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Properties

Label 19T5
Degree $19$
Order $171$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{19}:C_{9}$

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Show commands: Magma

magma: G := TransitiveGroup(19, 5);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$9$:  $C_9$

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

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $171=3^{2} \cdot 19$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  171.3
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 19A1 19A-1
Size 1 19 19 19 19 19 19 19 19 9 9
3 P 1A 3A-1 3A1 9A-1 9A-2 9A4 9A2 9A-4 9A1 19A-1 19A1
19 P 1A 1A 1A 3A1 3A-1 3A-1 3A1 3A1 3A-1 19A-1 19A1
Type
171.3.1a R 1 1 1 1 1 1 1 1 1 1 1
171.3.1b1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
171.3.1b2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
171.3.1c1 C 1 ζ9−3 ζ93 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 1 1
171.3.1c2 C 1 ζ93 ζ9−3 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 1 1
171.3.1c3 C 1 ζ9−3 ζ93 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 1 1
171.3.1c4 C 1 ζ93 ζ9−3 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 1 1
171.3.1c5 C 1 ζ9−3 ζ93 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 1 1
171.3.1c6 C 1 ζ93 ζ9−3 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 1 1
171.3.9a1 C 9 0 0 0 0 0 0 0 0 −ζ19−8−ζ19−3−ζ19−2−1−ζ19−ζ194−ζ195−ζ196−ζ197−ζ199 ζ19−8+ζ19−3+ζ19−2+ζ19+ζ194+ζ195+ζ196+ζ197+ζ199
171.3.9a2 C 9 0 0 0 0 0 0 0 0 ζ19−8+ζ19−3+ζ19−2+ζ19+ζ194+ζ195+ζ196+ζ197+ζ199 −ζ19−8−ζ19−3−ζ19−2−1−ζ19−ζ194−ζ195−ζ196−ζ197−ζ199

magma: CharacterTable(G);