↑

Properties

Label 20T35
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(20, 35);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
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:  (1,3)(2,4)(5,13)(6,14)(7,8)(9,10)(11,12)(15,17)(16,18), (1,6,10,14,18)(2,5,9,13,17)(3,8,11,16,19)(4,7,12,15,20)
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: None

Degree 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 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 $2^{9},1^{2}$ $10$ $2$ $9$ $( 1, 3)( 2, 4)( 5,13)( 6,14)( 7, 8)( 9,10)(11,12)(15,17)(16,18)$
2B $2^{10}$ $15$ $2$ $10$ $( 1, 2)( 3,16)( 4,15)( 5,12)( 6,11)( 7,13)( 8,14)( 9,19)(10,20)(17,18)$
3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1, 4, 8)( 2, 3, 7)( 5,13,19)( 6,14,20)(11,15,18)(12,16,17)$
4A $4^{5}$ $30$ $4$ $15$ $( 1,18, 2,17)( 3,10,16,20)( 4, 9,15,19)( 5, 7,12,13)( 6, 8,11,14)$
5A $5^{4}$ $24$ $5$ $16$ $( 1,19,18,12, 7)( 2,20,17,11, 8)( 3,16, 6, 9,14)( 4,15, 5,10,13)$
6A $6^{3},1^{2}$ $20$ $6$ $15$ $( 1, 5,16, 3,13,18)( 2, 6,15, 4,14,17)( 7,11, 9, 8,12,10)$

Malle's constant $a(G)$:     $1/9$

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);