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Magma
magma: G := TransitiveGroup(39, 12);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{39}:C_6$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,39,12)(2,37,10)(3,38,11)(4,26,19)(5,27,20)(6,25,21)(7,15,28)(8,13,29)(9,14,30)(16,17,18)(22,32,35)(23,33,36)(24,31,34), (1,2)(4,29,11,6,30,10)(5,28,12)(7,18,19,8,17,20)(9,16,21)(13,32,37)(14,31,38,15,33,39)(22,35,25,24,36,27)(23,34,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $39$: $C_{13}:C_3$ $78$: 26T5 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: $C_{13}:C_3$
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, 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, 3, 3, 3, 1, 1, 1 $ | $13$ | $3$ | $( 4,11,30)( 5,12,28)( 6,10,29)( 7,19,17)( 8,20,18)( 9,21,16)(13,37,32) (14,38,33)(15,39,31)(22,25,36)(23,26,34)(24,27,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $13$ | $3$ | $( 4,30,11)( 5,28,12)( 6,29,10)( 7,17,19)( 8,18,20)( 9,16,21)(13,32,37) (14,33,38)(15,31,39)(22,36,25)(23,34,26)(24,35,27)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 4, 5)( 8, 9)(11,12)(13,15)(16,18)(20,21)(22,23)(25,26)(28,30)(31,32) (34,36)(37,39)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $39$ | $6$ | $( 2, 3)( 4,12,30, 5,11,28)( 6,10,29)( 7,19,17)( 8,21,18, 9,20,16) (13,39,32,15,37,31)(14,38,33)(22,26,36,23,25,34)(24,27,35)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $39$ | $6$ | $( 2, 3)( 4,28,11, 5,30,12)( 6,29,10)( 7,17,19)( 8,16,20, 9,18,21) (13,31,37,15,32,39)(14,33,38)(22,34,25,23,36,26)(24,35,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1, 2, 3)( 4,12,29)( 5,10,30)( 6,11,28)( 7,20,16)( 8,21,17)( 9,19,18) (13,38,31)(14,39,32)(15,37,33)(22,26,35)(23,27,36)(24,25,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1, 2, 3)( 4,28,10)( 5,29,11)( 6,30,12)( 7,18,21)( 8,16,19)( 9,17,20) (13,33,39)(14,31,37)(15,32,38)(22,34,27)(23,35,25)(24,36,26)$ |
| $ 39 $ | $6$ | $39$ | $( 1, 4, 9,10,15,16,19,22,26,29,31,34,38, 2, 5, 7,11,13,17,20,23,27,30,32,35, 39, 3, 6, 8,12,14,18,21,24,25,28,33,36,37)$ |
| $ 26, 13 $ | $9$ | $26$ | $( 1, 4, 7,11,14,18,19,22,27,30,33,36,38, 2, 6, 8,10,15,17,20,24,25,29,31,35,39 )( 3, 5, 9,12,13,16,21,23,26,28,32,34,37)$ |
| $ 13, 13, 13 $ | $3$ | $13$ | $( 1, 6, 7,10,14,17,19,24,27,29,33,35,38)( 2, 4, 8,11,15,18,20,22,25,30,31,36, 39)( 3, 5, 9,12,13,16,21,23,26,28,32,34,37)$ |
| $ 13, 13, 13 $ | $3$ | $13$ | $( 1, 7,14,19,27,33,38, 6,10,17,24,29,35)( 2, 8,15,20,25,31,39, 4,11,18,22,30, 36)( 3, 9,13,21,26,32,37, 5,12,16,23,28,34)$ |
| $ 26, 13 $ | $9$ | $26$ | $( 1, 7,14,19,27,33,38, 6,10,17,24,29,35)( 2, 9,15,21,25,32,39, 5,11,16,22,28, 36, 3, 8,13,20,26,31,37, 4,12,18,23,30,34)$ |
| $ 39 $ | $6$ | $39$ | $( 1, 8,13,19,25,32,38, 4,12,17,22,28,35, 2, 9,14,20,26,33,39, 5,10,18,23,29, 36, 3, 7,15,21,27,31,37, 6,11,16,24,30,34)$ |
| $ 39 $ | $6$ | $39$ | $( 1,13,25,38,12,22,35, 9,20,33, 5,18,29, 3,15,27,37,11,24,34, 8,19,32, 4,17, 28, 2,14,26,39,10,23,36, 7,21,31, 6,16,30)$ |
| $ 26, 13 $ | $9$ | $26$ | $( 1,13,27,37,10,23,35, 9,19,32, 6,16,29, 3,14,26,38,12,24,34, 7,21,33, 5,17,28 )( 2,15,25,39,11,22,36, 8,20,31, 4,18,30)$ |
| $ 13, 13, 13 $ | $3$ | $13$ | $( 1,14,27,38,10,24,35, 7,19,33, 6,17,29)( 2,15,25,39,11,22,36, 8,20,31, 4,18, 30)( 3,13,26,37,12,23,34, 9,21,32, 5,16,28)$ |
| $ 39 $ | $6$ | $39$ | $( 1,22, 5,27, 8,28,10,31,13,35,18,37,19, 2,23, 6,25, 9,29,11,32,14,36,16,38, 20, 3,24, 4,26, 7,30,12,33,15,34,17,39,21)$ |
| $ 26, 13 $ | $9$ | $26$ | $( 1,22, 6,25, 7,30,10,31,14,36,17,39,19, 2,24, 4,27, 8,29,11,33,15,35,18,38,20 )( 3,23, 5,26, 9,28,12,32,13,34,16,37,21)$ |
| $ 13, 13, 13 $ | $3$ | $13$ | $( 1,24, 6,27, 7,29,10,33,14,35,17,38,19)( 2,22, 4,25, 8,30,11,31,15,36,18,39, 20)( 3,23, 5,26, 9,28,12,32,13,34,16,37,21)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $234=2 \cdot 3^{2} \cdot 13$ | 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: | 234.8 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);