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Magma
magma: G := TransitiveGroup(39, 30);
Group action invariants
Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^3:C_{13}:C_3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,12,39)(2,10,37)(3,11,38)(4,21,27)(5,19,25)(6,20,26)(7,29,15)(8,30,13)(9,28,14)(16,17,18)(22,36,33)(23,34,31)(24,35,32), (1,27,12,34,19,5,29,14,39,23,9,33,16)(2,25,10,35,20,6,30,15,37,24,7,31,17)(3,26,11,36,21,4,28,13,38,22,8,32,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $39$: $C_{13}:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 13: $C_{13}:C_3$
Low degree siblings
27T292, 39T29, 39T30Siblings 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, 1, 1, 1, 1, 1, 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, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $13$ | $3$ | $( 1, 3, 2)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,23,24) (31,32,33)(34,36,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $13$ | $3$ | $( 1, 2, 3)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,24,23) (31,33,32)(34,35,36)$ |
$ 13, 13, 13 $ | $81$ | $13$ | $( 1,24, 5,26, 8,29,11,33,14,34,17,37,21)( 2,22, 6,27, 9,30,12,31,15,35,18,38, 19)( 3,23, 4,25, 7,28,10,32,13,36,16,39,20)$ |
$ 13, 13, 13 $ | $81$ | $13$ | $( 1, 5, 8,11,14,17,21,24,26,29,33,34,37)( 2, 6, 9,12,15,18,19,22,27,30,31,35, 38)( 3, 4, 7,10,13,16,20,23,25,28,32,36,39)$ |
$ 13, 13, 13 $ | $81$ | $13$ | $( 1, 8,14,21,26,33,37, 5,11,17,24,29,34)( 2, 9,15,19,27,31,38, 6,12,18,22,30, 35)( 3, 7,13,20,25,32,39, 4,10,16,23,28,36)$ |
$ 13, 13, 13 $ | $81$ | $13$ | $( 1,14,26,37,11,24,34, 8,21,33, 5,17,29)( 2,15,27,38,12,22,35, 9,19,31, 6,18, 30)( 3,13,25,39,10,23,36, 7,20,32, 4,16,28)$ |
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ | $117$ | $9$ | $( 4,12,28, 5,10,29, 6,11,30)( 7,19,18, 8,20,16, 9,21,17)(13,37,32,15,39,31,14, 38,33)(22,25,36)(23,26,34)(24,27,35)$ |
$ 9, 9, 9, 3, 3, 3, 3 $ | $117$ | $9$ | $( 1, 3, 2)( 4,10,29, 6,12,28, 5,11,30)( 7,20,18, 9,19,17, 8,21,16) (13,37,33,14,38,31,15,39,32)(22,25,35)(23,26,36)(24,27,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $117$ | $3$ | $( 1, 2, 3)( 4,11,30)( 5,12,28)( 6,10,29)( 7,21,18)( 8,19,16)( 9,20,17) (13,37,31)(14,38,32)(15,39,33)(22,25,34)(23,26,35)(24,27,36)$ |
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ | $117$ | $9$ | $( 4,28,10, 6,30,12, 5,29,11)( 7,18,20, 9,17,19, 8,16,21)(13,32,39,14,33,37,15, 31,38)(22,36,25)(23,34,26)(24,35,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $117$ | $3$ | $( 1, 3, 2)( 4,28,11)( 5,29,12)( 6,30,10)( 7,17,20)( 8,18,21)( 9,16,19) (13,33,37)(14,31,38)(15,32,39)(22,35,27)(23,36,25)(24,34,26)$ |
$ 9, 9, 9, 3, 3, 3, 3 $ | $117$ | $9$ | $( 1, 2, 3)( 4,28,12, 5,29,10, 6,30,11)( 7,16,20, 8,17,21, 9,18,19) (13,31,38,15,33,37,14,32,39)(22,34,26)(23,35,27)(24,36,25)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1053=3^{4} \cdot 13$ | 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: | 1053.51 | magma: IdentifyGroup(G);
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Character table: |
3 4 4 4 . . . . 2 2 2 2 2 2 13 1 . . 1 1 1 1 . . . . . . 1a 3a 3b 13a 13b 13c 13d 9a 9b 3c 9c 3d 9d 2P 1a 3b 3a 13b 13c 13d 13a 9c 9d 3d 9a 3c 9b 3P 1a 1a 1a 13a 13b 13c 13d 3a 3b 1a 3b 1a 3a 5P 1a 3b 3a 13b 13c 13d 13a 9c 9d 3d 9a 3c 9b 7P 1a 3a 3b 13d 13a 13b 13c 9a 9b 3c 9c 3d 9d 11P 1a 3b 3a 13d 13a 13b 13c 9c 9d 3d 9a 3c 9b 13P 1a 3a 3b 1a 1a 1a 1a 9a 9b 3c 9c 3d 9d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 D D D /D /D /D X.3 1 1 1 1 1 1 1 /D /D /D D D D X.4 3 3 3 B C /B /C . . . . . . X.5 3 3 3 /B /C B C . . . . . . X.6 3 3 3 C /B /C B . . . . . . X.7 3 3 3 /C B C /B . . . . . . X.8 13 A /A . . . . /D 1 D D /D 1 X.9 13 /A A . . . . D 1 /D /D D 1 X.10 13 A /A . . . . 1 D /D 1 D /D X.11 13 /A A . . . . 1 /D D 1 /D D X.12 13 A /A . . . . D /D 1 /D 1 D X.13 13 /A A . . . . /D D 1 D 1 /D A = 2*E(3)-E(3)^2 = (-1+3*Sqrt(-3))/2 = 1+3b3 B = E(13)^2+E(13)^5+E(13)^6 C = E(13)^4+E(13)^10+E(13)^12 D = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 |
magma: CharacterTable(G);