 27 | 2 | [math]C_9 \times C_3[/math] | |||||
| 27 | 3 | [math]3_+^{1+2}[/math] | |||||
| 27 | 4 | [math]3_-^{1+2}[/math] | |||||
| 27 | 5 | [math]C_3 \times C_3 \times C_3[/math] |
Blocks for [math]p=5[/math]
| [math]5 \leq |D| \leq 125[/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] | ||
| 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] | |||||
| 125 | 1 | [math]C_{125}[/math] | |||||
| 125 | 2 | [math]C_{25} \times C_5[/math] | |||||
| 125 | 3 | [math]5_+^{1+2}[/math] | 62(62) | [math]\mathcal{O}[/math] | [AE23] | Inertial quotients are consistent within classes | |
| 125 | 4 | [math]5_-^{1+2}[/math] | |||||
| 125 | 5 | [math]C_5 \times C_5 \times C_5[/math] |
Blocks for [math]p\geq 7[/math]
| [math]|D|[/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] | ||
| 7 | 1 | [math]C_7[/math] | 14(14) | No | [math]\mathcal{O}[/math] | Max 21 classes | |
| 11 | 1 | [math]C_{11}[/math] | No | [math]\mathcal{O}[/math] | |||
| 13 | 1 | [math]C_{13}[/math] | No | [math]\mathcal{O}[/math] | |||
| 17 | 1 | [math]C_{17}[/math] | No | [math]\mathcal{O}[/math] | |||
| 19 | 1 | [math]C_{19}[/math] | No | [math]\mathcal{O}[/math] | |||
| 23 | 1 | [math]C_{23}[/math] | No | [math]\mathcal{O}[/math] |
