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Magma
magma: G := TransitiveGroup(39, 16);
Group action invariants
Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^3:C_{13}$ | ||
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: | (1,29,18,6,33,21,8,34,22,10,38,27,14)(2,30,16,4,31,19,9,35,23,11,39,25,15)(3,28,17,5,32,20,7,36,24,12,37,26,13), (1,6,7,12,15,18,20,22,25,30,33,35,38)(2,4,8,10,13,16,21,23,26,28,31,36,39)(3,5,9,11,14,17,19,24,27,29,32,34,37) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $13$: $C_{13}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 13: $C_{13}$
Low degree siblings
27T134Siblings 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 $ | $13$ | $3$ | $( 7, 8, 9)(13,14,15)(16,17,18)(19,20,21)(22,24,23)(25,27,26)(31,32,33) (34,36,35)(37,38,39)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $13$ | $3$ | $( 7, 9, 8)(13,15,14)(16,18,17)(19,21,20)(22,23,24)(25,26,27)(31,33,32) (34,35,36)(37,39,38)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1, 4, 7,10,13,16,19,24,25,28,32,36,38)( 2, 5, 8,11,14,17,20,22,26,29,33,34, 39)( 3, 6, 9,12,15,18,21,23,27,30,31,35,37)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1, 7,13,19,25,32,38, 4,10,16,24,28,36)( 2, 8,14,20,26,33,39, 5,11,17,22,29, 34)( 3, 9,15,21,27,31,37, 6,12,18,23,30,35)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,10,19,28,38, 7,16,25,36, 4,13,24,32)( 2,11,20,29,39, 8,17,26,34, 5,14,22, 33)( 3,12,21,30,37, 9,18,27,35, 6,15,23,31)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,13,25,38,10,24,36, 7,19,32, 4,16,28)( 2,14,26,39,11,22,34, 8,20,33, 5,17, 29)( 3,15,27,37,12,23,35, 9,21,31, 6,18,30)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,16,33, 9,24,37,13,29, 5,20,35,11,26)( 2,17,31, 7,22,38,14,30, 6,21,36,12, 27)( 3,18,32, 8,23,39,15,28, 4,19,34,10,25)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,19,37,18,36,14,31,11,30, 7,25, 4,22)( 2,20,38,16,34,15,32,12,28, 8,26, 5, 23)( 3,21,39,17,35,13,33,10,29, 9,27, 6,24)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,22, 6,27, 8,29,10,33,14,34,18,38,21)( 2,23, 4,25, 9,30,11,31,15,35,16,39, 19)( 3,24, 5,26, 7,28,12,32,13,36,17,37,20)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,25,11,34,21, 5,29,14,38,23, 7,31,17)( 2,26,12,35,19, 6,30,15,39,24, 8,32, 18)( 3,27,10,36,20, 4,28,13,37,22, 9,33,16)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,28,18, 6,33,19, 9,36,22,11,38,26,13)( 2,29,16, 4,31,20, 7,34,23,12,39,27, 14)( 3,30,17, 5,32,21, 8,35,24,10,37,25,15)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,31,24,15, 6,36,26,16, 9,39,29,19,10)( 2,32,22,13, 4,34,27,17, 7,37,30,20, 11)( 3,33,23,14, 5,35,25,18, 8,38,28,21,12)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,34,29,23,17,11, 5,38,31,25,21,14, 7)( 2,35,30,24,18,12, 6,39,32,26,19,15, 8)( 3,36,28,22,16,10, 4,37,33,27,20,13, 9)$ |
$ 13, 13, 13 $ | $27$ | $13$ | $( 1,37,36,31,30,25,22,19,18,14,11, 7, 4)( 2,38,34,32,28,26,23,20,16,15,12, 8, 5)( 3,39,35,33,29,27,24,21,17,13,10, 9, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $351=3^{3} \cdot 13$ | 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: | 351.12 | magma: IdentifyGroup(G);
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Character table: |
3 3 3 3 . . . . . . . . . . . . 13 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1a 3a 3b 13a 13b 13c 13d 13e 13f 13g 13h 13i 13j 13k 13l 2P 1a 3b 3a 13b 13d 13f 13h 13j 13l 13a 13c 13e 13g 13i 13k 3P 1a 1a 1a 13c 13f 13i 13l 13b 13e 13h 13k 13a 13d 13g 13j 5P 1a 3b 3a 13e 13j 13b 13g 13l 13d 13i 13a 13f 13k 13c 13h 7P 1a 3a 3b 13g 13a 13h 13b 13i 13c 13j 13d 13k 13e 13l 13f 11P 1a 3b 3a 13k 13i 13g 13e 13c 13a 13l 13j 13h 13f 13d 13b 13P 1a 3a 3b 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 B C D E F G /G /F /E /D /C /B X.3 1 1 1 C E G /F /D /B B D F /G /E /C X.4 1 1 1 D G /E /B C F /F /C B E /G /D X.5 1 1 1 E /F /B D /G /C C G /D B F /E X.6 1 1 1 F /D C /G /B E /E B G /C D /F X.7 1 1 1 G /B F /C E /D D /E C /F B /G X.8 1 1 1 /G B /F C /E D /D E /C F /B G X.9 1 1 1 /F D /C G B /E E /B /G C /D F X.10 1 1 1 /E F B /D G C /C /G D /B /F E X.11 1 1 1 /D /G E B /C /F F C /B /E G D X.12 1 1 1 /C /E /G F D B /B /D /F G E C X.13 1 1 1 /B /C /D /E /F /G G F E D C B X.14 13 A /A . . . . . . . . . . . . X.15 13 /A A . . . . . . . . . . . . A = -E(3)+2*E(3)^2 = (-1-3*Sqrt(-3))/2 = -2-3b3 B = E(13)^12 C = E(13)^11 D = E(13)^10 E = E(13)^9 F = E(13)^8 G = E(13)^7 |
magma: CharacterTable(G);