Difference between revisions of "Classification by p-group"

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(→‎Blocks for p=2: Added quaternion consequences of [EEKLS26])
(Added order 64 table.)
 
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|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]]]).
 
|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 || [[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]]]  
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|}
 
|}
  
−
<!--
+
Fusion trivial groups of order 64 were determined by Pete Gautam.
 +
 
 
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 
| <strong><math>|D|=64</math> &nbsp;</strong>
 
| <strong><math>|D|=64</math> &nbsp;</strong>
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! scope="col"| References
 
! scope="col"| References
 
! 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 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> || 2(2) || <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> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>C_8:C_8</math>. Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 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)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>C_8:C_8</math>. Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>C_8:C_8</math>. Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || <math>\mathcal{O}</math> ||  ||  ||  ||  || Maxperm (as in the supplied data); no class count or completeness assertion supplied
 +
|-
 +
| 64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>M_{16}:C_4</math>. Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>M_{16}:C_4</math>. Fusion trivial
 +
|-
 +
| 64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>C_{16}:C_4</math>. Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>C_{16}:C_4</math>. Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>M_{32}:C_2</math>. Fusion trivial
 +
|-
 +
| 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)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,34)]] || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || <math>C_{16}:C_4</math>. Fusion trivial by [[References#C|[CG12], [Sa12b]]]
 +
|-
 +
| 64 || [[SmallGroup(64,49)|49]] || [[SmallGroup(64,49)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No ||  ||  ||  ||  ||
 +
|-
 +
| 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> ||  || [[References#W|[WZZ18]]], [[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>]] || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</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)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No ||  ||  ||  ||  ||
 
|-
 
|-
−
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||
+
| 64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || || FT
+
| 64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|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(64,76)|76]] || [[SmallGroup(64,76)]] || No || || || || ||  
 
|-
 
|-
−
|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(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math> FT
+
| 64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || || Maxperm
+
| 64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math>
+
| 64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || || [[References#E|[Ea24]]] || Suzuki <math>2</math>-group of type A
 
|-
 
|-
−
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || || FT
+
| 64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || || FT
+
| 64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math> FT
+
| 64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math> FT
+
| 64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|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> || || FT
+
| 64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]] || No || || || || ||  
 
|-
 
|-
−
|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> FT
+
| 64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || ||  
 
|-
 
|-
−
|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> FT
+
| 64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||
+
| 64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math> FT
+
| 64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math>  
+
| 64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||  
+
| 64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || FT
+
| 64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math>
+
| 64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math>
+
| 64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?
+
| 64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?
+
| 64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math>
+
| 64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||
+
| 64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || ||
+
| 64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || ||
+
| 64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial?
+
| 64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || 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(64,105)|105]] || [[SmallGroup(64,105)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|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(64,108)|108]] || [[SmallGroup(64,108)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|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 || [[M4(2):C4|109]] || [[M4(2):C4|<math>M_4(2):C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
−
|-
 
−
|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 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || ||
+
| 64 || [[SmallGroup(64,112)|112]] || [[SmallGroup(64,112)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || ||
+
| 64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || ||
+
| 64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
−
|-
 
−
|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 || [[D8xC8|115]] || [[D8xC8|<math>D_8 \times C_8</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || ||
+
| 64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || 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(64,117)|117]] || [[SmallGroup(64,117)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || ||
+
| 64 || [[D16xC4|118]] || [[D16xC4|<math>D_{16} \times C_4</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||
+
| 64 || [[SD16xC4|119]] || [[SD16xC4|<math>SD_{16} \times C_4</math>]] || 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 || [[Q16xC4|120]] || [[Q16xC4|<math>Q_{16} \times C_4</math>]] || No || ||  || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || ||
+
| 64 || [[SD16:C4|121]] || [[SD16:C4|<math>SD_{16}:C_4</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||
+
| 64 || [[Q16:C4|122]] || [[Q16:C4|<math>Q_{16}:C_4</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial?
+
| 64 || [[D16:C4|123]] || [[D16:C4|<math>D_{16}:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial?
+
| 64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || ||  
 
|-
 
|-
−
|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(64,125)|125]] || [[SmallGroup(64,125)]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial?
+
| 64 || [[Q8xC8|126]] || [[Q8xC8|<math>Q_{8} \times C_8</math>]] || No || || || || || Invariants known by [[References#S|[Sa14,9.28]]]
 
|-
 
|-
−
|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(64,127)|127]] || [[SmallGroup(64,127)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial?
+
| 64 || [[(C2xC2):D16|128]] || [[(C2xC2):D16|<math>(C_2 \times C_2):D_{16}</math>]] || No || || || || ||  
 
|-
 
|-
−
|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 || [[Q8:D8|129]] || [[Q8:D8|<math>Q_8:D_{8}</math>]] || No || ||  || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||
+
| 64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]] || No || || || || ||  
 
|-
 
|-
−
|64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||
+
| 64 || 131 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial?
+
| 64 || 132 || || No || || || || ||  
 
|-
 
|-
−
|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 || 133 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||
+
| 64 || 134 || <math>\operatorname{Syl}_2(M_{12})</math> || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial?
+
| 64 || 135 || || No || || || || ||  
 
|-
 
|-
−
|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 || 136 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||
+
| 64 || 137 || || 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 || 138 || || No || || || || ||  
 
|-
 
|-
−
|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 || 139 || || No || || || || ||  
 
|-
 
|-
−
|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 || 140 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial?
+
| 64 || 141 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || ||
+
| 64 || 142 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial?
+
| 64 || 143 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || ||
+
| 64 || 144 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial?
+
| 64 || 145 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial?
+
| 64 || 146 || || No || || || || ||  
 
|-
 
|-
−
|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 || 147 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial?
+
| 64 || 148 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial?
+
| 64 || 149 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial?
+
| 64 || 150 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial?
+
| 64 || 151 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || ||
+
| 64 || 152 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || ||
+
| 64 || 153 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || ||  
+
| 64 || 154 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || ||  
+
| 64 || 155 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || ||  
+
| 64 || 156 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || ||  
+
| 64 || 157 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || ||
+
| 64 || 158 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || ||
+
| 64 || 159 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial?
+
| 64 || 160 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial?
+
| 64 || 161 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial?
+
| 64 || 162 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||
+
| 64 || 163 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || No || || || || || Fusion trivial?
+
| 64 || 164 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,108)|108]] || [[SmallGroup(64,108)]] || No || || || || ||
+
| 64 || 165 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[M4(2):C4|109]] || [[M4(2):C4|<math>M_4(2):C_4</math>]]|| No || || || || ||
+
| 64 || 166 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || No || || || || ||
+
| 64 || 167 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || No || || || || ||
+
| 64 || 168 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|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 || 169 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || No || || || || ||
+
| 64 || 170 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || No || || || || ||
+
| 64 || 171 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[D8xC8|115]] || [[D8xC8|<math>D_8 \times C_8</math>]]|| No || || || || ||
+
| 64 || 172 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || ||
+
| 64 || 173 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || No || || || || ||
+
| 64 || 174 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[D16xC4|118]] || [[D16xC4|<math>D_{16} \times C_4</math>]]|| No || || || || ||
+
| 64 || 175 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SD16xC4|119]] || [[SD16xC4|<math>SD_{16} \times C_4</math>]]|| No || || || || ||
+
| 64 || 176 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[Q16xC4|120]] || [[Q16xC4|<math>Q_{16} \times C_4</math>]]|| No || || || || ||
+
| 64 || 177 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SD16:C4|121]] || [[SD16:C4|<math>SD_{16}:C_4</math>]]|| No || || || || || Fusion trivial?
+
| 64 || 178 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[Q16:C4|122]] || [[Q16:C4|<math>Q_{16}:C_4</math>]]|| No || || || || ||
+
| 64 || 179 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[D16:C4|123]] || [[D16:C4|<math>D_{16}:C_4</math>]]|| No || || || || ||
+
| 64 || 180 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || ||
+
| 64 || 181 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[SmallGroup(64,125)|125]] || [[SmallGroup(64,125)]] || No || || || || ||
+
| 64 || 182 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|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 || 183 || <math>C_{16} \times C_2^2</math> || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || || [[References#E|[EL18a]]] ||  
 
|-
 
|-
−
|64 || [[SmallGroup(64,127)|127]] || [[SmallGroup(64,127)]] || No || || || || ||
+
| 64 || 184 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial
 
|-
 
|-
−
|64 || [[(C2xC2):D16|128]] || [[(C2xC2)D16|<math>(C_2 \times C_2):D_{16}</math>]]|| No || || || || ||
+
| 64 || 185 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[Q8:D8|129]] || [[Q8:D8|<math>Q_8:D_{8}</math>]]|| No || || || || ||
+
| 64 || 186 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]]|| No || || || || ||
+
| 64 || 187 || || No || || || || ||  
 
|-
 
|-
−
|64 || [[Q8xQ8|239]] || [[Q8xQ8|<math>Q_{8} \times Q_8</math>]]|| <math>\mathcal{O}</math> || || || || [[References#E|[EL20]]] ||  
+
| 64 || 188 || || No || || || || ||  
 
|-
 
|-
−
|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>
+
| 64 || 189 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 190 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 191 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 192 || <math>C_4^2 \times C_2^2</math> || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> ||  || [[References#E|[EL23]]] || Abelian
 +
|-
 +
| 64 || 193 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 194 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 195 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 196 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 197 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 198 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 199 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 200 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 201 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 202 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 203 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 204 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 205 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 206 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 207 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 208 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 209 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 210 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 211 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 212 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 213 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 214 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 215 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 216 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 217 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 218 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 219 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 220 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 221 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 222 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 223 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 224 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 225 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 226 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 227 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 228 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 229 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 230 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 231 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 232 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 233 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 234 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 235 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 236 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 237 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 238 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || [[Q8xQ8|239]] || [[Q8xQ8|<math>Q_{8} \times Q_8</math>]] || <math>\mathcal{O}</math> ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 240 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 241 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 242 || <math>\operatorname{Syl}_2(L_3(4))</math> || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 243 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || 244 ||  || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> ||  ||  || Fusion trivial
 +
|-
 +
| 64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> ||  || [[References#E|[Ea24]]] || Suzuki <math>2</math>-group of type B
 +
|-
 +
| 64 || 246 || <math>C_8 \times C_2^3</math> || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> ||  || [[References#E|[EL23]]] ||
 +
|-
 +
| 64 || 247 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 248 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 249 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 250 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 251 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 252 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 253 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 254 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 255 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 256 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 257 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 258 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 259 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 260 || <math>C_4 \times C_2^4</math> || <math>\mathcal{O}</math> ||  || No ||  || [[References#E|[EEL18]]] ||  
 +
|-
 +
| 64 || 261 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 262 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 263 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 264 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 265 ||  || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 266 || <math>2_+^{1+4} * C_4</math> || No ||  ||  ||  ||  ||
 +
|-
 +
| 64 || 267 || <math>(C_2)^6</math> || <math>\mathcal{O}</math> || ?(81) || No ||  || [[References#A|[Ar21]]], [[References#E|[EEL18]]] || 81 known <math>\mathcal{O}</math>-classes containing a principal block; full classification not known
 +
|}
  
−
|}-->
 
  
 
==Blocks for <math>p=3</math>==
 
==Blocks for <math>p=3</math>==

Latest revision as of 10:19, 19 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.

Blocks for [math] p=2 [/math]

The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [Sa14].

Fusion trivial groups of order 64 were determined by Pete Gautam.


Blocks for [math]p=3[/math]

Blocks for [math]p=5[/math]

Blocks for [math]p\geq 7[/math]

Retrieved from ‘http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&oldid=1257’