Difference between revisions of "Classification by p-group"
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'''Classification of Morita equivalences for blocks with a given defect group''' | '''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. [[ | + | On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]]. |
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions. | See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions. | ||
| − | == Blocks | + | == Blocks for <math> p=2 </math> == |
{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" | ||
| − | | <strong><math>|D| | + | | <strong><math>1 \leq |D| \leq 8</math> </strong> |
|- | |- | ||
! scope="col"| <math>|D|</math> | ! scope="col"| <math>|D|</math> | ||
| Line 20: | Line 20: | ||
|- | |- | ||
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | | 1 || 1 || <math>1</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> || || | ||
| + | |- | ||
| + | | 4 || [[C4|1]] || [[C4|<math>C_4</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] ]] || | ||
| + | |- | ||
| + | |8 || [[C8|1]] || [[C8|<math>C_8</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 || [[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 || [[Q8|4]] || [[Q8|<math>Q_8</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 | ||
|} | |} | ||
| − | == | + | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" |
| + | | <strong><math>|D|=16</math> </strong> | ||
| + | |- | ||
| + | ! scope="col"| <math>|D|</math> | ||
| + | ! scope="col"| SmallGroup | ||
| + | ! scope="col"| Isotype | ||
| + | ! scope="col"| Donovan (w.r.t)? | ||
| + | ! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes | ||
| + | ! scope="col"| Complete (w.r.t.)? | ||
| + | ! scope="col"| Derived equiv classes (w.r.t)? | ||
| + | ! scope="col"| References | ||
| + | ! scope="col"| Notes | ||
| + | |- | ||
| + | |16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| + | |- | ||
| + | |16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] || | ||
| + | |- | ||
| + | |16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known | ||
| + | |- | ||
| + | |16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] || | ||
| + | |- | ||
| + | |16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| + | |- | ||
| + | |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>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || 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 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[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]]]. 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 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known | ||
| + | |- | ||
| + | |16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]] | ||
| + | |- | ||
| + | |16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known | ||
| + | |- | ||
| + | |16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] || | ||
| + | |} | ||
| + | |||
| + | The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]]. | ||
{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" | ||
| − | | <strong><math> | + | | <strong><math>|D|=32</math> </strong> |
|- | |- | ||
! scope="col"| <math>|D|</math> | ! scope="col"| <math>|D|</math> | ||
! 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 36: | Line 95: | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
|- | |- | ||
| − | | 2 || [[C2| | + | |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>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] || |
| − | |- | + | |- |
| − | | 4 || [[ | + | |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>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || |
| + | |- | ||
| + | |32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known | ||
| + | |- | ||
| + | |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#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math> | ||
| + | |- | ||
| + | |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#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || 6(6) || No || || [[References#K|[Ku80]]], [[References#K|[KoLaSa23]]] || Invariants known. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLaSa23]]] | ||
| + | |- | ||
| + | |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#C|[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#C|[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#C|[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#C|[CG12], [Sa12b]]] || | ||
| + | |- | ||
| + | |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>]] || <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>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] | ||
| + | |- | ||
| + | |32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || | ||
| + | |- | ||
| + | |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]]]. 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 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known | ||
| + | |- | ||
| + | |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#S|[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#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || | ||
| + | Invariants known | ||
| + | |- | ||
| + | |32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]] | ||
| + | |- | ||
| + | |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)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[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>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || | ||
| + | |- | ||
| + | |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#E|[EL18a]]] || | ||
| + | |- | ||
| + | |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#S|[Sa14]]] || | ||
| + | |- | ||
| + | |32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known | ||
| + | |- | ||
| + | |32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || 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)]] || No || || || || || | ||
| + | |- | ||
| + | |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>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL23]]] || | ||
| + | |- | ||
| + | |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>]] || No || || || || || | ||
|- | |- | ||
| − | | | + | |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>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known |
|- | |- | ||
| − | | | + | |32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known |
|- | |- | ||
| − | | | + | |32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. |
|} | |} | ||
| + | <!-- | ||
{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" | ||
| − | | <strong><math>|D|= | + | | <strong><math>|D|=64</math> </strong> |
|- | |- | ||
! scope="col"| <math>|D|</math> | ! scope="col"| <math>|D|</math> | ||
! 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 65: | Line 213: | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
|- | |- | ||
| − | | | + | |64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || |
| + | |- | ||
| + | |64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]|| | ||
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math> |
|- | |- | ||
| − | | | + | |64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || || |
|- | |- | ||
| − | | | + | |64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || || |
|- | |- | ||
| − | |16 || [[(C2)^ | + | |64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math> |
| − | |} | + | |- |
| + | |64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math> | ||
| + | |- | ||
| + | |64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math> | ||
| + | |- | ||
| + | |64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math> | ||
| + | |- | ||
| + | |64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_4:C_{16}</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math> | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| + | |- | ||
| + | |64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || || | ||
| + | |- | ||
| + | |64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || || | ||
| + | |- | ||
| + | |64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[C4xC4xC4|55]] || [[C4xC4xC4|<math>C_4 \times C_4 \times C_4</math>]] || <math>\mathcal{O}</math> ||4(4) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]]|| | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,56)|56]] || [[SmallGroup(64,56)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[C4x(C2xC2):C4|58]] || [[C4x(C2xC2):C4|<math>C_{4} \times (C_2 \times C_2):C_4</math>]] || || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,60)|60]] || [[SmallGroup(64,60)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>Sz(8)</math> | ||
| + | |- | ||
| + | |64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| + | |- | ||
| + | |64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,108)|108]] || [[SmallGroup(64,108)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[M4(2):C4|109]] || [[M4(2):C4|<math>M_4(2):C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,112)|112]] || [[SmallGroup(64,112)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[D8xC8|115]] || [[D8xC8|<math>D_8 \times C_8</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[D16xC4|118]] || [[D16xC4|<math>D_{16} \times C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SD16xC4|119]] || [[SD16xC4|<math>SD_{16} \times C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[Q16xC4|120]] || [[Q16xC4|<math>Q_{16} \times C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SD16:C4|121]] || [[SD16:C4|<math>SD_{16}:C_4</math>]]|| No || || || || || Fusion trivial? | ||
| + | |- | ||
| + | |64 || [[Q16:C4|122]] || [[Q16:C4|<math>Q_{16}:C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[D16:C4|123]] || [[D16:C4|<math>D_{16}:C_4</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,125)|125]] || [[SmallGroup(64,125)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[Q8xC8|126]] || [[Q8xC8|<math>Q_{8} \times C_8</math>]]|| <math>\mathcal{O}</math> || 3(3) || || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]] | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,127)|127]] || [[SmallGroup(64,127)]] || No || || || || || | ||
| + | |- | ||
| + | |64 || [[(C2xC2):D16|128]] || [[(C2xC2)D16|<math>(C_2 \times C_2):D_{16}</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[Q8:D8|129]] || [[Q8:D8|<math>Q_8:D_{8}</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]]|| No || || || || || | ||
| + | |- | ||
| + | |64 || [[Q8xQ8|239]] || [[Q8xQ8|<math>Q_{8} \times Q_8</math>]]|| <math>\mathcal{O}</math> || || || || [[References#E|[EL20]]] || | ||
| + | |- | ||
| + | |64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math> | ||
| + | |||
| + | |}--> | ||
==Blocks for <math>p=3</math>== | ==Blocks for <math>p=3</math>== | ||
{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" | ||
| − | | <strong><math> | + | | <strong><math>1 \leq |D| \leq 27</math> </strong> |
|- | |- | ||
! scope="col"| <math>|D|</math> | ! scope="col"| <math>|D|</math> | ||
| Line 107: | Line 492: | ||
! scope="col"| References | ! scope="col"| References | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
| + | |- | ||
| + | | 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
|- | |- | ||
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | | 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| − | |||
|- | |- | ||
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | |9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| Line 129: | Line 515: | ||
{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" | ||
| − | | <strong><math>5 \leq |D| \leq | + | | <strong><math>5 \leq |D| \leq 125</math> </strong> |
|- | |- | ||
! scope="col"| <math>|D|</math> | ! scope="col"| <math>|D|</math> | ||
| Line 139: | Line 525: | ||
! scope="col"| References | ! scope="col"| References | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
| + | |- | ||
| + | | 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
|- | |- | ||
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | |5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
| Line 144: | Line 532: | ||
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes | |25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes | ||
|- | |- | ||
| − | |25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || || | + | |25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || || |
| + | |- | ||
| + | |125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || | ||
| + | |- | ||
| + | |125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || | ||
| + | |- | ||
| + | |125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes | ||
| + | |- | ||
| + | |125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || || | ||
| + | |- | ||
| + | |125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || || | ||
|} | |} | ||
| Line 160: | Line 558: | ||
! scope="col"| References | ! scope="col"| References | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
| + | |- | ||
| + | | 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
|- | |- | ||
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes | |7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes | ||
Latest revision as of 18:59, 10 January 2024
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] | Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 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 | 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]k[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | 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 | ||
| 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]. 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]k[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 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]. 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] |
