Classification by p-group - Revision history

Pete Gautam: /* Blocks for p=2 */

2025-09-09T11:50:13Z

‎Blocks for p=2

Charles Eaton: /* Blocks for p=2 */

2024-01-10T18:59:52Z

‎Blocks for p=2

Charles Eaton: /* Blocks for p=2 */

2024-01-10T14:31:50Z

‎Blocks for p=2

Charles Eaton at 18:49, 9 January 2024

2024-01-09T18:49:39Z

Charles Eaton: /* Blocks for p=2 */

2024-01-09T17:36:39Z

‎Blocks for p=2

Charles Eaton: /* Blocks for p=2 */ SmallGroup(32,11), SmallGroup(32,45)

2023-10-24T08:58:13Z

‎Blocks for p=2: SmallGroup(32,11), SmallGroup(32,45)

Charles Eaton: /* Blocks for p=5 */

2023-10-03T09:46:22Z

‎Blocks for p=5

Charles Eaton: /* Blocks for p=5 */

2023-10-03T08:57:05Z

‎Blocks for p=5

Charles Eaton: /* Blocks for p=5 */

2023-10-03T08:56:07Z

‎Blocks for p=5

Charles Eaton: /* Blocks for p=2 */ Updated number of classes for SD_{16}

2022-08-04T13:21:23Z

‎Blocks for p=2: Updated number of classes for SD_{16}

@@ Line 219: / Line 219: @@
 |64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math>
 |-
-|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||
+|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||
+|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || || FT
 |-
-|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||
+|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || || FT
 |-
 |64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||
@@ Line 231: / Line 231: @@
 |64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||
 |-
-|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||
+|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||
+|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||
+|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||
+|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||
+|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || || FT
 |-
 |64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math>
@@ Line 245: / Line 245: @@
 |64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math>
 |-
-|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||
+|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math>
+|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math> FT
 |-
-|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||
+|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || || Maxperm
 |-
 |64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math>
 |-
-|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||
+|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||
+|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||
+|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || || FT
 |-
-|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math>
+|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math> FT
 |-
-|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math>
+|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math> FT
 |-
-|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || FT
 |-
-|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math>
+|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math> FT
 |-
-|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math>
+|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math> FT
 |-
 |64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||
 |-
-|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math>
+|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math> FT
 |-
 |64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math>
 |-
 |64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||
 |-
-|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || ||
+|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || FT
 |-
 |64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || ||  <math>(C_4 \times C_4):C_4</math>

@@ Line 441: / Line 441: @@
 |64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || No || || || || ||
 |-
-|64 || [[D8xC8|115]] || [[D8xC8|<math>(D_8 \times C_8</math>]]|| No || || || || ||
+|64 || [[D8xC8|115]] || [[D8xC8|<math>D_8 \times C_8</math>]]|| No || || || || ||
 |-
 |64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || ||
@@ Line 447: / Line 447: @@
 |64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || No || || || || ||
 |-
-|64 || [[D16xC4|118]] || [[D16xC4|<math>(D_{16} \times C_4</math>]]|| 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 || [[SD16xC4|119]] || [[SD16xC4|<math>SD_{16} \times C_4</math>]]|| No || || || || ||
 |-
-|64 || [[Q16xC4|120]] || [[Q16xC4|<math>(Q_{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 || [[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 || [[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 || [[D16:C4|123]] || [[D16:C4|<math>D_{16}:C_4</math>]]|| No || || || || ||
 |-
 |64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || ||
@@ Line 463: / Line 463: @@
 |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 || [[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 || || || || ||
@@ Line 472: / Line 472: @@
 |-
 |64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]]|| No || || || || ||
 |-
 |64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math>

@@ Line 424: / Line 424: @@
 |-
 |64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||
 |-
 |64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math>
 |}-->

@@ Line 337: / Line 337: @@
 |64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||
 |-
-|64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || 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 || || || || ||

@@ Line 292: / Line 292: @@
 |-
 |64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||
 |-
 |64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
 |-
 |64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group
-|}
+|-
--->
+|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>
 ==Blocks for <math>p=3</math>==

@@ 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 || [[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]]] ||
@@ Line 184: / Line 184: @@
 |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 || || || || ||

@@ Line 358: / Line 358: @@
 |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]]] ||
+|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>]] || || || || ||

@@ Line 335: / Line 335: @@
 {| 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>
@@ Line 352: / Line 352: @@
 |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>]] || ||  || || ||
 |}

@@ Line 65: / Line 65: @@
 |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]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]