Difference between revisions of "Classification by p-group"

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(→‎Blocks for p=2: Updated number of classes for SD_{16})
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|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||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] ]] ||  
+
|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 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] ||  
Line 63: Line 63:
 
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] ||  
 
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] ||  
 
|-
 
|-
−
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] ||  
+
|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> || 8(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes
+
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes
 
|-
 
|-
−
|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]]]
+
|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 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||
Line 73: Line 73:
 
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known
 
|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>]] || No || || || || [[References|[Sa13] ]] || 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 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known
Line 80: Line 80:
 
|}
 
|}
  
−
The following takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].
+
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"
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|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 || [[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 || [[M16:C2|7]] || [[M16:C2|<math>M_{16}:C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#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 || [[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]]] ||
Line 129: Line 129:
 
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||
 
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||
 
|-
 
|-
−
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] ||
+
|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 || [[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]]]
+
|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 || [[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]]]  
Line 146: Line 146:
 
Invariants known
 
Invariants known
 
|-
 
|-
−
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || 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(32,27)|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 || || || || ||
Line 198: Line 198:
 
|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.  
 
|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.  
 
|}
 
|}
−
 
  
 
<!--
 
<!--
Line 216: Line 215:
 
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 
|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(12 || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||
+
|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 || [[C8:C8|3]] || [[C8:C8|<math>C_8:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||
+
|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 || [[SmallGroup(64,4)|4]] || [[SmallGroup(64,4)]] || No || || || || ||
+
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||
 
|-
 
|-
 
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || 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 || || || || ||
 +
|-
 +
|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(64,79)|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(64,81)|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
 
|}
 
|}
 
-->
 
-->

Revision as of 14:21, 4 August 2022

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].


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=1193’