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Magma
magma: G := TransitiveGroup(39, 10);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}:C_{12}$ | ||
| 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: | (1,21,9,28,3,20,8,30,2,19,7,29)(4,6,5)(10,15,38,34,12,14,37,36,11,13,39,35)(16,23,33,27,18,22,32,26,17,24,31,25), (1,34,28,22,17,10,5,37,33,25,19,13,8)(2,35,29,23,18,11,6,38,31,26,20,14,9)(3,36,30,24,16,12,4,39,32,27,21,15,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $12$: $C_{12}$ $52$: $C_{13}:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 13: $C_{13}:C_4$
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 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $13$ | $4$ | $( 4,16,39,27)( 5,17,37,25)( 6,18,38,26)( 7,32,36,12)( 8,33,34,10)( 9,31,35,11) (13,22,28,19)(14,23,29,20)(15,24,30,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $13$ | $4$ | $( 4,27,39,16)( 5,25,37,17)( 6,26,38,18)( 7,12,36,32)( 8,10,34,33)( 9,11,35,31) (13,19,28,22)(14,20,29,23)(15,21,30,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $13$ | $2$ | $( 4,39)( 5,37)( 6,38)( 7,36)( 8,34)( 9,35)(10,33)(11,31)(12,32)(13,28)(14,29) (15,30)(16,27)(17,25)(18,26)(19,22)(20,23)(21,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$ |
| $ 12, 12, 12, 3 $ | $13$ | $12$ | $( 1, 2, 3)( 4,17,38,27, 5,18,39,25, 6,16,37,26)( 7,33,35,12, 8,31,36,10, 9,32, 34,11)(13,23,30,19,14,24,28,20,15,22,29,21)$ |
| $ 12, 12, 12, 3 $ | $13$ | $12$ | $( 1, 2, 3)( 4,25,38,16, 5,26,39,17, 6,27,37,18)( 7,10,35,32, 8,11,36,33, 9,12, 34,31)(13,20,30,22,14,21,28,23,15,19,29,24)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $13$ | $6$ | $( 1, 2, 3)( 4,37, 6,39, 5,38)( 7,34, 9,36, 8,35)(10,31,12,33,11,32) (13,29,15,28,14,30)(16,25,18,27,17,26)(19,23,21,22,20,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)$ |
| $ 12, 12, 12, 3 $ | $13$ | $12$ | $( 1, 3, 2)( 4,18,37,27, 6,17,39,26, 5,16,38,25)( 7,31,34,12, 9,33,36,11, 8,32, 35,10)(13,24,29,19,15,23,28,21,14,22,30,20)$ |
| $ 12, 12, 12, 3 $ | $13$ | $12$ | $( 1, 3, 2)( 4,26,37,16, 6,25,39,18, 5,27,38,17)( 7,11,34,32, 9,10,36,31, 8,12, 35,33)(13,21,29,22,15,20,28,24,14,19,30,23)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $13$ | $6$ | $( 1, 3, 2)( 4,38, 5,39, 6,37)( 7,35, 8,36, 9,34)(10,32,11,33,12,31) (13,30,14,28,15,29)(16,26,17,27,18,25)(19,24,20,22,21,23)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 4, 9,10,15,18,19,24,26,28,32,35,37, 3, 6, 8,12,14,17,21,23,25,30,31,34, 39, 2, 5, 7,11,13,16,20,22,27,29,33,36,38)$ |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1, 5, 8,10,13,17,19,22,25,28,33,34,37)( 2, 6, 9,11,14,18,20,23,26,29,31,35, 38)( 3, 4, 7,12,15,16,21,24,27,30,32,36,39)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 6, 7,10,14,16,19,23,27,28,31,36,37, 2, 4, 8,11,15,17,20,24,25,29,32,34, 38, 3, 5, 9,12,13,18,21,22,26,30,33,35,39)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 7,14,19,27,31,37, 4,11,17,24,29,34, 3, 9,13,21,26,33,39, 6,10,16,23,28, 36, 2, 8,15,20,25,32,38, 5,12,18,22,30,35)$ |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1, 8,13,19,25,33,37, 5,10,17,22,28,34)( 2, 9,14,20,26,31,38, 6,11,18,23,29, 35)( 3, 7,15,21,27,32,39, 4,12,16,24,30,36)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 9,15,19,26,32,37, 6,12,17,23,30,34, 2, 7,13,20,27,33,38, 4,10,18,24,28, 35, 3, 8,14,21,25,31,39, 5,11,16,22,29,36)$ |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1,13,25,37,10,22,34, 8,19,33, 5,17,28)( 2,14,26,38,11,23,35, 9,20,31, 6,18, 29)( 3,15,27,39,12,24,36, 7,21,32, 4,16,30)$ |
| $ 39 $ | $4$ | $39$ | $( 1,14,27,37,11,24,34, 9,21,33, 6,16,28, 2,15,25,38,12,22,35, 7,19,31, 4,17, 29, 3,13,26,39,10,23,36, 8,20,32, 5,18,30)$ |
| $ 39 $ | $4$ | $39$ | $( 1,15,26,37,12,23,34, 7,20,33, 4,18,28, 3,14,25,39,11,22,36, 9,19,32, 6,17, 30, 2,13,27,38,10,24,35, 8,21,31, 5,16,29)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $156=2^{2} \cdot 3 \cdot 13$ | 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: | 156.9 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);