Difference between revisions of "Classification by p-group"
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(→Blocks for p=2) |
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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 94: | Line 95: | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
|- | |- | ||
| − | |32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | |32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | |32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKS12]]] || | + | |32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKS12]]] || |
|- | |- | ||
| − | |32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | |32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | |32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || | + | |32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || |
|- | |- | ||
| − | |32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || || || || [[References|[Sa11] ]] || Invariants known | + | |32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References|[Sa11] ]] || Invariants known |
|- | |- | ||
| − | |32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[M16:C2|7]] || [[M16:C2|<math>M_{16}:C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[M16:C2|7]] || [[M16:C2|<math>M_{16}:C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || || || || [[References|[Sa14,10.23]]] || Invariants known | + | |32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References|[Sa14,10.23]]] || Invariants known |
|- | |- | ||
| − | |32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || || || || [[References|[Sa14,10.25]]] || Invariants known | + | |32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References|[Sa14,10.25]]] || Invariants known |
|- | |- | ||
| − | |32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || || || || [[References|[Ku80]]] || Invariants known | + | |32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || || || || [[References|[Ku80]]] || Invariants known |
|- | |- | ||
| − | |32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || | + | |32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || |
|- | |- | ||
| − | |32 || [[C8:C4a|13]] || [[C8:C4a|<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> || [[References|[CG12], [Sa12b]]] || | + | |32 || [[C8:C4a|13]] || [[C8:C4a|<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> || [[References|[CG12], [Sa12b]]] || |
|- | |- | ||
| − | |32 || [[C8:C4b|14]] || [[C8:C4b|<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> || [[References|[CG12], [Sa12b]]] || | + | |32 || [[C8:C4b|14]] || [[C8:C4b|<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> || [[References|[CG12], [Sa12b]]] || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || | + | |32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || |
|- | |- | ||
| − | |32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | |32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
|- | |- | ||
| − | |32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[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|[CG12], [Sa12b]]] || |
|- | |- | ||
| − | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || | + | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || |
|- | |- | ||
| − | |32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || || || || || | + | |32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || |
|- | |- | ||
| − | |32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]] | + | |32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]] |
|- | |- | ||
| − | |32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[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|[EKKS14]]] |
|- | |- | ||
| − | |32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || || || || [[References|[Sa14,10.25]]] || Invariants known | + | |32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References|[Sa14,10.25]]] || Invariants known |
|- | |- | ||
| − | |32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || || || || [[References|[Sa14,9.7]]] || | + | |32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References|[Sa14,9.7]]] || |
Invariants known | Invariants known | ||
|- | |- | ||
| − | |32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || || || || [[References|[Sa14,9.28]]] || Invariants known | + | |32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || No || || || || [[References|[Sa14,9.28]]] || Invariants known |
|- | |- | ||
| − | |32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || || || || || | + | |32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || || || || [[References|[Sa14,13.11]]] || Invariants known | + | |32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References|[Sa14,13.11]]] || Invariants known |
|- | |- | ||
| − | |32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || || || || [[References|[Sa14,13.11]]] || Invariants known | + | |32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References|[Sa14,13.11]]] || Invariants known |
|- | |- | ||
| − | |32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || || || || [[References|[Sa14,13.12]]] || Invariants partly known | + | |32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known |
|- | |- | ||
| − | |32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || || || || [[References|[Sa14,13.12]]] || Invariants partly known | + | |32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known |
|- | |- | ||
| − | |32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] || | + | |32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] || |
|- | |- | ||
| − | |32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || | + | |32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Sa14]]] || |
|- | |- | ||
| − | |32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || || || || [[References|[Sa14,9.18]]] || Invariants known | + | |32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known |
|- | |- | ||
| − | |32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || || || || [[References|[Sa14,9.7]]] || Invariants known | + | |32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References|[Sa14,9.7]]] || Invariants known |
|- | |- | ||
| − | |32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || || || || [[References|[Sa14,9.37]]] || Invariants known | + | |32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References|[Sa14,9.37]]] || Invariants known |
|- | |- | ||
| − | |32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || || || || [[References|[Sa14,9.28]]] || Invariants known | + | |32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References|[Sa14,9.28]]] || Invariants known |
|- | |- | ||
| − | |32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || || || || [[References|[Sa14,9.18]]] || Invariants known | + | |32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known |
|- | |- | ||
| − | |32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || || || || || | + | |32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || || |
|- | |- | ||
| − | |32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || || || || || | + | |32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || || |
|- | |- | ||
| − | |32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || || || || [[References|[Sa14, 13.9]]] || Invariants known | + | |32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || No || || || || [[References|[Sa14, 13.9]]] || Invariants known |
|- | |- | ||
| − | |32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || || || || || | + | |32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || || |
|- | |- | ||
| − | |32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || || || || || | + | |32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || || |
|- | |- | ||
| − | |32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || || || || || | + | |32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || || |
|- | |- | ||
| − | |32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || || || || [[References|[Sa13c]]] || Invariants partly known | + | |32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References|[Sa13c]]] || Invariants partly known |
|- | |- | ||
| − | |32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || || || || [[References|[Sa13c]]] || Invariants partly known | + | |32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References|[Sa13c]]] || Invariants partly known |
|- | |- | ||
| − | |32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || || || || || | + | |32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || || || || || |
|} | |} | ||
Revision as of 16:21, 28 March 2019
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 | 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] | ||
| 2 | 1 | [math]C_2[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 4 | 1 | [math]C_4[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 4 | 2 | [math]C_2 \times C_2[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Er82], [Li94] | |
| 8 | 1 | [math]C_8[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 8 | 2 | [math]C_4 \times C_2[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 8 | 3 | [math]D_8[/math] | 6(?) | [math]k[/math] | [math]k[/math] | [Er87] | |
| 8 | 4 | [math]Q_8[/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] | 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 | [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]k[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | |
| 16 | 8 | [math]SD_{16}[/math] | [math]k[/math] | 8(?) | [Er88c], [Er90b] | Two other possible classes | ||
| 16 | 9 | [math]Q_{16}[/math] | No | 6(?) | [math]k[/math] | [Er88a], [Er88b], [Ho97] | Two possibly infinite families when [math]l(B)=2[/math]. Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16] | |
| 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] | No | [Sa13] | Block invariants known | |||
| 16 | 13 | [math]D_8*C_4[/math] | 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 following 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 | [math]M_{16}:C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 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 | [Ku80] | Invariants known | |||
| 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]k[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | |
| 32 | 19 | [math]SD_{32}[/math] | [math]k[/math] | |||||
| 32 | 20 | [math]Q_{32}[/math] | No | [Er88a], [Er88b], [Ho97] | Two possibly infinite families when [math]l(B)=2[/math]. Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16] | |||
| 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] | No | [Sa14,9.28] | Invariants known | |||
| 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] | No | [Sa14, 13.9] | Invariants known | |||
| 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] |
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 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 |
|---|---|---|---|---|---|---|---|
| 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] |
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] |
