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. Information on broad classes of p-groups can be found [[p-group classes|here]].
+
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]].
  
−
We use the following notation for Morita equivalence classes of blocks of finite groups with respect to an algebraically closed field k.  
+
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.
  
−
<math>M(x,y,z)</math> is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.
+
== Blocks for <math> p=2 </math> ==
  
−
Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that <math>M(x,y1,z1)=M(x,y2,z2)</math> for some <math>(y1,z1) \neq (y2,z2)</math>.
+
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong>
 +
|-
 +
! scope="col"| <math>|D|</math>
 +
! scope="col"| SmallGroup
 +
! scope="col"| Isotype
 +
! 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
 +
|-
 +
| 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
 +
|}
  
−
Also, at present there is no known example of a k-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to the d.v.r.
+
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong><math>|D|=16</math> &nbsp;</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"
 +
| <strong><math>|D|=32</math> &nbsp;</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
 +
|-
 +
|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]]] ||
 +
|-
 +
|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.  
 +
|}
  
−
{| class="wikitable"
+
<!--
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong><math>|D|=64</math> &nbsp;</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
 +
|-
 +
|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 || || || || ||
 +
|-
 +
|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>==
 +
 
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong>
 
|-
 
|-
 
! scope="col"| <math>|D|</math>
 
! scope="col"| <math>|D|</math>
Line 24: Line 495:
 
| 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> || ||
+
| 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 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||
 +
|-
 +
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 +
|-
 +
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||
 +
|-
 +
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||
 +
|-
 +
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||
 +
|-
 +
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||
 +
|}
 +
 
 +
==Blocks for <math>p=5</math>==
 +
 
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong>
 +
|-
 +
! scope="col"| <math>|D|</math>
 +
! scope="col"| SmallGroup
 +
! scope="col"| Isotype
 +
! 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
 
|-  
 
|-  
−
| 4 || [[C4|1]] || [[C4|<math>C_4</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> || ||  
 
|-  
 
|-  
−
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <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 ||1 ||<math>C_5</math> ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+
|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>]] || ||  || || ||
 
|-  
 
|-  
−
|7 ||1 ||<math>C_7</math> ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 19 classes
+
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || ||  
 
|-
 
|-
−
|8 ||1 ||<math>C_8</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || ||  
 
|-
 
|-
−
|8 ||2 ||<math>C_4 \times C_2</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</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
 
|-
 
|-
−
|8 ||3 ||<math>D_8</math> ||4(?) || <math>k</math> || || ||
+
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||
 
|-
 
|-
−
|8 ||4 ||<math>Q_8</math> ||3(?) || <math>k</math> || || ||
+
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||
 +
|}
 +
 
 +
==Blocks for <math>p\geq 7</math>==
 +
 
 +
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 +
| <strong><math>|D|</math> &nbsp;</strong>
 
|-
 
|-
−
|8 ||5 ||<math>C_2 \times C_2 \times C_2</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Uses CFSG
+
! scope="col"| <math>|D|</math>
 +
! scope="col"| SmallGroup
 +
! scope="col"| Isotype
 +
! 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
 +
|-
 +
| 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
 +
|-
 +
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 +
|-
 +
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
|-
 
|-
−
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||  
+
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 +
|-
 +
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
|-
 
|-
−
|9 ||2 ||<math>C_3 \times C_3</math> || || || || ||  
+
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||
 
|}
 
|}

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.

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=1240’