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Magma
magma: G := TransitiveGroup(13, 3);
Group action invariants
Degree $n$: | $13$ | 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: | $C_{13}:C_3$ | ||
CHM label: | $F_{39}(13)=13:3$ | ||
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), (1,3,9)(2,6,5)(4,12,10)(7,8,11) | 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
39T2Siblings are shown 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 1 $ | $13$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$ |
$ 3, 3, 3, 3, 1 $ | $13$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$ |
$ 13 $ | $3$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$ |
$ 13 $ | $3$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$ |
$ 13 $ | $3$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$ |
$ 13 $ | $3$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $39=3 \cdot 13$ | 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: | 39.1 | magma: IdentifyGroup(G);
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Character table: |
3 1 1 1 . . . . 13 1 . . 1 1 1 1 1a 3a 3b 13a 13b 13c 13d 2P 1a 3b 3a 13b 13c 13d 13a 3P 1a 1a 1a 13a 13b 13c 13d 5P 1a 3b 3a 13b 13c 13d 13a 7P 1a 3a 3b 13d 13a 13b 13c 11P 1a 3b 3a 13d 13a 13b 13c 13P 1a 3a 3b 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 1 1 X.3 1 /A A 1 1 1 1 X.4 3 . . B C /B /C X.5 3 . . C /B /C B X.6 3 . . /B /C B C X.7 3 . . /C B C /B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(13)+E(13)^3+E(13)^9 C = E(13)^2+E(13)^5+E(13)^6 |
magma: CharacterTable(G);