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Properties

Label 19T6
Degree $19$
Order $342$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $F_{19}$

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Show commands: Magma

magma: G := TransitiveGroup(19, 6);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$18$:  $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{19}$ $1$ $1$ $0$ $()$
2A $2^{9},1$ $19$ $2$ $9$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
3A1 $3^{6},1$ $19$ $3$ $12$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$
3A-1 $3^{6},1$ $19$ $3$ $12$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$
6A1 $6^{3},1$ $19$ $6$ $15$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)$
6A-1 $6^{3},1$ $19$ $6$ $15$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)$
9A1 $9^{2},1$ $19$ $9$ $16$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)$
9A-1 $9^{2},1$ $19$ $9$ $16$ $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)$
9A2 $9^{2},1$ $19$ $9$ $16$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)$
9A-2 $9^{2},1$ $19$ $9$ $16$ $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)$
9A4 $9^{2},1$ $19$ $9$ $16$ $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)$
9A-4 $9^{2},1$ $19$ $9$ $16$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)$
18A1 $18,1$ $19$ $18$ $17$ $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)$
18A-1 $18,1$ $19$ $18$ $17$ $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)$
18A5 $18,1$ $19$ $18$ $17$ $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)$
18A-5 $18,1$ $19$ $18$ $17$ $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)$
18A7 $18,1$ $19$ $18$ $17$ $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)$
18A-7 $18,1$ $19$ $18$ $17$ $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)$
19A $19$ $18$ $19$ $18$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$:     $1/9$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $342=2 \cdot 3^{2} \cdot 19$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  342.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7 19A
Size 1 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18
2 P 1A 1A 3A-1 3A1 3A1 3A-1 9A-2 9A-4 9A4 9A2 9A-1 9A1 9A2 9A-2 9A-4 9A-1 9A1 9A4 19A
3 P 1A 2A 1A 1A 2A 2A 3A-1 3A1 3A-1 3A1 3A1 3A-1 6A-1 6A1 6A-1 6A-1 6A1 6A1 19A
19 P 1A 2A 3A1 3A-1 6A1 6A-1 9A-1 9A-2 9A2 9A1 9A4 9A-4 18A-7 18A7 18A5 18A-1 18A1 18A-5 1A
Type
342.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
342.7.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1
342.7.1c1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1
342.7.1c2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1
342.7.1d1 C 1 −1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1
342.7.1d2 C 1 −1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1
342.7.1e1 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4 1
342.7.1e2 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94 1
342.7.1e3 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92 1
342.7.1e4 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2 1
342.7.1e5 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1 1
342.7.1e6 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9 1
342.7.1f1 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4 1
342.7.1f2 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94 1
342.7.1f3 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92 1
342.7.1f4 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2 1
342.7.1f5 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1 1
342.7.1f6 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9 1
342.7.18a R 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1

magma: CharacterTable(G);