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Magma
magma: G := TransitiveGroup(39, 18);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{39}:C_{12}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,33,8,36,28,26,5,14,37,11,16,21)(2,32,9,35,30,25,6,13,38,12,17,20)(3,31,7,34,29,27,4,15,39,10,18,19)(23,24), (1,3)(4,16,39,25)(5,18,37,27)(6,17,38,26)(7,32,34,12)(8,31,35,10)(9,33,36,11)(13,22,28,19)(14,23,30,21)(15,24,29,20) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $S_3$, $C_6$ $12$: $C_{12}$, $C_3 : C_4$ $18$: $S_3\times C_3$ $36$: $C_3\times (C_3 : C_4)$ $156$: $F_{13}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: $F_{13}$
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 $ | $13$ | $3$ | $( 4,10,29)( 5,12,28)( 6,11,30)( 7,19,18)( 8,20,16)( 9,21,17)(13,37,32) (14,38,33)(15,39,31)(22,27,34)(23,26,36)(24,25,35)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1, 1 $ | $13$ | $6$ | $( 4,15,10,39,29,31)( 5,13,12,37,28,32)( 6,14,11,38,30,33)( 7,27,19,34,18,22) ( 8,25,20,35,16,24)( 9,26,21,36,17,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $13$ | $3$ | $( 4,29,10)( 5,28,12)( 6,30,11)( 7,18,19)( 8,16,20)( 9,17,21)(13,32,37) (14,33,38)(15,31,39)(22,34,27)(23,36,26)(24,35,25)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1, 1 $ | $13$ | $6$ | $( 4,31,29,39,10,15)( 5,32,28,37,12,13)( 6,33,30,38,11,14)( 7,22,18,34,19,27) ( 8,24,16,35,20,25)( 9,23,17,36,21,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $13$ | $2$ | $( 4,39)( 5,37)( 6,38)( 7,34)( 8,35)( 9,36)(10,31)(11,33)(12,32)(13,28)(14,30) (15,29)(16,25)(17,26)(18,27)(19,22)(20,24)(21,23)$ |
| $ 12, 12, 12, 2, 1 $ | $39$ | $12$ | $( 2, 3)( 4, 9,15,26,10,21,39,36,29,17,31,23)( 5, 8,13,25,12,20,37,35,28,16,32, 24)( 6, 7,14,27,11,19,38,34,30,18,33,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $39$ | $4$ | $( 2, 3)( 4,17,39,26)( 5,16,37,25)( 6,18,38,27)( 7,33,34,11)( 8,32,35,12) ( 9,31,36,10)(13,24,28,20)(14,22,30,19)(15,23,29,21)$ |
| $ 12, 12, 12, 2, 1 $ | $39$ | $12$ | $( 2, 3)( 4,21,31,26,29, 9,39,23,10,17,15,36)( 5,20,32,25,28, 8,37,24,12,16,13, 35)( 6,19,33,27,30, 7,38,22,11,18,14,34)$ |
| $ 12, 12, 12, 2, 1 $ | $39$ | $12$ | $( 2, 3)( 4,23,31,17,29,36,39,21,10,26,15, 9)( 5,24,32,16,28,35,37,20,12,25,13, 8)( 6,22,33,18,30,34,38,19,11,27,14, 7)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $39$ | $4$ | $( 2, 3)( 4,26,39,17)( 5,25,37,16)( 6,27,38,18)( 7,11,34,33)( 8,12,35,32) ( 9,10,36,31)(13,20,28,24)(14,19,30,22)(15,21,29,23)$ |
| $ 12, 12, 12, 2, 1 $ | $39$ | $12$ | $( 2, 3)( 4,36,15,17,10,23,39, 9,29,26,31,21)( 5,35,13,16,12,24,37, 8,28,25,32, 20)( 6,34,14,18,11,22,38, 7,30,27,33,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,14,15)(16,17,18)(19,20,21) (22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,35,36)(37,38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1, 2, 3)( 4,12,30)( 5,11,29)( 6,10,28)( 7,20,17)( 8,21,18)( 9,19,16) (13,38,31)(14,39,32)(15,37,33)(22,25,36)(23,27,35)(24,26,34)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $26$ | $6$ | $( 1, 2, 3)( 4,13,11,39,28,33)( 5,14,10,37,30,31)( 6,15,12,38,29,32) ( 7,25,21,34,16,23)( 8,26,19,35,17,22)( 9,27,20,36,18,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1, 2, 3)( 4,28,11)( 5,30,10)( 6,29,12)( 7,16,21)( 8,17,19)( 9,18,20) (13,33,39)(14,31,37)(15,32,38)(22,35,26)(23,34,25)(24,36,27)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $26$ | $6$ | $( 1, 2, 3)( 4,32,30,39,12,14)( 5,33,29,37,11,15)( 6,31,28,38,10,13) ( 7,24,17,34,20,26)( 8,23,18,35,21,27)( 9,22,16,36,19,25)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $26$ | $6$ | $( 1, 2, 3)( 4,37, 6,39, 5,38)( 7,35, 9,34, 8,36)(10,32,11,31,12,33) (13,30,15,28,14,29)(16,26,18,25,17,27)(19,24,21,22,20,23)$ |
| $ 39 $ | $12$ | $39$ | $( 1, 4, 9,12,15,17,20,22,26,28,31,36,37, 3, 6, 8,10,14,16,19,23,25,29,33,35, 39, 2, 5, 7,11,13,18,21,24,27,30,32,34,38)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 5, 8,12,13,16,20,24,25,28,32,35,37)( 2, 6, 9,11,14,17,21,23,26,30,33,36, 38)( 3, 4, 7,10,15,18,19,22,27,29,31,34,39)$ |
| $ 39 $ | $12$ | $39$ | $( 1, 6, 7,12,14,18,20,23,27,28,33,34,37, 2, 4, 8,11,15,16,21,22,25,30,31,35, 38, 3, 5, 9,10,13,17,19,24,26,29,32,36,39)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $468=2^{2} \cdot 3^{2} \cdot 13$ | 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: | 468.30 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);