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Properties

Label 20T36
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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Show commands: Magma

magma: G := TransitiveGroup(20, 36);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times A_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,20,13)(2,19,14)(3,6,7)(4,5,8)(9,17,12)(10,18,11), (1,20,18,11,8,2,19,17,12,7)(3,16,6,9,14,4,15,5,10,13)
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: None

Degree 4: None

Degree 5: None

Degree 10: $A_{5}$

Low degree siblings

10T11, 12T75, 12T76, 20T31, 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 $2^{8},1^{4}$ $15$ $2$ $8$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,13)( 6,14)( 7,11)( 8,12)$
2C $2^{10}$ $15$ $2$ $10$ $( 1,17)( 2,18)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$
3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1,20,13)( 2,19,14)( 3, 6, 7)( 4, 5, 8)( 9,17,12)(10,18,11)$
5A1 $5^{4}$ $12$ $5$ $16$ $( 1,12, 3,16, 6)( 2,11, 4,15, 5)( 7, 9,18,19,13)( 8,10,17,20,14)$
5A2 $5^{4}$ $12$ $5$ $16$ $( 1, 3, 6,12,16)( 2, 4, 5,11,15)( 7,18,13, 9,19)( 8,17,14,10,20)$
6A $6^{3},2$ $20$ $6$ $16$ $( 1,19,13, 2,20,14)( 3, 5, 7, 4, 6, 8)( 9,18,12,10,17,11)(15,16)$
10A1 $10^{2}$ $12$ $10$ $18$ $( 1, 4, 6,11,16, 2, 3, 5,12,15)( 7,17,13,10,19, 8,18,14, 9,20)$
10A3 $10^{2}$ $12$ $10$ $18$ $( 1,11, 3,15, 6, 2,12, 4,16, 5)( 7,10,18,20,13, 8, 9,17,19,14)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.35
magma: IdentifyGroup(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

magma: CharacterTable(G);