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Properties

Label 20T31
20T31 1 10 1->10 12 1->12 2 9 2->9 11 2->11 3 8 3->8 15 3->15 4 7 4->7 16 4->16 5 5->2 6 5->6 6->1 7->10 8->9 18 9->18 17 10->17 11->4 20 11->20 12->3 19 12->19 13 13->8 13->18 14 14->7 14->17 15->5 15->16 16->6 17->20 18->19 19->14 20->13
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times A_5$

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Copy content magma:G := TransitiveGroup(20, 31);
 

Group invariants

Abstract group:  $C_2\times A_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $120=2^{3} \cdot 3 \cdot 5$
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 $n$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
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:  $(1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)$, $(1,12,3,15,5,2,11,4,16,6)(7,10,17,20,13,8,9,18,19,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $A_{5}$

Low degree siblings

10T11, 12T75, 12T76, 20T36, 24T203, 30T29, 30T30, 40T61

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

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3
Size 1 1 15 15 20 12 12 20 12 12
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 10A3 10A1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A
Type
120.35.1a R 1 1 1 1 1 1 1 1 1 1
120.35.1b R 1 −1 −1 1 1 1 1 −1 −1 −1
120.35.3a1 R 3 3 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5
120.35.3a2 R 3 3 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52
120.35.3b1 R 3 −3 1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5
120.35.3b2 R 3 −3 1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52
120.35.4a R 4 4 0 0 1 −1 −1 1 −1 −1
120.35.4b R 4 −4 0 0 1 −1 −1 −1 1 1
120.35.5a R 5 5 1 1 −1 0 0 −1 0 0
120.35.5b R 5 −5 −1 1 −1 0 0 1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed