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Properties

Label 22T11
22T11 1 18 1->18 1->18 2 14 2->14 20 2->20 3 21 3->21 22 3->22 4 13 4->13 17 4->17 5 5->13 15 5->15 6 6->17 6->20 7 16 7->16 19 7->19 8 12 8->12 8->21 9 9->12 9->19 10 10->14 10->15 11 11->16 11->22 12->5 12->6 13->3 13->11 14->6 14->11 15->1 15->8 16->5 17->2 17->2 18->8 18->10 19->3 20->4 20->9 21->1 21->4 22->9 22->10
Degree $22$
Order $1210$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{11}:F_{11}$

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Copy content magma:G := TransitiveGroup(22, 11);
 

Group invariants

Abstract group:  $C_{11}:F_{11}$
Copy content magma:IdentifyGroup(G);
 
Order:  $1210=2 \cdot 5 \cdot 11^{2}$
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 $t$:  $11$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,18,10,14,11,16,5,15,8,21)(2,20,4,13,3,22,9,12,6,17)(7,19)$, $(1,18,8,12,5,13,11,22,10,15)(2,14,6,20,9,19,3,21,4,17)(7,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$55$:  $C_{11}:C_5$
$110$:  $F_{11}$, 22T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

22T11 x 4

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 $2^{11}$ $11$ $2$ $11$ $( 1,14)( 2,20)( 3,15)( 4,21)( 5,16)( 6,22)( 7,17)( 8,12)( 9,18)(10,13)(11,19)$
5A1 $5^{4},1^{2}$ $121$ $5$ $16$ $( 1, 6, 4, 7, 8)( 2,10, 9, 5,11)(12,14,22,21,17)(13,18,16,19,20)$
5A-1 $5^{4},1^{2}$ $121$ $5$ $16$ $( 1, 8, 7, 4, 6)( 2,11, 5, 9,10)(12,17,21,22,14)(13,20,19,16,18)$
5A2 $5^{4},1^{2}$ $121$ $5$ $16$ $( 1, 4, 8, 6, 7)( 2, 9,11,10, 5)(12,22,17,14,21)(13,16,20,18,19)$
5A-2 $5^{4},1^{2}$ $121$ $5$ $16$ $( 1, 7, 6, 8, 4)( 2, 5,10,11, 9)(12,21,14,17,22)(13,19,18,20,16)$
10A1 $10^{2},2$ $121$ $10$ $19$ $( 1,17, 6,12, 4,14, 7,22, 8,21)( 2,16,10,19, 9,20, 5,13,11,18)( 3,15)$
10A-1 $10^{2},2$ $121$ $10$ $19$ $( 1,21, 8,22, 7,14, 4,12, 6,17)( 2,18,11,13, 5,20, 9,19,10,16)( 3,15)$
10A3 $10^{2},2$ $121$ $10$ $19$ $( 1,12, 7,21, 6,14, 8,17, 4,22)( 2,19, 5,18,10,20,11,16, 9,13)( 3,15)$
10A-3 $10^{2},2$ $121$ $10$ $19$ $( 1,22, 4,17, 8,14, 6,21, 7,12)( 2,13, 9,16,11,20,10,18, 5,19)( 3,15)$
11A1 $11^{2}$ $5$ $11$ $20$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,17,22,16,21,15,20,14,19,13,18)$
11A-1 $11^{2}$ $5$ $11$ $20$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,18,13,19,14,20,15,21,16,22,17)$
11B $11^{2}$ $10$ $11$ $20$ $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,19,15,22,18,14,21,17,13,20,16)$
11C1 $11^{2}$ $10$ $11$ $20$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,16,20,13,17,21,14,18,22,15,19)$
11C-1 $11^{2}$ $10$ $11$ $20$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,20,17,14,22,19,16,13,21,18,15)$
11D1 $11^{2}$ $10$ $11$ $20$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,13,14,15,16,17,18,19,20,21,22)$
11D-1 $11^{2}$ $10$ $11$ $20$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,14,16,18,20,22,13,15,17,19,21)$
11E1 $11,1^{11}$ $10$ $11$ $10$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
11E-1 $11,1^{11}$ $10$ $11$ $10$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)$
11F1 $11^{2}$ $10$ $11$ $20$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,21,19,17,15,13,22,20,18,16,14)$
11F-1 $11^{2}$ $10$ $11$ $20$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,16,20,13,17,21,14,18,22,15,19)$
11G1 $11^{2}$ $10$ $11$ $20$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,19,15,22,18,14,21,17,13,20,16)$
11G-1 $11^{2}$ $10$ $11$ $20$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,19,15,22,18,14,21,17,13,20,16)$
22A1 $22$ $55$ $22$ $21$ $( 1,14,11,19,10,13, 9,18, 8,12, 7,17, 6,22, 5,16, 4,21, 3,15, 2,20)$
22A-1 $22$ $55$ $22$ $21$ $( 1,20, 2,15, 3,21, 4,16, 5,22, 6,17, 7,12, 8,18, 9,13,10,19,11,14)$

Malle's constant $a(G)$:     $1/10$

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Character table

1A 2A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 11A1 11A-1 11B 11C1 11C-1 11D1 11D-1 11E1 11E-1 11F1 11F-1 11G1 11G-1 22A1 22A-1
Size 1 11 121 121 121 121 121 121 121 121 5 5 10 10 10 10 10 10 10 10 10 10 10 55 55
2 P 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 11A-1 11A1 11B 11C-1 11C1 11D-1 11D1 11E-1 11E1 11F-1 11F1 11G-1 11G1 11A1 11A-1
5 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 11A1 11A-1 11B 11C1 11C-1 11D1 11D-1 11E1 11E-1 11F1 11F-1 11G1 11G-1 22A1 22A-1
11 P 1A 2A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
1210.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1210.12.1b R 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
1210.12.1c1 C 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1210.12.1c2 C 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1210.12.1c3 C 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1210.12.1c4 C 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1210.12.1d1 C 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
1210.12.1d2 C 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
1210.12.1d3 C 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
1210.12.1d4 C 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
1210.12.5a1 C 5 5 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 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 5 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
1210.12.5a2 C 5 5 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −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 5 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
1210.12.5b1 C 5 −5 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 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 5 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115
1210.12.5b2 C 5 −5 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −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 5 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115
1210.12.10a R 10 0 0 0 0 0 0 0 0 0 10 10 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 0 0
1210.12.10b1 C 10 0 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −1 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 0 0
1210.12.10b2 C 10 0 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 −1 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 0 0
1210.12.10c1 C 10 0 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 −1 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 0 0
1210.12.10c2 C 10 0 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −1 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 0 0
1210.12.10d1 C 10 0 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 −1 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −1 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −1 0 0
1210.12.10d2 C 10 0 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −1 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 −1 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 −1 0 0
1210.12.10e1 C 10 0 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −1 −1 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 0 0
1210.12.10e2 C 10 0 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 −1 −1 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 0 0
1210.12.10f1 C 10 0 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 −1 −1 −1 −1 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −1 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 0 0
1210.12.10f2 C 10 0 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −ζ11−2+4−ζ11−ζ113−ζ114−ζ115 −1 −1 −1 −1 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 −1 ζ11−2+5+ζ11+ζ113+ζ114+ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 0 0

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Regular extensions

Data not computed