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Properties

Label 22T6
22T6 1 2 1->2 19 1->19 20 2->20 3 10 3->10 12 3->12 4 9 4->9 11 4->11 5 21 5->21 22 5->22 6 6->21 6->22 7 7->9 7->11 8 8->10 8->12 9->2 9->19 10->1 10->20 11->8 13 11->13 12->7 14 12->14 13->3 18 13->18 14->4 17 14->17 15 15->5 16 16->6 17->6 17->16 18->5 18->15 19->4 19->17 20->3 20->18 21->7 21->14 22->8 22->13
Degree $22$
Order $220$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_{11}$

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Copy content magma:G := TransitiveGroup(22, 6);
 

Group invariants

Abstract group:  $C_2\times F_{11}$
Copy content magma:IdentifyGroup(G);
 
Order:  $220=2^{2} \cdot 5 \cdot 11$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $6$
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:  $(1,19,17,6,21,7,11,13,3,10)(2,20,18,5,22,8,12,14,4,9)$, $(1,2)(3,12,7,9,19,4,11,8,10,20)(5,21,14,17,16,6,22,13,18,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $C_{10}$ x 3
$20$:  20T3
$110$:  $F_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $F_{11}$

Low degree siblings

22T6, 44T14

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

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 $2^{10},1^{2}$ $11$ $2$ $10$ $( 3,21)( 4,22)( 5,20)( 6,19)( 7,17)( 8,18)( 9,16)(10,15)(11,13)(12,14)$
2C $2^{11}$ $11$ $2$ $11$ $( 1,22)( 2,21)( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,13)(10,14)(11,12)$
5A1 $5^{4},1^{2}$ $11$ $5$ $16$ $( 3,10,11,19, 7)( 4, 9,12,20, 8)( 5,18,22,16,14)( 6,17,21,15,13)$
5A-1 $5^{4},1^{2}$ $11$ $5$ $16$ $( 3, 7,19,11,10)( 4, 8,20,12, 9)( 5,14,16,22,18)( 6,13,15,21,17)$
5A2 $5^{4},1^{2}$ $11$ $5$ $16$ $( 3,11, 7,10,19)( 4,12, 8, 9,20)( 5,22,14,18,16)( 6,21,13,17,15)$
5A-2 $5^{4},1^{2}$ $11$ $5$ $16$ $( 3,19,10, 7,11)( 4,20, 9, 8,12)( 5,16,18,14,22)( 6,15,17,13,21)$
10A1 $10^{2},1^{2}$ $11$ $10$ $18$ $( 3, 6,10,17,11,21,19,15, 7,13)( 4, 5, 9,18,12,22,20,16, 8,14)$
10A-1 $10^{2},1^{2}$ $11$ $10$ $18$ $( 3,13, 7,15,19,21,11,17,10, 6)( 4,14, 8,16,20,22,12,18, 9, 5)$
10A3 $10^{2},1^{2}$ $11$ $10$ $18$ $( 3,17,19,13,10,21, 7, 6,11,15)( 4,18,20,14, 9,22, 8, 5,12,16)$
10A-3 $10^{2},1^{2}$ $11$ $10$ $18$ $( 3,15,11, 6, 7,21,10,13,19,17)( 4,16,12, 5, 8,22, 9,14,20,18)$
10B1 $10^{2},2$ $11$ $10$ $19$ $( 1,14,17,12,10, 2,13,18,11, 9)( 3,22, 6, 8,15, 4,21, 5, 7,16)(19,20)$
10B-1 $10^{2},2$ $11$ $10$ $19$ $( 1,14, 6, 4,19, 2,13, 5, 3,20)( 7, 9,15,12,21, 8,10,16,11,22)(17,18)$
10B3 $10^{2},2$ $11$ $10$ $19$ $( 1,14,11,16, 7, 2,13,12,15, 8)( 3, 9,19,22,17, 4,10,20,21,18)( 5, 6)$
10B-3 $10^{2},2$ $11$ $10$ $19$ $( 1,14, 7,22, 3, 2,13, 8,21, 4)( 5,11,20,15,18, 6,12,19,16,17)( 9,10)$
10C1 $10^{2},2$ $11$ $10$ $19$ $( 1,14,15,20, 6,22,10, 8, 3,18)( 2,13,16,19, 5,21, 9, 7, 4,17)(11,12)$
10C-1 $10^{2},2$ $11$ $10$ $19$ $( 1,14,19,12, 7, 5,15, 9,17,22)( 2,13,20,11, 8, 6,16,10,18,21)( 3, 4)$
10C3 $10^{2},2$ $11$ $10$ $19$ $( 1,14,21,20, 3, 8,17, 9,11, 5)( 2,13,22,19, 4, 7,18,10,12, 6)(15,16)$
10C-3 $10^{2},2$ $11$ $10$ $19$ $( 1,14,10, 4, 6,20, 7,12,17,16)( 2,13, 9, 3, 5,19, 8,11,18,15)(21,22)$
11A $11^{2}$ $10$ $11$ $20$ $( 1, 3, 6, 7,10,11,13,15,17,19,21)( 2, 4, 5, 8, 9,12,14,16,18,20,22)$
22A $22$ $10$ $22$ $21$ $( 1,14, 3,16, 6,18, 7,20,10,22,11, 2,13, 4,15, 5,17, 8,19, 9,21,12)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3 11A 22A
Size 1 1 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 10 10
2 P 1A 1A 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 5A2 5A-2 5A1 5A-1 5A1 5A-1 5A-2 5A2 11A 11A
5 P 1A 2A 2B 2C 1A 1A 1A 1A 2B 2B 2B 2B 2A 2A 2A 2A 2C 2C 2C 2C 11A 22A
11 P 1A 2A 2B 2C 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3 1A 2A
Type
220.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
220.7.1b R 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 −1 1 −1
220.7.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 −1
220.7.1d R 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 −1 1 1 1 1 1
220.7.1e1 C 1 1 1 1 ζ5−2 ζ5 ζ52 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ52 1 1
220.7.1e2 C 1 1 1 1 ζ52 ζ5−1 ζ5−2 ζ5 ζ5 ζ5−1 ζ52 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5−2 1 1
220.7.1e3 C 1 1 1 1 ζ5−1 ζ5−2 ζ5 ζ52 ζ52 ζ5−2 ζ5−1 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ5 1 1
220.7.1e4 C 1 1 1 1 ζ5 ζ52 ζ5−1 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−1 1 1
220.7.1f1 C 1 −1 −1 1 ζ5−2 ζ5 ζ52 ζ5−1 ζ5−1 ζ5 ζ5−2 −ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ52 −ζ5−2 −ζ5 −ζ52 1 −1
220.7.1f2 C 1 −1 −1 1 ζ52 ζ5−1 ζ5−2 ζ5 ζ5 ζ5−1 ζ52 −ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−2 −ζ52 −ζ5−1 −ζ5−2 1 −1
220.7.1f3 C 1 −1 −1 1 ζ5−1 ζ5−2 ζ5 ζ52 ζ52 ζ5−2 ζ5−1 −ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5 −ζ5−1 −ζ5−2 −ζ5 1 −1
220.7.1f4 C 1 −1 −1 1 ζ5 ζ52 ζ5−1 ζ5−2 ζ5−2 ζ52 ζ5 −ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−1 −ζ5 −ζ52 −ζ5−1 1 −1
220.7.1g1 C 1 −1 1 −1 ζ5−2 ζ5 ζ52 ζ5−1 −ζ5−1 −ζ5 −ζ5−2 ζ5−1 ζ52 ζ5−2 ζ5 −ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ52 1 −1
220.7.1g2 C 1 −1 1 −1 ζ52 ζ5−1 ζ5−2 ζ5 −ζ5 −ζ5−1 −ζ52 ζ5 ζ5−2 ζ52 ζ5−1 −ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5−2 1 −1
220.7.1g3 C 1 −1 1 −1 ζ5−1 ζ5−2 ζ5 ζ52 −ζ52 −ζ5−2 −ζ5−1 ζ52 ζ5 ζ5−1 ζ5−2 −ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ5 1 −1
220.7.1g4 C 1 −1 1 −1 ζ5 ζ52 ζ5−1 ζ5−2 −ζ5−2 −ζ52 −ζ5 ζ5−2 ζ5−1 ζ5 ζ52 −ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ5−1 1 −1
220.7.1h1 C 1 1 −1 −1 ζ5−2 ζ5 ζ52 ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ5−1 −ζ52 −ζ5−2 −ζ5 ζ5−1 −ζ52 ζ5−2 ζ5 ζ52 1 1
220.7.1h2 C 1 1 −1 −1 ζ52 ζ5−1 ζ5−2 ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5 −ζ5−2 −ζ52 −ζ5−1 ζ5 −ζ5−2 ζ52 ζ5−1 ζ5−2 1 1
220.7.1h3 C 1 1 −1 −1 ζ5−1 ζ5−2 ζ5 ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ52 −ζ5 −ζ5−1 −ζ5−2 ζ52 −ζ5 ζ5−1 ζ5−2 ζ5 1 1
220.7.1h4 C 1 1 −1 −1 ζ5 ζ52 ζ5−1 ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−2 −ζ5−1 −ζ5 −ζ52 ζ5−2 −ζ5−1 ζ5 ζ52 ζ5−1 1 1
220.7.10a R 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1
220.7.10b R 10 −10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed