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Properties

Label 20T7
20T7 1 8 1->8 11 1->11 2 7 2->7 12 2->12 3 9 3->9 3->9 4 10 4->10 4->10 5 5->8 5->11 6 6->7 6->12 7->5 14 7->14 8->6 13 8->13 9->4 15 9->15 10->3 16 10->16 11->2 17 11->17 12->1 18 12->18 19 13->19 20 13->20 14->19 14->20 15->2 15->17 16->1 16->18 17->3 17->16 18->4 18->15 19->5 19->14 20->6 20->13
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, 7);
 

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 $t$:  $7$
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:  $(1,11,2,12)(3,9,4,10)(5,8,6,7)(13,19,14,20)(15,17,16,18)$, $(1,8,13,20,6,12,18,4,10,16)(2,7,14,19,5,11,17,3,9,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T11, 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 $2^{10}$ $2$ $2$ $10$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,16)( 6,15)( 7,18)( 8,17)( 9,20)(10,19)$
2C $2^{9},1^{2}$ $10$ $2$ $9$ $( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,14)(10,13)(11,12)$
4A $4^{5}$ $10$ $4$ $15$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,20,14,19)(15,18,16,17)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1,10,18, 6,13)( 2, 9,17, 5,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1,18,13,10, 6)( 2,17,14, 9, 5)( 3,19,15,11, 7)( 4,20,16,12, 8)$
10A1 $10^{2}$ $2$ $10$ $18$ $( 1, 5,10,14,18, 2, 6, 9,13,17)( 3, 8,11,16,19, 4, 7,12,15,20)$
10A3 $10^{2}$ $2$ $10$ $18$ $( 1, 9,18, 5,13, 2,10,17, 6,14)( 3,12,19, 8,15, 4,11,20, 7,16)$
10B1 $10^{2}$ $2$ $10$ $18$ $( 1,15,10, 3,18,11, 6,19,13, 7)( 2,16, 9, 4,17,12, 5,20,14, 8)$
10B-1 $10^{2}$ $2$ $10$ $18$ $( 1, 7,13,19, 6,11,18, 3,10,15)( 2, 8,14,20, 5,12,17, 4, 9,16)$
10B3 $10^{2}$ $2$ $10$ $18$ $( 1,20,18,16,13,12,10, 8, 6, 4)( 2,19,17,15,14,11, 9, 7, 5, 3)$
10B-3 $10^{2}$ $2$ $10$ $18$ $( 1, 4, 6, 8,10,12,13,16,18,20)( 2, 3, 5, 7, 9,11,14,15,17,19)$

Malle's constant $a(G)$:     $1/9$

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