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Properties

Label 20T362
Degree $20$
Order $6840$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PGL(2,19)$

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Show commands: Magma

magma: G := TransitiveGroup(20, 362);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $362$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PGL(2,19)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,20,7,8,16,15,2,13,17,5,18,4,9,3,10,14,6,12,11,19), (1,18,8,7,16,14,2,17,13)(4,10,5,11,15,9,20,6,19)
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: None

Low degree siblings

40T5409

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 $2^{10}$ $171$ $2$ $10$ $( 1,12)( 2, 5)( 3, 8)( 4,18)( 6,14)( 7,11)( 9,10)(13,20)(15,19)(16,17)$
2B $2^{9},1^{2}$ $190$ $2$ $9$ $( 1, 6)( 2,15)( 3,13)( 5,18)( 7,17)( 8,11)( 9,12)(10,20)(14,19)$
3A $3^{6},1^{2}$ $380$ $3$ $12$ $( 1, 8,11)( 2, 7, 9)( 3,13,12)( 4,16,18)( 5,14,17)( 6,19,20)$
4A $4^{5}$ $342$ $4$ $15$ $( 1,15,12,19)( 2,13, 5,20)( 3, 9, 8,10)( 4,16,18,17)( 6, 7,14,11)$
5A1 $5^{4}$ $342$ $5$ $16$ $( 1, 4, 9,13,12)( 2,16, 6,15,10)( 3, 8,11,19,18)( 5, 7,17,20,14)$
5A2 $5^{4}$ $342$ $5$ $16$ $( 1, 9, 3, 2,19)( 4,11, 7,17, 8)( 5,16,18,10,20)( 6,14,15,13,12)$
6A $6^{3},1^{2}$ $380$ $6$ $15$ $( 1,17, 8, 6, 7,11)( 2, 5,20,15,18,10)( 3,19, 9,13,14,12)$
9A1 $9^{2},1^{2}$ $380$ $9$ $16$ $( 1,10,15, 3, 9, 6,17, 2,19)( 4,18,11, 5, 8,16,12,14,13)$
9A2 $9^{2},1^{2}$ $380$ $9$ $16$ $( 1,13,18, 8,12, 2, 7,19,20)( 3, 5,11, 9,15,17,14,10, 6)$
9A4 $9^{2},1^{2}$ $380$ $9$ $16$ $( 1, 6,10,17,15, 2, 3,19, 9)( 4,16,18,12,11,14, 5,13, 8)$
10A1 $10^{2}$ $342$ $10$ $18$ $( 1,19, 4,18, 9, 3,13, 8,12,11)( 2,14,16, 5, 6, 7,15,17,10,20)$
10A3 $10^{2}$ $342$ $10$ $18$ $( 1,18,13,11, 4, 3,12,19, 9, 8)( 2, 5,15,20,16, 7,10,14, 6,17)$
18A1 $18,1^{2}$ $380$ $18$ $17$ $( 1,14, 2,11,13,10, 7, 9,18, 6,19,15, 8, 3,20,17,12, 5)$
18A5 $18,1^{2}$ $380$ $18$ $17$ $( 1,10,19,17, 2, 9, 8, 5,13, 6,20,14, 7,15,12,11,18, 3)$
18A7 $18,1^{2}$ $380$ $18$ $17$ $( 1,15,13,17,18,14, 8,10,12, 6, 2, 3, 7, 5,19,11,20, 9)$
19A $19,1$ $360$ $19$ $18$ $( 1,12, 7,20, 8, 3,14,10,19, 6,13, 4,16,18,11, 2, 9,15, 5)$
20A1 $20$ $342$ $20$ $19$ $( 1, 6,11, 5,12,16, 8,14,13, 2, 3,20, 9,10,18,17, 4,15,19, 7)$
20A3 $20$ $342$ $20$ $19$ $( 1, 5, 8, 2, 9,17,19, 6,12,14, 3,10, 4, 7,11,16,13,20,18,15)$
20A7 $20$ $342$ $20$ $19$ $( 1,14,18, 6,13,17,11, 2, 4, 5, 3,15,12,20,19,16, 9, 7, 8,10)$
20A9 $20$ $342$ $20$ $19$ $( 1, 2,19,14, 4,16,18, 5, 9, 6, 3, 7,13,15, 8,17,12,10,11,20)$

Malle's constant $a(G)$:     $1/9$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $6840=2^{3} \cdot 3^{2} \cdot 5 \cdot 19$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  6840.b
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 5A1 5A2 6A 9A1 9A2 9A4 10A1 10A3 18A1 18A5 18A7 19A 20A1 20A3 20A7 20A9
Size 1 171 190 380 342 342 342 380 380 380 380 342 342 380 380 380 360 342 342 342 342
2 P 1A 1A 1A 3A 2A 5A2 5A1 3A 9A4 9A1 9A2 5A1 5A2 9A2 9A1 9A4 19A 10A1 10A3 10A3 10A1
3 P 1A 2A 2B 1A 4A 5A2 5A1 2B 3A 3A 3A 10A3 10A1 6A 6A 6A 19A 20A3 20A9 20A1 20A7
5 P 1A 2A 2B 3A 4A 1A 1A 6A 9A1 9A2 9A4 2A 2A 18A1 18A5 18A7 19A 4A 4A 4A 4A
19 P 1A 2A 2B 3A 4A 5A1 5A2 6A 9A2 9A4 9A1 10A1 10A3 18A7 18A1 18A5 1A 20A1 20A3 20A7 20A9
Type

magma: CharacterTable(G);