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Properties

Label 22T13
22T13 1 20 1->20 21 1->21 2 19 2->19 22 2->22 3 8 3->8 17 3->17 4 7 4->7 18 4->18 5 14 5->14 5->22 6 13 6->13 6->21 9 7->9 15 7->15 10 8->10 16 8->16 9->2 9->10 10->1 11 11->6 11->6 12 12->5 12->5 13->8 13->12 14->7 14->11 15->3 15->3 16->4 16->4 17->13 17->20 18->14 18->19 19->11 20->12 21->1 21->15 22->2 22->16
Degree $22$
Order $1320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times \PSL(2,11)$

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Copy content magma:G := TransitiveGroup(22, 13);
 

Group invariants

Abstract group:  $C_2\times \PSL(2,11)$
Copy content magma:IdentifyGroup(G);
 
Order:  $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $22$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,20,12,5,22,2,19,11,6,21)(3,17,13,8,16,4,18,14,7,15)(9,10)$, $(1,21,15,3,8,10)(2,22,16,4,7,9)(5,14,11,6,13,12)(17,20)(18,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$660$:  $\PSL(2,11)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $\PSL(2,11)$

Low degree siblings

22T13, 24T2948

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{22}$ $1$ $1$ $0$ $()$
2A $2^{11}$ $1$ $2$ $11$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
2B $2^{11}$ $55$ $2$ $11$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5, 9)( 6,10)(11,12)(15,20)(16,19)(17,18)(21,22)$
2C $2^{8},1^{6}$ $55$ $2$ $8$ $( 1,22)( 2,21)( 3, 7)( 4, 8)( 5,11)( 6,12)( 9,15)(10,16)$
3A $3^{6},1^{4}$ $110$ $3$ $12$ $( 1, 9,19)( 2,10,20)( 3, 7,11)( 4, 8,12)(13,21,16)(14,22,15)$
5A1 $5^{4},1^{2}$ $132$ $5$ $16$ $( 1, 6,15, 4, 9)( 2, 5,16, 3,10)(11,21,13,20,18)(12,22,14,19,17)$
5A2 $5^{4},1^{2}$ $132$ $5$ $16$ $( 1,15, 9, 6, 4)( 2,16,10, 5, 3)(11,13,18,21,20)(12,14,17,22,19)$
6A $6^{3},2^{2}$ $110$ $6$ $17$ $( 1,20, 9, 2,19,10)( 3,12, 7, 4,11, 8)( 5, 6)(13,15,21,14,16,22)(17,18)$
6B $6^{3},2^{2}$ $110$ $6$ $17$ $( 1,13,15, 7, 4,20)( 2,14,16, 8, 3,19)( 5, 9)( 6,10)(11,17,21,12,18,22)$
6C $6^{2},3^{2},2^{2}$ $110$ $6$ $16$ $( 1,12, 4,22, 6, 8)( 2,11, 3,21, 5, 7)( 9,15)(10,16)(13,20,18)(14,19,17)$
10A1 $10^{2},2$ $132$ $10$ $19$ $( 1, 3, 6,10,15, 2, 4, 5, 9,16)( 7, 8)(11,19,21,17,13,12,20,22,18,14)$
10A3 $10^{2},2$ $132$ $10$ $19$ $( 1,10, 4,16, 6, 2, 9, 3,15, 5)( 7, 8)(11,17,20,14,21,12,18,19,13,22)$
11A1 $11^{2}$ $60$ $11$ $20$ $( 1,15, 4,19, 9,14,17, 6,12,22, 8)( 2,16, 3,20,10,13,18, 5,11,21, 7)$
11A-1 $11^{2}$ $60$ $11$ $20$ $( 1, 8,22,12, 6,17,14, 9,19, 4,15)( 2, 7,21,11, 5,18,13,10,20, 3,16)$
22A1 $22$ $60$ $22$ $21$ $( 1,18,15, 5, 4,11,19,21, 9, 7,14, 2,17,16, 6, 3,12,20,22,10, 8,13)$
22A-1 $22$ $60$ $22$ $21$ $( 1,13, 8,10,22,20,12, 3, 6,16,17, 2,14, 7, 9,21,19,11, 4, 5,15,18)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 5A1 5A2 6A 6B 6C 10A1 10A3 11A1 11A-1 22A1 22A-1
Size 1 1 55 55 110 132 132 110 110 110 132 132 60 60 60 60
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 3A 3A 5A1 5A2 11A-1 11A1 11A1 11A-1
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 2B 2C 10A3 10A1 11A1 11A-1 22A1 22A-1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 6B 6C 2A 2A 11A1 11A-1 22A1 22A-1
11 P 1A 2A 2B 2C 3A 5A1 5A2 6A 6B 6C 10A1 10A3 1A 1A 2A 2A
Type
1320.134.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1320.134.1b R 1 −1 −1 1 1 1 1 −1 1 −1 −1 −1 1 1 −1 −1
1320.134.5a1 C 5 5 1 1 −1 0 0 1 1 −1 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
1320.134.5a2 C 5 5 1 1 −1 0 0 1 1 −1 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
1320.134.5b1 C 5 −5 −1 1 −1 0 0 −1 1 1 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115
1320.134.5b2 C 5 −5 −1 1 −1 0 0 −1 1 1 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115
1320.134.10a R 10 10 2 2 1 0 0 −1 −1 1 0 0 −1 −1 −1 −1
1320.134.10b R 10 −10 −2 2 1 0 0 1 −1 −1 0 0 −1 −1 1 1
1320.134.10c R 10 10 −2 −2 1 0 0 1 1 1 0 0 −1 −1 −1 −1
1320.134.10d R 10 −10 2 −2 1 0 0 −1 1 −1 0 0 −1 −1 1 1
1320.134.11a R 11 11 −1 −1 −1 1 1 −1 −1 −1 1 1 0 0 0 0
1320.134.11b R 11 −11 1 −1 −1 1 1 1 −1 1 −1 −1 0 0 0 0
1320.134.12a1 R 12 12 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 1 1 1 1
1320.134.12a2 R 12 12 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 1 1 1 1
1320.134.12b1 R 12 −12 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 1 1 −1 −1
1320.134.12b2 R 12 −12 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 1 1 −1 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed