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Magma
magma: G := TransitiveGroup(19, 2);
Group action invariants
Degree $n$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_{19}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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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);
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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
Label | Cycle Type | Size | Order | Representative |
$ 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);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 38.1 | magma: IdentifyGroup(G);
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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 | |||||||||||
38.1.1b | R | |||||||||||
38.1.2a1 | R | |||||||||||
38.1.2a2 | R | |||||||||||
38.1.2a3 | R | |||||||||||
38.1.2a4 | R | |||||||||||
38.1.2a5 | R | |||||||||||
38.1.2a6 | R | |||||||||||
38.1.2a7 | R | |||||||||||
38.1.2a8 | R | |||||||||||
38.1.2a9 | R |
magma: CharacterTable(G);