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Properties

Label 20T11
20T11 1 4 1->4 17 1->17 2 3 2->3 18 2->18 5 3->5 16 3->16 6 4->6 15 4->15 8 5->8 14 5->14 7 6->7 13 6->13 10 7->10 12 7->12 9 8->9 11 8->11 9->2 20 9->20 10->1 19 10->19 11->7 11->13 12->8 12->14 13->5 13->16 14->6 14->15 15->3 15->18 16->4 16->17 17->2 17->20 18->1 18->19 19->9 19->11 20->10 20->12
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:D_4$

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Copy content magma:G := TransitiveGroup(20, 11);
 

Group invariants

Abstract group:  $C_5:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $40=2^{3} \cdot 5$
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 $n$:  $20$
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);
 
$\card{\Aut(F/K)}$:  $10$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,17,2,18)(3,16,4,15)(5,14,6,13)(7,12,8,11)(9,20,10,19)$, $(1,4,6,7,10)(2,3,5,8,9)(11,13,16,17,20,12,14,15,18,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_5$

Low degree siblings

20T7, 40T11

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{5},1^{10}$ $2$ $2$ $5$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)$
2C $2^{10}$ $10$ $2$ $10$ $( 1,17)( 2,18)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,20)(10,19)$
4A $4^{5}$ $10$ $4$ $15$ $( 1,18, 2,17)( 3,15, 4,16)( 5,13, 6,14)( 7,11, 8,12)( 9,19,10,20)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1, 7, 4,10, 6)( 2, 8, 3, 9, 5)(11,18,14,20,16)(12,17,13,19,15)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,14,16,18,20)(12,13,15,17,19)$
10A1 $10^{2}$ $2$ $10$ $18$ $( 1, 9, 7, 5, 4, 2,10, 8, 6, 3)(11,19,18,15,14,12,20,17,16,13)$
10A3 $10^{2}$ $2$ $10$ $18$ $( 1, 8, 4, 9, 6, 2, 7, 3,10, 5)(11,17,14,19,16,12,18,13,20,15)$
10B1 $10,5^{2}$ $2$ $10$ $17$ $( 1, 9, 7, 5, 4, 2,10, 8, 6, 3)(11,20,18,16,14)(12,19,17,15,13)$
10B-1 $10,5^{2}$ $2$ $10$ $17$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,14,16,18,20)(12,13,15,17,19)$
10B3 $10,5^{2}$ $2$ $10$ $17$ $( 1, 7, 4,10, 6)( 2, 8, 3, 9, 5)(11,17,14,19,16,12,18,13,20,15)$
10B-3 $10,5^{2}$ $2$ $10$ $17$ $( 1, 6,10, 4, 7)( 2, 5, 9, 3, 8)(11,15,20,13,18,12,16,19,14,17)$

Malle's constant $a(G)$:     $1/5$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 5A1 5A2 10A1 10A3 10B1 10B-1 10B3 10B-3
Size 1 1 2 10 10 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 5A2 5A1 5A1 5A2 5A1 5A1 5A2 5A2
5 P 1A 2A 2B 2C 4A 1A 1A 2A 2A 2B 2B 2B 2B
Type
40.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
40.8.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
40.8.1c R 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
40.8.1d R 1 1 1 −1 −1 1 1 1 1 1 1 1 1
40.8.2a R 2 −2 0 0 0 2 2 −2 −2 0 0 0 0
40.8.2b1 R 2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
40.8.2b2 R 2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
40.8.2c1 R 2 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
40.8.2c2 R 2 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
40.8.2d1 C 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52
40.8.2d2 C 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52
40.8.2d3 C 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52
40.8.2d4 C 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed