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Properties

Label 20T32
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 32);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_5$
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:  (1,5,9,14,18)(2,6,10,13,17)(3,7,12,16,20)(4,8,11,15,19), (1,3)(2,4)(5,13)(6,14)(7,8)(9,10)(11,12)(15,17)(16,18)(19,20)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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 $2^{10}$ $10$ $2$ $10$ $( 1, 2)( 3,19)( 4,20)( 5, 6)( 7,14)( 8,13)( 9,16)(10,15)(11,12)(17,18)$
2B $2^{8},1^{4}$ $15$ $2$ $8$ $( 1, 5)( 2, 6)( 7,20)( 8,19)( 9,18)(10,17)(11,15)(12,16)$
3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1, 9,15)( 2,10,16)( 3,17,20)( 4,18,19)( 7,12,13)( 8,11,14)$
4A $4^{4},2^{2}$ $30$ $4$ $14$ $( 1, 7, 5,20)( 2, 8, 6,19)( 3,14)( 4,13)( 9,12,18,16)(10,11,17,15)$
5A $5^{4}$ $24$ $5$ $16$ $( 1, 4,18,11,14)( 2, 3,17,12,13)( 5,15, 8,19, 9)( 6,16, 7,20,10)$
6A $6^{3},2$ $20$ $6$ $16$ $( 1, 2)( 3,14,10,19, 7,15)( 4,13, 9,20, 8,16)( 5,12,18, 6,11,17)$

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:  120.34
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 5A 6A
Size 1 10 15 20 30 24 20
2 P 1A 1A 1A 3A 2B 5A 3A
3 P 1A 2A 2B 1A 4A 5A 2A
5 P 1A 2A 2B 3A 4A 1A 6A
Type
120.34.1a R 1 1 1 1 1 1 1
120.34.1b R 1 −1 1 1 −1 1 −1
120.34.4a R 4 2 0 1 0 −1 −1
120.34.4b R 4 −2 0 1 0 −1 1
120.34.5a R 5 −1 1 −1 1 0 −1
120.34.5b R 5 1 1 −1 −1 0 1
120.34.6a R 6 0 −2 0 0 1 0

magma: CharacterTable(G);