Difference between revisions of "Classification by p-group"
(Added order 64 table.) |
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! scope="col"| SmallGroup | ! scope="col"| SmallGroup | ||
! scope="col"| Isotype | ! scope="col"| Isotype | ||
| + | ! scope="col"| Donovan (w.r.t.)? | ||
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes | ! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes | ||
! scope="col"| Complete (w.r.t.)? | ! scope="col"| Complete (w.r.t.)? | ||
| Line 19: | Line 20: | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
|- | |- | ||
| − | | 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | + | | 1 || 1 || <math>1</math> || <math>\mathcal{O}</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> || || | + | | 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || <math>\mathcal{O}</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 || [[C4|1]] || [[C4|<math>C_4</math>]] || <math>\mathcal{O}</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] ]] || | + | | 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || <math>\mathcal{O}</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 || [[C8|1]] || [[C8|<math>C_8</math>]] || <math>\mathcal{O}</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 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] || <math>\mathcal{O}</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>]] || <math>\mathcal{O}</math> ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. 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>]] || <math>\mathcal{O}</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 | + | |8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG |
|} | |} | ||
| Line 63: | Line 64: | ||
|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> | + | |16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>\mathcal{O}</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
| − | |16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || | + | |16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes. 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 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || | + | |16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || <math>\mathcal{O}</math> || 6(?) || || <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]]] |
|- | |- | ||
|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 74: | ||
|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>]] || | + | |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 81: | ||
|} | |} | ||
| − | The | + | 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" | ||
| Line 107: | Line 108: | ||
|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 || [[ | + | |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 115: | Line 116: | ||
|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 130: | ||
|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> | + | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>\mathcal{O}</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. 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> || || || || || 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 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || | + | |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]]] | ||
| Line 146: | Line 147: | ||
Invariants known | Invariants known | ||
|- | |- | ||
| − | |32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || No || || || | | + | |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 185: | ||
|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# | + | |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 199: | Line 200: | ||
|} | |} | ||
| + | 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> </strong> | | <strong><math>|D|=64</math> </strong> | ||
| Line 213: | Line 214: | ||
! scope="col"| References | ! scope="col"| References | ||
! scope="col"| Notes | ! scope="col"| Notes | ||
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | |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 || [[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 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || || | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 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)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || No || || || || || | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || || | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 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> || || || Fusion trivial | ||
| + | |- | ||
| + | | 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 || [[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 || [[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 || [[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(64,86)|86]] || [[SmallGroup(64,86)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || 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 || || || || || | ||
| + | |- | ||
| + | | 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 || [[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(64,91)|91]] || [[SmallGroup(64,91)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || <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 || [[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)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 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>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || 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 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,108)|108]] || [[SmallGroup(64,108)]] || <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>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || <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> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || <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 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || || | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || <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 || [[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 || || || || || | ||
| + | |- | ||
| + | | 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>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 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>]] || No || || || || || Invariants known by [[References#S|[Sa14,9.28]]] | ||
| + | |- | ||
| + | | 64 || [[SmallGroup(64,127)|127]] || [[SmallGroup(64,127)]] || <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 || [[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 || 131 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 132 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 133 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 134 || <math>\operatorname{Syl}_2(M_{12})</math> || No || || || || || | ||
| + | |- | ||
| + | | 64 || 135 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 136 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 137 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 138 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 139 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 140 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 141 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 142 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 143 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 144 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 145 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 146 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 147 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 148 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 149 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 150 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 151 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 152 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 153 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 154 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 155 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 156 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 157 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 158 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 159 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 160 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 161 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 162 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 163 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 164 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 165 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 166 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 167 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 168 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 169 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 170 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 171 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 172 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 173 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 174 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 175 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 176 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 177 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 178 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 179 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 180 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 181 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 182 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 183 || <math>C_{16} \times C_2^2</math> || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || || [[References#E|[EL18a]]] || | ||
| + | |- | ||
| + | | 64 || 184 || || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || || || Fusion trivial | ||
| + | |- | ||
| + | | 64 || 185 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 186 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 187 || || No || || || || || | ||
| + | |- | ||
| + | | 64 || 188 || || No || || || || || | ||
| + | |- | ||
| + | | 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 || | + | | 64 || 266 || <math>2_+^{1+4} * C_4</math> || No || || || || || |
|- | |- | ||
| − | |64 || | + | | 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>== | ||
| Line 262: | Line 787: | ||
{| role="presentation" class="wikitable mw-collapsible mw-collapsed" | {| role="presentation" class="wikitable mw-collapsible mw-collapsed" | ||
| − | | <strong><math>5 \leq |D| \leq | + | | <strong><math>5 \leq |D| \leq 125</math> </strong> |
|- | |- | ||
! scope="col"| <math>|D|</math> | ! scope="col"| <math>|D|</math> | ||
| Line 279: | Line 804: | ||
|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 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.
Contents
Blocks for [math] p=2 [/math]
| [math]1 \leq |D| \leq 8[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t.)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | [math]1[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 2 | 1 | [math]C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 4 | 1 | [math]C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 4 | 2 | [math]C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Er82], [Li94] | |
| 8 | 1 | [math]C_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 8 | 2 | [math]C_4 \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 8 | 3 | [math]D_8[/math] | [math]\mathcal{O}[/math] | 6(?) | [math]k[/math] | [math]k[/math] | [Er87] | See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 8 | 4 | [math]Q_8[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]k[/math] | [Er88a], [Er88b], [HKL07], [Ei16] | |
| 8 | 5 | [math]C_2 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 8(8) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Ea16] | Uses CFSG |
| [math]|D|=16[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|---|
| 16 | 1 | [math]C_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 16 | 2 | [math]C_4 \times C_4[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKKS14] | |
| 16 | 3 | MNA(2,1) | No | 3(?) | No | [Sa11] | Block invariants known | |
| 16 | 4 | [math]C_4:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 16 | 5 | [math]C_8 \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 16 | 6 | [math]M_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 16 | 7 | [math]D_{16}[/math] | [math]\mathcal{O}[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 16 | 8 | [math]SD_{16}[/math] | [math]k[/math] | 7(?) | [Er88c], [Er90b] | Two other possible classes. Donovan's conjecture known over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/math] by [La16] (see discussion in [EEKLS26]). | ||
| 16 | 9 | [math]Q_{16}[/math] | [math]\mathcal{O}[/math] | 6(?) | [math]k[/math] | [Er88a], [Er88b], [EEKLS26], [Ho97] | Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [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 [EEKLS26]. Principal blocks classified up to source algebra equivalence in [KoLa20b] | |
| 16 | 10 | [math]C_4 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL18a] | |
| 16 | 11 | [math]D_8 \times C_2[/math] | No | [Sa12] | Block invariants known | |||
| 16 | 12 | [math]Q_8 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | No | [EL20] | Block invariants known by [Sa13] | |
| 16 | 13 | [math]D_8*C_4[/math] | No | 3(?) | No | [Sa13b] | Block invariants known | |
| 16 | 14 | [math](C_2)^4[/math] | [math]\mathcal{O}[/math] | 16(16) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Ea18] |
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [Sa14].
| [math]|D|=32[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|---|
| 32 | 1 | [math]C_{32}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 32 | 2 | [math]MNA(2,2)[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKS12] | |
| 32 | 3 | [math]C_8 \times C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 32 | 4 | [math]C_8:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 5 | [math]MNA(3,1)[/math] | No | [Sa11] | Invariants known | |||
| 32 | 6 | [math]MNA(2,1):C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 7 | SmallGroup(32,7) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | [math]M_{16}:C_2[/math] |
| 32 | 8 | [math]2.MNA(2,1)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 9 | [math]D_8:C_4[/math] | No | [Sa14,10.23] | Invariants known | |||
| 32 | 10 | [math]Q_8:C_4[/math] | No | [Sa14,10.25] | Invariants known | |||
| 32 | 11 | [math]C_4 \wr C_2[/math] | No | 6(6) | No | [Ku80], [KoLaSa23] | Invariants known. Principal blocks classified up to source algebra equivalence in [KoLaSa23] | |
| 32 | 12 | [math]C_4:C_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 13 | [math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 14 | [math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 15 | SmallGroup(32,15) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 16 | [math]C_{16} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 32 | 17 | [math]M_{32}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
| 32 | 18 | [math]D_{32}[/math] | [math]\mathcal{O}[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | See discussion in [EEKLS26] regarding Donovan over [math]\mathcal{O}[/math]. Principal blocks classified up to source algebra equivalence in [KoLa20] |
| 32 | 19 | [math]SD_{32}[/math] | [math]k[/math] | Donovan's conjecture known over [math]\mathcal{O}[/math] for [math]l(B) \neq 2[/math] by [La16] (see discussion in [EEKLS26]). | ||||
| 32 | 20 | [math]Q_{32}[/math] | [math]\mathcal{O}[/math] | [math]k[/math] | [Er88a], [Er88b], [EEKLS26], [Ho97] | Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [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 [EEKLS26]. Principal blocks classified up to source algebra equivalence in [KoLa20b] | ||
| 32 | 21 | [math]C_4 \times C_4 \times C_2[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKKS14] | |
| 32 | 22 | [math]MNA(2,1) \times C_2[/math] | No | [Sa14,10.25] | Invariants known | |||
| 32 | 23 | [math](C_4:C_4) \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 24 | SmallGroup(32,24) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 25 | [math]D_8 \times C_4[/math] | No | [Sa14,9.7] |
Invariants known | |||
| 32 | 26 | [math]Q_8 \times C_4[/math] | [math]\mathcal{O}[/math] | 3(3) | No | [EL20] | Invariants known by [Sa14,9.28] | |
| 32 | 27 | SmallGroup(32,27) | No | |||||
| 32 | 28 | SmallGroup(32,28) | No | [Sa14,13.11] | Invariants known | |||
| 32 | 29 | SmallGroup(32,29) | No | [Sa14,13.11] | Invariants known | |||
| 32 | 30 | SmallGroup(32,30) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 31 | SmallGroup(32,31) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 32 | SmallGroup(32,32) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 33 | SmallGroup(32,33) | No | [Sa14,13.12] | Invariants partly known | |||
| 32 | 34 | SmallGroup(32,34) | No | [Sa14,13.12] | Invariants partly known | |||
| 32 | 35 | [math]C_4:Q_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 36 | [math]C_8 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL18a] | |
| 32 | 37 | [math]M_{16} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
| 32 | 38 | [math]D_8*C_8[/math] | No | [Sa14,9.18] | Invariants known | |||
| 32 | 39 | [math]D_{16} \times C_2[/math] | No | [Sa14,9.7] | Invariants known | |||
| 32 | 40 | [math]SD_{16} \times C_2[/math] | No | [Sa14,9.37] | Invariants known | |||
| 32 | 41 | [math]Q_{16} \times C_2[/math] | No | [Sa14,9.28] | Invariants known | |||
| 32 | 42 | [math]D_{16}*C_4[/math] | No | [Sa14,9.18] | Invariants known | |||
| 32 | 43 | SmallGroup(32,43) | No | |||||
| 32 | 44 | SmallGroup(32,44) | No | |||||
| 32 | 45 | [math]C_4 \times C_2 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 8(8) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL23] | |
| 32 | 46 | [math]D_8 \times C_2 \times C_2[/math] | No | |||||
| 32 | 47 | [math]Q_8 \times C_2 \times C_2[/math] | No | |||||
| 32 | 48 | [math](D_8*C_4) \times C_2[/math] | No | |||||
| 32 | 49 | [math]D_8*D_8[/math] | No | [Sa13c] | Invariants partly known | |||
| 32 | 50 | [math]D_8*Q_8[/math] | No | [Sa13c] | Invariants partly known | |||
| 32 | 51 | [math](C_2)^5[/math] | [math]\mathcal{O}[/math] | 34 (34) | [math]\mathcal{O}[/math] | [Ar19] | Derived eq. classes determined for 30 of the 34 Morita eq. classes. |
Fusion trivial groups of order 64 were determined by Pete Gautam.
| [math]|D|=64[/math] | ||||||||
| [math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|---|
| 64 | 1 | [math]C_{64}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 2 | [math]C_8 \times C_8[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [EKKS14] | ||
| 64 | 3 | SmallGroup(64,3) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 4 | [math](C_2)^3:C_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 5 | SmallGroup(64,5) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]C_8:C_8[/math]. Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 6 | [math]D_8:C_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 7 | [math]Q_8:C_8[/math] | No | |||||
| 64 | 8 | SmallGroup(64,8) | No | |||||
| 64 | 9 | SmallGroup(64,9) | No | |||||
| 64 | 10 | SmallGroup(64,10) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 11 | SmallGroup(64,11) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 12 | SmallGroup(64,12) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 13 | SmallGroup(64,13) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 14 | SmallGroup(64,14) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 15 | SmallGroup(64,15) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]C_8:C_8[/math]. Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 16 | SmallGroup(64,16) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]C_8:C_8[/math]. Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 17 | SmallGroup(64,17) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 18 | SmallGroup(64,18) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 19 | SmallGroup(64,19) | [math]\mathcal{O}[/math] | Maxperm (as in the supplied data); no class count or completeness assertion supplied | ||||
| 64 | 20 | SmallGroup(64,20) | No | |||||
| 64 | 21 | SmallGroup(64,21) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 22 | SmallGroup(64,22) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 23 | SmallGroup(64,23) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 24 | SmallGroup(64,24) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]M_{16}:C_4[/math]. Fusion trivial | ||
| 64 | 25 | SmallGroup(64,25) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]M_{16}:C_4[/math]. Fusion trivial | ||
| 64 | 26 | [math]C_{16} \times C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 27 | SmallGroup(64,27) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]C_{16}:C_4[/math]. Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 28 | SmallGroup(64,28) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]C_{16}:C_4[/math]. Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 29 | [math](C_2)^2:C_{16}[/math] | No | |||||
| 64 | 30 | SmallGroup(64,30) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]M_{32}:C_2[/math]. Fusion trivial | ||
| 64 | 31 | SmallGroup(64,31) | No | [math]M_{32}:C_2[/math] | ||||
| 64 | 32 | [math]C_2 \wr C_4[/math] | No | |||||
| 64 | 33 | SmallGroup(64,33) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 34 | SmallGroup(64,34) | No | |||||
| 64 | 35 | SmallGroup(64,35) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 36 | SmallGroup(64,36) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 37 | SmallGroup(64,37) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 38 | SmallGroup(64,38) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 39 | SmallGroup(64,39) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 40 | SmallGroup(64,40) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 41 | SmallGroup(64,41) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 42 | SmallGroup(64,42) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 43 | SmallGroup(64,43) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 44 | [math]C_4:C_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 45 | SmallGroup(64,45) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 46 | SmallGroup(64,46) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 47 | SmallGroup(64,47) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 48 | SmallGroup(64,48) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]C_{16}:C_4[/math]. Fusion trivial by [CG12], [Sa12b] | ||
| 64 | 49 | SmallGroup(64,49) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 50 | [math]C_{32} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 51 | [math]M_6(2)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 52 | [math]D_{64}[/math] | No | |||||
| 64 | 53 | [math]SD_{64}[/math] | No | |||||
| 64 | 54 | [math]Q_{64}[/math] | No | |||||
| 64 | 55 | [math]C_4 \times C_4 \times C_4[/math] | [math]\mathcal{O}[/math] | 4(4) | [math]\mathcal{O}[/math] | [WZZ18], [EL18a] | ||
| 64 | 56 | SmallGroup(64,56) | No | |||||
| 64 | 57 | SmallGroup(64,57) | No | |||||
| 64 | 58 | [math]C_{4} \times (C_2 \times C_2):C_4[/math] | No | |||||
| 64 | 59 | [math]C_{4} \times (C_4:C_4)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 60 | SmallGroup(64,60) | No | |||||
| 64 | 61 | SmallGroup(64,61) | No | |||||
| 64 | 62 | SmallGroup(64,62) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 63 | SmallGroup(64,63) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 64 | SmallGroup(64,64) | No | |||||
| 64 | 65 | SmallGroup(64,65) | No | |||||
| 64 | 66 | SmallGroup(64,66) | No | |||||
| 64 | 67 | SmallGroup(64,67) | No | |||||
| 64 | 68 | SmallGroup(64,68) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 69 | SmallGroup(64,69) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 70 | SmallGroup(64,70) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 71 | SmallGroup(64,71) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 72 | SmallGroup(64,72) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 73 | SmallGroup(64,73) | No | |||||
| 64 | 74 | [math](C_2)^3:Q_8[/math] | No | |||||
| 64 | 75 | SmallGroup(64,75) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 76 | SmallGroup(64,76) | No | |||||
| 64 | 77 | SmallGroup(64,77) | No | |||||
| 64 | 78 | SmallGroup(64,78) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 79 | SmallGroup(64,79) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 80 | SmallGroup(64,80) | No | |||||
| 64 | 81 | SmallGroup(64,81) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 82 | SmallGroup(64,82) | [math]\mathcal{O}[/math] | 6(6) | [math]\mathcal{O}[/math] | [Ea24] | Suzuki [math]2[/math]-group of type A | |
| 64 | 83 | [math]C_{8} \times C_4 \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 84 | [math]C_{2} \times (C_8:C_4)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 85 | [math]M_4(2) \times C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 86 | SmallGroup(64,86) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 87 | [math]C_{2} \times (C_2 \times C_2):C_8[/math] | No | |||||
| 64 | 88 | SmallGroup(64,88) | No | |||||
| 64 | 89 | [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 | 90 | [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 | 91 | SmallGroup(64,91) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 92 | SmallGroup(64,92) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 93 | SmallGroup(64,93) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 94 | SmallGroup(64,94) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 95 | [math]C_{2} \times (D_8:C_4)[/math] | No | |||||
| 64 | 96 | [math]C_{2} \times (Q_8:C_4)[/math] | No | |||||
| 64 | 97 | SmallGroup(64,97) | No | |||||
| 64 | 98 | SmallGroup(64,98) | No | |||||
| 64 | 99 | SmallGroup(64,99) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 100 | SmallGroup(64,100) | No | |||||
| 64 | 101 | [math]C_{2} \times (C_4 \wr C_2)[/math] | No | |||||
| 64 | 102 | [math](C_4 \times C_4):(C_2 \times C_2)[/math] | No | |||||
| 64 | 103 | [math]C_{2} \times (C_4:C_8)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 104 | [math]C_{4}:M_4(2)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 105 | SmallGroup(64,105) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 106 | SmallGroup(64,106) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 107 | SmallGroup(64,107) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 108 | SmallGroup(64,108) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 109 | [math]M_4(2):C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 110 | SmallGroup(64,110) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 111 | SmallGroup(64,111) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 112 | SmallGroup(64,112) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 113 | SmallGroup(64,113) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 114 | SmallGroup(64,114) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 115 | [math]D_8 \times C_8[/math] | No | |||||
| 64 | 116 | SmallGroup(64,116) | No | |||||
| 64 | 117 | SmallGroup(64,117) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 118 | [math]D_{16} \times C_4[/math] | No | |||||
| 64 | 119 | [math]SD_{16} \times C_4[/math] | No | |||||
| 64 | 120 | [math]Q_{16} \times C_4[/math] | No | |||||
| 64 | 121 | [math]SD_{16}:C_4[/math] | No | |||||
| 64 | 122 | [math]Q_{16}:C_4[/math] | No | |||||
| 64 | 123 | [math]D_{16}:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 124 | SmallGroup(64,124) | No | |||||
| 64 | 125 | SmallGroup(64,125) | No | |||||
| 64 | 126 | [math]Q_{8} \times C_8[/math] | No | Invariants known by [Sa14,9.28] | ||||
| 64 | 127 | SmallGroup(64,127) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | ||
| 64 | 128 | [math](C_2 \times C_2):D_{16}[/math] | No | |||||
| 64 | 129 | [math]Q_8:D_{8}[/math] | No | |||||
| 64 | 130 | [math]D_8:D_{8}[/math] | No | |||||
| 64 | 131 | No | ||||||
| 64 | 132 | No | ||||||
| 64 | 133 | No | ||||||
| 64 | 134 | [math]\operatorname{Syl}_2(M_{12})[/math] | No | |||||
| 64 | 135 | No | ||||||
| 64 | 136 | No | ||||||
| 64 | 137 | No | ||||||
| 64 | 138 | No | ||||||
| 64 | 139 | No | ||||||
| 64 | 140 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 141 | No | ||||||
| 64 | 142 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 143 | No | ||||||
| 64 | 144 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 145 | No | ||||||
| 64 | 146 | No | ||||||
| 64 | 147 | No | ||||||
| 64 | 148 | No | ||||||
| 64 | 149 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 150 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 151 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 152 | No | ||||||
| 64 | 153 | No | ||||||
| 64 | 154 | No | ||||||
| 64 | 155 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 156 | No | ||||||
| 64 | 157 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 158 | No | ||||||
| 64 | 159 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 160 | No | ||||||
| 64 | 161 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 162 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 163 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 164 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 165 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 166 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 167 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 168 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 169 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 170 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 171 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 172 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 173 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 174 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 175 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 176 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 177 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 178 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 179 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 180 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 181 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 182 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 183 | [math]C_{16} \times C_2^2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [EL18a] | ||
| 64 | 184 | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | Fusion trivial | |||
| 64 | 185 | No | ||||||
| 64 | 186 | No | ||||||
| 64 | 187 | No | ||||||
| 64 | 188 | No | ||||||
| 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] | [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 | 239 | [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 | 245 | SmallGroup(64,245) | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [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] | [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 | [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 | [Ar21], [EEL18] | 81 known [math]\mathcal{O}[/math]-classes containing a principal block; full classification not known |
Blocks for [math]p=3[/math]
| [math]1 \leq |D| \leq 27[/math] | |||||||
| [math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|
| 1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 3 | 1 | [math]C_3[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 9 | 1 | [math]C_9[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 9 | 2 | [math]C_3 \times C_3[/math] | |||||
| 27 | 1 | [math]C_{27}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 27 | 2 | [math]C_9 \times C_3[/math] | |||||
| 27 | 3 | [math]3_+^{1+2}[/math] | |||||
| 27 | 4 | [math]3_-^{1+2}[/math] | |||||
| 27 | 5 | [math]C_3 \times C_3 \times C_3[/math] |
Blocks for [math]p=5[/math]
| [math]5 \leq |D| \leq 125[/math] | |||||||
| [math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|
| 1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 5 | 1 | [math]C_5[/math] | 6(6) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 25 | 1 | [math]C_{25}[/math] | 6(6) | No | [math]\mathcal{O}[/math] | Max 12 classes | |
| 25 | 2 | [math]C_5 \times C_5[/math] | |||||
| 125 | 1 | [math]C_{125}[/math] | |||||
| 125 | 2 | [math]C_{25} \times C_5[/math] | |||||
| 125 | 3 | [math]5_+^{1+2}[/math] | 62(62) | [math]\mathcal{O}[/math] | [AE23] | Inertial quotients are consistent within classes | |
| 125 | 4 | [math]5_-^{1+2}[/math] | |||||
| 125 | 5 | [math]C_5 \times C_5 \times C_5[/math] |
Blocks for [math]p\geq 7[/math]
| [math]|D|[/math] | |||||||
| [math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
|---|---|---|---|---|---|---|---|
| 1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
| 7 | 1 | [math]C_7[/math] | 14(14) | No | [math]\mathcal{O}[/math] | Max 21 classes | |
| 11 | 1 | [math]C_{11}[/math] | No | [math]\mathcal{O}[/math] | |||
| 13 | 1 | [math]C_{13}[/math] | No | [math]\mathcal{O}[/math] | |||
| 17 | 1 | [math]C_{17}[/math] | No | [math]\mathcal{O}[/math] | |||
| 19 | 1 | [math]C_{19}[/math] | No | [math]\mathcal{O}[/math] | |||
| 23 | 1 | [math]C_{23}[/math] | No | [math]\mathcal{O}[/math] |
