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Properties

Label 22T27
22T27 1 4 1->4 22 1->22 2 3 2->3 21 2->21 7 3->7 9 3->9 8 4->8 10 4->10 5 5->3 19 5->19 6 6->4 20 6->20 14 7->14 17 7->17 13 8->13 18 8->18 9->6 9->19 10->5 10->20 11 11->6 15 11->15 12 12->5 16 12->16 13->8 13->10 14->7 14->9 15->11 16->12 17->13 17->16 18->14 18->15 19->2 19->22 20->1 20->21 21->2 21->18 22->1 22->17
Degree $22$
Order $15840$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times M_{11}$

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Copy content magma:G := TransitiveGroup(22, 27);
 

Group invariants

Abstract group:  $C_2\times M_{11}$
Copy content magma:IdentifyGroup(G);
 
Order:  $15840=2^{5} \cdot 3^{2} \cdot 5 \cdot 11$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  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 $t$:  $27$
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:  $(1,22,17,16,12,5,3,7,14,9,19,2,21,18,15,11,6,4,8,13,10,20)$, $(1,4,10,5,19,22)(2,3,9,6,20,21)(7,17,13,8,18,14)(11,15)(12,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$7920$:  $M_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $M_{11}$

Low degree siblings

22T26, 24T12204, 44T140

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 $2^{8},1^{6}$ $165$ $2$ $8$ $( 1,14)( 2,13)( 5,11)( 6,12)( 7,16)( 8,15)(17,21)(18,22)$
2C $2^{11}$ $165$ $2$ $11$ $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,10)( 6, 9)(11,12)(15,20)(16,19)(17,18)(21,22)$
3A $3^{6},1^{4}$ $440$ $3$ $12$ $( 3,14,10)( 4,13, 9)( 5,22,11)( 6,21,12)( 7,18,16)( 8,17,15)$
4A $4^{4},1^{6}$ $990$ $4$ $12$ $( 1,21,14,17)( 2,22,13,18)( 5, 7,11,16)( 6, 8,12,15)$
4B $4^{4},2^{3}$ $990$ $4$ $15$ $( 1, 2)( 3, 4)( 5,14,18, 8)( 6,13,17, 7)( 9,19,11,15)(10,20,12,16)(21,22)$
5A $5^{4},1^{2}$ $1584$ $5$ $16$ $( 1,12, 3,21,15)( 2,11, 4,22,16)( 5,20,13, 7, 9)( 6,19,14, 8,10)$
6A $6^{3},2^{2}$ $440$ $6$ $17$ $( 1, 9, 8, 2,10, 7)( 3,11,19, 4,12,20)( 5,21,18, 6,22,17)(13,14)(15,16)$
6B $6^{2},3^{2},2^{2}$ $1320$ $6$ $16$ $( 1,19)( 2,20)( 3, 6,14,21,10,12)( 4, 5,13,22, 9,11)( 7,16,18)( 8,15,17)$
6C $6^{3},2^{2}$ $1320$ $6$ $17$ $( 1,20, 3, 7,15,13)( 2,19, 4, 8,16,14)( 5,10)( 6, 9)(11,21,18,12,22,17)$
8A1 $8^{2},2^{3}$ $990$ $8$ $17$ $( 1,16,21, 5,14, 7,17,11)( 2,15,22, 6,13, 8,18,12)( 3, 9)( 4,10)(19,20)$
8A-1 $8^{2},2^{3}$ $990$ $8$ $17$ $( 1,11,17, 7,14, 5,21,16)( 2,12,18, 8,13, 6,22,15)( 3, 9)( 4,10)(19,20)$
8B1 $8^{2},2^{2},1^{2}$ $990$ $8$ $16$ $( 1,21)( 2,22)( 5,13,20, 9,18, 7,16,11)( 6,14,19,10,17, 8,15,12)$
8B-1 $8^{2},2^{2},1^{2}$ $990$ $8$ $16$ $( 1,21)( 2,22)( 5,11,16, 7,18, 9,20,13)( 6,12,15, 8,17,10,19,14)$
10A $10^{2},2$ $1584$ $10$ $19$ $( 1,22,12,16, 3, 2,21,11,15, 4)( 5, 8,20,10,13, 6, 7,19, 9,14)(17,18)$
11A1 $11^{2}$ $720$ $11$ $20$ $( 1,15,14, 6, 3,12,21,17,10,19, 8)( 2,16,13, 5, 4,11,22,18, 9,20, 7)$
11A-1 $11^{2}$ $720$ $11$ $20$ $( 1, 8,19,10,17,21,12, 3, 6,14,15)( 2, 7,20, 9,18,22,11, 4, 5,13,16)$
22A1 $22$ $720$ $22$ $21$ $( 1,22,15,18,14, 9, 6,20, 3, 7,12, 2,21,16,17,13,10, 5,19, 4, 8,11)$
22A-1 $22$ $720$ $22$ $21$ $( 1,11, 8, 4,19, 5,10,13,17,16,21, 2,12, 7, 3,20, 6, 9,14,18,15,22)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A 4B 5A 6A 6B 6C 8A1 8A-1 8B1 8B-1 10A 11A1 11A-1 22A1 22A-1
Size 1 1 165 165 440 990 990 1584 440 1320 1320 990 990 990 990 1584 720 720 720 720
2 P 1A 1A 1A 1A 3A 2B 2B 5A 3A 3A 3A 4A 4A 4A 4A 5A 11A-1 11A1 11A1 11A-1
3 P 1A 2A 2B 2C 1A 4A 4B 5A 2A 2B 2C 8A1 8A-1 8B1 8B-1 10A 11A1 11A-1 22A1 22A-1
5 P 1A 2A 2B 2C 3A 4A 4B 1A 6A 6B 6C 8A-1 8A1 8B-1 8B1 2A 11A1 11A-1 22A1 22A-1
11 P 1A 2A 2B 2C 3A 4A 4B 5A 6A 6B 6C 8A1 8A-1 8B1 8B-1 10A 1A 1A 2A 2A
Type
15840.q.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
15840.q.1b R 1 −1 1 −1 1 1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 1 −1 −1
15840.q.10a R 10 10 2 2 1 2 2 0 1 −1 −1 0 0 0 0 0 −1 −1 −1 −1
15840.q.10b R 10 −10 2 −2 1 2 −2 0 −1 −1 1 0 0 0 0 0 −1 −1 1 1
15840.q.10c1 C 10 −10 −2 2 1 0 0 0 −1 1 −1 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 0 −1 −1 1 1
15840.q.10c2 C 10 −10 −2 2 1 0 0 0 −1 1 −1 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 0 −1 −1 1 1
15840.q.10d1 C 10 10 −2 −2 1 0 0 0 1 1 1 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 0 −1 −1 −1 −1
15840.q.10d2 C 10 10 −2 −2 1 0 0 0 1 1 1 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 0 −1 −1 −1 −1
15840.q.11a R 11 11 3 3 2 −1 −1 1 2 0 0 −1 −1 −1 −1 1 0 0 0 0
15840.q.11b R 11 −11 3 −3 2 −1 1 1 −2 0 0 −1 1 1 −1 −1 0 0 0 0
15840.q.16a1 C 16 16 0 0 −2 0 0 1 −2 0 0 0 0 0 0 1 −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
15840.q.16a2 C 16 16 0 0 −2 0 0 1 −2 0 0 0 0 0 0 1 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
15840.q.16b1 C 16 −16 0 0 −2 0 0 1 2 0 0 0 0 0 0 −1 −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
15840.q.16b2 C 16 −16 0 0 −2 0 0 1 2 0 0 0 0 0 0 −1 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
15840.q.44a R 44 44 4 4 −1 0 0 −1 −1 1 1 0 0 0 0 −1 0 0 0 0
15840.q.44b R 44 −44 4 −4 −1 0 0 −1 1 1 −1 0 0 0 0 1 0 0 0 0
15840.q.45a R 45 −45 −3 3 0 1 −1 0 0 0 0 −1 1 1 −1 0 1 1 −1 −1
15840.q.45b R 45 45 −3 −3 0 1 1 0 0 0 0 −1 −1 −1 −1 0 1 1 1 1
15840.q.55a R 55 −55 −1 1 1 −1 1 0 −1 −1 1 1 −1 −1 1 0 0 0 0 0
15840.q.55b R 55 55 −1 −1 1 −1 −1 0 1 −1 −1 1 1 1 1 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed