Difference between revisions of "Classification by p-group"

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Line 31: Line 31:
 
|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 115: Line 115:
 
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || 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 || || || || [[References#K|[Ku80]]] || 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 || [[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]]] ||
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 184: Line 184:
 
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || 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> || || || || [[References#S|[Sa14, 13.9]]] || Invariants known
+
|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 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||
Line 292: Line 292:
 
|-
 
|-
 
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||
 
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||
−
|}
+
|-
−
-->
+
|64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_4:C_{16}</math>
 +
|-
 +
|64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math>
 +
|-
 +
|64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math>
 +
|-
 +
|64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || ||
 +
|-
 +
|64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 +
|-
 +
|64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || No || || || || ||
 +
|-
 +
|64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || ||
 +
|-
 +
|64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || ||
 +
|-
 +
|64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || ||
 +
|-
 +
|64 || [[C4xC4xC4|55]] || [[C4xC4xC4|<math>C_4 \times C_4 \times C_4</math>]] || <math>\mathcal{O}</math> ||4(4) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]]||
 +
|-
 +
|64 || [[SmallGroup(64,56)|56]] || [[SmallGroup(64,56)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || ||
 +
|-
 +
|64 || [[C4x(C2xC2):C4|58]] || [[C4x(C2xC2):C4|<math>C_{4} \times (C_2 \times C_2):C_4</math>]] || || || || || || Fusion trivial?
 +
|-
 +
|64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,60)|60]] || [[SmallGroup(64,60)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||
 +
|-
 +
|64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>Sz(8)</math>
 +
|-
 +
|64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 +
|-
 +
|64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || ||
 +
|-
 +
|64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[SmallGroup(64,108)|108]] || [[SmallGroup(64,108)]] || No || || || || ||
 +
|-
 +
|64 || [[M4(2):C4|109]] || [[M4(2):C4|<math>M_4(2):C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,112)|112]] || [[SmallGroup(64,112)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 +
|-
 +
|64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || No || || || || ||
 +
|-
 +
|64 || [[D8xC8|115]] || [[D8xC8|<math>D_8 \times C_8</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || No || || || || ||
 +
|-
 +
|64 || [[D16xC4|118]] || [[D16xC4|<math>D_{16} \times C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SD16xC4|119]] || [[SD16xC4|<math>SD_{16} \times C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[Q16xC4|120]] || [[Q16xC4|<math>Q_{16} \times C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SD16:C4|121]] || [[SD16:C4|<math>SD_{16}:C_4</math>]]|| No || || || || || Fusion trivial?
 +
|-
 +
|64 || [[Q16:C4|122]] || [[Q16:C4|<math>Q_{16}:C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[D16:C4|123]] || [[D16:C4|<math>D_{16}:C_4</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || ||
 +
|-
 +
|64 || [[SmallGroup(64,125)|125]] || [[SmallGroup(64,125)]] || No || || || || ||
 +
|-
 +
|64 || [[Q8xC8|126]] || [[Q8xC8|<math>Q_{8} \times C_8</math>]]|| <math>\mathcal{O}</math> || 3(3) || || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]
 +
|-
 +
|64 || [[SmallGroup(64,127)|127]] || [[SmallGroup(64,127)]] || No || || || || ||
 +
|-
 +
|64 || [[(C2xC2):D16|128]] || [[(C2xC2)D16|<math>(C_2 \times C_2):D_{16}</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[Q8:D8|129]] || [[Q8:D8|<math>Q_8:D_{8}</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]]|| No || || || || ||
 +
|-
 +
|64 || [[Q8xQ8|239]] || [[Q8xQ8|<math>Q_{8} \times Q_8</math>]]|| <math>\mathcal{O}</math> || || || || [[References#E|[EL20]]] ||
 +
|-
 +
|64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math>
 +
 
 +
|}-->
  
 
==Blocks for <math>p=3</math>==
 
==Blocks for <math>p=3</math>==
Line 331: Line 515:
  
 
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
 
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
−
| <strong><math>5 \leq |D| \leq 25</math> &nbsp;</strong>
+
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong>
 
|-
 
|-
 
! scope="col"| <math>|D|</math>
 
! scope="col"| <math>|D|</math>
Line 348: Line 532:
 
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes  
 
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes  
 
|-
 
|-
−
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||
+
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || ||  || || ||
 +
|-
 +
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || ||
 +
|-
 +
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || ||
 +
|-
 +
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes
 +
|-
 +
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||
 +
|-
 +
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||
 
|}
 
|}
  

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’