(C2)^4 - Revision history

CesareGArdito: Removed possibly incorrect Picard groups

2019-12-09T17:39:54Z

Removed possibly incorrect Picard groups

CesareGArdito: /* Blocks with defect group (C_2)^4 */

2019-12-09T17:38:12Z

‎Blocks with defect group (C_2)^4

CesareGArdito: /* Blocks with defect group (C_2)^4 */

2019-11-28T14:35:44Z

‎Blocks with defect group (C_2)^4

Elliotmckernon at 16:16, 22 August 2019

2019-08-22T16:16:04Z

Charles Eaton at 16:57, 18 December 2018

2018-12-18T16:57:33Z

CesareGArdito at 15:22, 3 December 2018

2018-12-03T15:22:49Z

Charles Eaton: Added some O-Picard groups

2018-10-26T07:03:35Z

Added some O-Picard groups

Charles Eaton: Switched M(16,14,2) and M(16,14,3) to match C2xC2 and C3xC3xC3 labellings. Changed reference for k(B), l(B)

2018-10-04T09:31:14Z

Switched M(16,14,2) and M(16,14,3) to match C2xC2 and C3xC3xC3 labellings. Changed reference for k(B), l(B)

CesareGArdito at 11:50, 3 October 2018

2018-10-03T11:50:06Z

CesareGArdito: /* Blocks with defect group (C_2)^4 */

2018-10-03T11:49:44Z

‎Blocks with defect group (C_2)^4

@@ Line 26: / Line 26: @@
 |[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times S_3</math> || ||1 ||1 ||
 |-
-|[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_3</math> || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math>
+|[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math>
 |-
-|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || <math>C_5</math> || ||1 ||1 || The action comes from the 3rd power of a Singer cycle for <math>\mathbb{F}_{16}</math>
+|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || || ||1 ||1 || The action comes from the 3rd power of a Singer cycle for <math>\mathbb{F}_{16}</math>
 |-
 |[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||

@@ Line 40: / Line 40: @@
 |[[M(16,14,10)]] || <math>B_0(k(A_5 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||
 |-
-|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}</math> || ||1 ||1 ||
+|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 ||
 |-
-|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 ||
+|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_4</math> || ||1 ||1 ||
 |-
 |[[M(16,14,13)]] || <math>k(C_2 \times ((C_2)^3:(C_7:C_3)))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||

@@ Line 50: / Line 50: @@
 |[[M(16,14,15)]] || <math>B_0(k(C_2 \times{\rm Aut}(SL_2(8))))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||
 |-
-|[[M(16,14,16)]] || <math>b(k((C_2)^4 : 3^{1+2}_{+}))</math> || 1 ||8 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal faithful block. Cannot be Morita equivalent to a principal block of any finite group.
+|[[M(16,14,16)]] || <math>b_2(k((C_2)^4 : 3^{1+2}_{+}))</math> || 1 ||8 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal faithful block. Cannot be Morita equivalent to a principal block of any finite group.
 |}

@@ Line 28: / Line 28: @@
 |[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_3</math> || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math>
 |-
-|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || <math>C_5</math> || ||1 ||1 ||
+|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || <math>C_5</math> || ||1 ||1 || The action comes from the 3rd power of a Singer cycle for <math>\mathbb{F}_{16}</math>
 |-
 |[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>(C_2)^4</math> ==
-These were classified in [[References|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References|[KS13] and [EKKS14]]].
+These were classified in [[References#E|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References#K|[KS13]]] and [[References#E|[EKKS14]]].
 {| class="wikitable"

@@ Line 2: / Line 2: @@
 == Blocks with defect group <math>(C_2)^4</math> ==
 These were classified in [[References|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References|[KS13] and [EKKS14]]].
@@ Line 22: / Line 20: @@
 |-
-|[[M(16,14,1)]] || <math>k((C_2)^4)</math> || 1 ||16 ||1 ||<math>1</math> || || ||1 ||1 ||
+|[[M(16,14,1)]] || <math>k((C_2)^4)</math> || 1 ||16 ||1 ||<math>1</math> || <math>(C_2)^4:GL_4(2)</math> || ||1 ||1 ||
 |-
-|[[M(16,14,2)]] || <math>B_0(k(C_2 \times C_2 \times A_5))</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 ||
+|[[M(16,14,2)]] || <math>B_0(k(C_2 \times C_2 \times A_5))</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times C_2</math> || ||1 ||1 ||
 |-
-|[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 ||
+|[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times S_3</math> || ||1 ||1 ||
 |-
-|[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math>
+|[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_3</math> || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math>
 |-
-|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || || ||1 ||1 ||
+|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || <math>C_5</math> || ||1 ||1 ||
 |-
 |[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||
@@ Line 36: / Line 34: @@
 |[[M(16,14,7)]] || <math>B_0(k(C_2 \times SL_2(8)))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||
 |-
-|[[M(16,14,8)]] || <math>k(A_4 \times A_4)</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||
+|[[M(16,14,8)]] || <math>k(A_4 \times A_4)</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \wr C_2</math>|| ||1 ||1 ||
 |-
-|[[M(16,14,9)]] || <math>B_0(k(A_4 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||
+|[[M(16,14,9)]] || <math>B_0(k(A_4 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \times C_2</math> || ||1 ||1 ||
 |-
 |[[M(16,14,10)]] || <math>B_0(k(A_5 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||
 |-
-|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || || ||1 ||1 ||
+|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}</math> || ||1 ||1 ||
 |-
-|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || || ||1 ||1 ||
+|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 ||
 |-
 |[[M(16,14,13)]] || <math>k(C_2 \times ((C_2)^3:(C_7:C_3)))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||

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