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Properties

Label 19T2
Degree $19$
Order $38$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_{19}$

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Show commands: Magma

magma: G := TransitiveGroup(19, 2);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $19$ $2$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 19 $ $2$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)$
$ 19 $ $2$ $19$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)$
$ 19 $ $2$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)$
$ 19 $ $2$ $19$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)$
$ 19 $ $2$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)$
$ 19 $ $2$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)$
$ 19 $ $2$ $19$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)$
$ 19 $ $2$ $19$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)$
$ 19 $ $2$ $19$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $38=2 \cdot 19$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  38.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 19A1 19A2 19A3 19A4 19A5 19A6 19A7 19A8 19A9
Size 1 19 2 2 2 2 2 2 2 2 2
2 P 1A 1A 19A4 19A7 19A6 19A3 19A9 19A8 19A1 19A5 19A2
19 P 1A 2A 19A6 19A1 19A9 19A5 19A4 19A7 19A8 19A2 19A3
Type
38.1.1a R 1 1 1 1 1 1 1 1 1 1 1
38.1.1b R 1 −1 1 1 1 1 1 1 1 1 1
38.1.2a1 R 2 0 ζ19−9+ζ199 ζ19−1+ζ19 ζ19−8+ζ198 ζ19−2+ζ192 ζ19−7+ζ197 ζ19−3+ζ193 ζ19−6+ζ196 ζ19−4+ζ194 ζ19−5+ζ195
38.1.2a2 R 2 0 ζ19−8+ζ198 ζ19−3+ζ193 ζ19−5+ζ195 ζ19−6+ζ196 ζ19−2+ζ192 ζ19−9+ζ199 ζ19−1+ζ19 ζ19−7+ζ197 ζ19−4+ζ194
38.1.2a3 R 2 0 ζ19−7+ζ197 ζ19−5+ζ195 ζ19−2+ζ192 ζ19−9+ζ199 ζ19−3+ζ193 ζ19−4+ζ194 ζ19−8+ζ198 ζ19−1+ζ19 ζ19−6+ζ196
38.1.2a4 R 2 0 ζ19−6+ζ196 ζ19−7+ζ197 ζ19−1+ζ19 ζ19−5+ζ195 ζ19−8+ζ198 ζ19−2+ζ192 ζ19−4+ζ194 ζ19−9+ζ199 ζ19−3+ζ193
38.1.2a5 R 2 0 ζ19−5+ζ195 ζ19−9+ζ199 ζ19−4+ζ194 ζ19−1+ζ19 ζ19−6+ζ196 ζ19−8+ζ198 ζ19−3+ζ193 ζ19−2+ζ192 ζ19−7+ζ197
38.1.2a6 R 2 0 ζ19−4+ζ194 ζ19−8+ζ198 ζ19−7+ζ197 ζ19−3+ζ193 ζ19−1+ζ19 ζ19−5+ζ195 ζ19−9+ζ199 ζ19−6+ζ196 ζ19−2+ζ192
38.1.2a7 R 2 0 ζ19−3+ζ193 ζ19−6+ζ196 ζ19−9+ζ199 ζ19−7+ζ197 ζ19−4+ζ194 ζ19−1+ζ19 ζ19−2+ζ192 ζ19−5+ζ195 ζ19−8+ζ198
38.1.2a8 R 2 0 ζ19−2+ζ192 ζ19−4+ζ194 ζ19−6+ζ196 ζ19−8+ζ198 ζ19−9+ζ199 ζ19−7+ζ197 ζ19−5+ζ195 ζ19−3+ζ193 ζ19−1+ζ19
38.1.2a9 R 2 0 ζ19−1+ζ19 ζ19−2+ζ192 ζ19−3+ζ193 ζ19−4+ζ194 ζ19−5+ζ195 ζ19−6+ζ196 ζ19−7+ζ197 ζ19−8+ζ198 ζ19−9+ζ199

magma: CharacterTable(G);