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Magma
magma: G := TransitiveGroup(19, 3);
Group action invariants
Degree $n$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{19}:C_{3}$ | ||
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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ 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
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 1 $ | $19$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$ |
$ 3, 3, 3, 3, 3, 3, 1 $ | $19$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ |
$ 19 $ | $3$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)$ |
$ 19 $ | $3$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)$ |
$ 19 $ | $3$ | $19$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)$ |
$ 19 $ | $3$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)$ |
$ 19 $ | $3$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)$ |
$ 19 $ | $3$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $57=3 \cdot 19$ | 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: | 57.1 | magma: IdentifyGroup(G);
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Character table: |
3 1 1 1 . . . . . . 19 1 . . 1 1 1 1 1 1 1a 3a 3b 19a 19b 19c 19d 19e 19f 2P 1a 3b 3a 19b 19c 19e 19f 19d 19a 3P 1a 1a 1a 19b 19c 19e 19f 19d 19a 5P 1a 3b 3a 19d 19f 19a 19c 19b 19e 7P 1a 3a 3b 19a 19b 19c 19d 19e 19f 11P 1a 3b 3a 19a 19b 19c 19d 19e 19f 13P 1a 3a 3b 19f 19a 19b 19e 19c 19d 17P 1a 3b 3a 19d 19f 19a 19c 19b 19e 19P 1a 3a 3b 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 1 1 1 1 X.3 1 /A A 1 1 1 1 1 1 X.4 3 . . B C /D /C /B D X.5 3 . . /B /C D C B /D X.6 3 . . C /D /B D /C B X.7 3 . . D B C /B /D /C X.8 3 . . /C D B /D C /B X.9 3 . . /D /B /C B D C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(19)^2+E(19)^3+E(19)^14 C = E(19)^4+E(19)^6+E(19)^9 D = E(19)+E(19)^7+E(19)^11 |
magma: CharacterTable(G);