MNA(2,1) - Revision history

Charles Eaton at 11:47, 9 November 2022

2022-11-09T11:47:28Z

Charles Eaton at 09:35, 15 August 2019

2019-08-15T09:35:14Z

Charles Eaton: Added M(16,3,3). Renamed M(16,3,2) to fit with Klein four classification

2019-08-15T09:25:45Z

Added M(16,3,3). Renamed M(16,3,2) to fit with Klein four classification

Charles Eaton at 17:10, 14 August 2019

2019-08-14T17:10:54Z

Charles Eaton at 17:05, 14 August 2019

2019-08-14T17:05:12Z

Charles Eaton: [Sa16] reference

2019-01-02T22:17:20Z

[Sa16] reference

Charles Eaton at 17:00, 18 December 2018

2018-12-18T17:00:00Z

Charles Eaton at 17:04, 24 November 2018

2018-11-24T17:04:33Z

Charles Eaton at 08:46, 24 November 2018

2018-11-24T08:46:36Z

Charles Eaton: Created page with "NOTITLE == Blocks with defect group $MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle$ == The defect groups are minimal nonabelian $2$
2018-11-24T08:03:21Z

Created page with "NOTITLE == Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> == The defect groups are minimal nonabelian <math>2</..."

New page
NOTITLE

== Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> ==

The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but there exists at least one non-nilpotent fusion system for blocks with this defect group.

@@ Line 4: / Line 4: @@
 The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>. By [[References#S|[Sa16]]] all non-nilpotent blocks with this defect group are [[Glossary#Isotypy|isotypic]].
-'''<pre style="color: red">CLASSES NOT CLASSIFIED</pre>'''
+'''<pre style="color: red">CLASSIFICATION NOT COMPLETE</pre>'''
 {| class="wikitable"

@@ Line 23: / Line 23: @@
 |[[M(16,3,1)]] || <math>k(MNA(2,1))</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(16,3,2)]] || <math>k(A_5:C_4)</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 ||
+|[[M(16,3,2)]] || <math>B_0(k(A_5:C_4))</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 ||
 |-
 |[[M(16,3,3)]] || <math>k(A_4:C_4)</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 ||

@@ Line 23: / Line 23: @@
 |[[M(16,3,1)]] || <math>k(MNA(2,1))</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(16,3,2)]] || <math>k(SmallGroup(48,30))</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 ||
+|[[M(16,3,2)]] || <math>k(A_5:C_4)</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 ||
 |}
 If <math>B</math> is not nilpotent, then <math>k(B)=10, k_1(B)=2, l(B)=2</math>.

@@ Line 4: / Line 4: @@
 The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>. By [[References#S|[Sa16]]] all non-nilpotent blocks with this defect group are [[Glossary#Isotypy|isotypic]].
+'''<pre style="color: red">CLASSES NOT CLASSIFIED</pre>'''
 {| class="wikitable"

← Older revision		Revision as of 17:05, 14 August 2019
Line 26:		Line 26:
	\|[[M(16,3,2)]] \|\| <math>k(SmallGroup(48,30))</math> \|\| ? \|\|10 \|\|2 \|\|<math>1</math> \|\| \|\| \|\|1 \|\|1 \|\|		\|[[M(16,3,2)]] \|\| <math>k(SmallGroup(48,30))</math> \|\| ? \|\|10 \|\|2 \|\|<math>1</math> \|\| \|\| \|\|1 \|\|1 \|\|
	\|}		\|}
		+
		+	If <math>B</math> is not nilpotent, then <math>k(B)=10, k_1(B)=2, l(B)=2</math>.

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> ==
-The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>.
+The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>. By [[References#S|[Sa16]]] all non-nilpotent blocks with this defect group are [[Glossary#Isotypy|isotypic]].
 '''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''

@@ Line 3: / Line 3: @@
 == Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> ==
-The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but there exists at least one non-nilpotent fusion system for blocks with this defect group.
+The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but there exists at least one non-nilpotent fusion system for blocks with this defect group (realised in SmallGroup(48,30)<math>\cong A_4:C_4</math>).
-<!-- {| class="wikitable"
+'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''
 |-
 ! scope="col"| Class
@@ Line 20: / Line 22: @@
 |-
-|[[M(16,2,1)]] || <math>k(C_4 \times C_4)</math> || 1 ||16 ||1 ||<math>1</math> ||<math>(C_4 \times C_4):({\rm Aut}(C_4 \times C_4))</math> || ||1 ||1 ||
+|[[M(16,3,1)]] || <math>k(MNA(2,1))</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 ||
 |-
-|[[M(16,2,2)]] || <math>k((C_4 \times C_4):C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> ||<math>C_2 \times S_3</math> || ||1 ||1 ||
+|[[M(16,3,2)]] || <math>k(SmallGroup(48,30))</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 ||
-|}-->
+|}