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Magma
magma: G := TransitiveGroup(39, 47);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^3:C_{39}:C_6$ | ||
| 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: | (1,4,33)(2,6,31,3,5,32)(7,21,10)(8,20,11,9,19,12)(13,34,30,15,35,29)(14,36,28)(16,22,39)(17,24,37,18,23,38)(25,26), (1,7,26)(2,8,27)(3,9,25)(4,16,14,5,17,15,6,18,13)(10,35,28,11,36,29,12,34,30)(19,22,33,21,24,32,20,23,31) | 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 $234$: 39T12 $2106$: 27T422 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 13: $C_{13}:C_3$
Low degree siblings
39T47Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | 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 $ | $26$ | $3$ | $( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(22,23,24)(25,27,26) (28,29,30)(37,38,39)$ | |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,21,39,16,36,14,33,10,28, 7,27, 4,22)( 2,19,37,17,34,15,31,11,29, 8,25, 5, 23)( 3,20,38,18,35,13,32,12,30, 9,26, 6,24)$ | |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,39,36,33,28,27,22,21,16,14,10, 7, 4)( 2,37,34,31,29,25,23,19,17,15,11, 8, 5)( 3,38,35,32,30,26,24,20,18,13,12, 9, 6)$ | |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,36,28,22,16,10, 4,39,33,27,21,14, 7)( 2,34,29,23,17,11, 5,37,31,25,19,15, 8)( 3,35,30,24,18,12, 6,38,32,26,20,13, 9)$ | |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,28,16, 4,33,21, 7,36,22,10,39,27,14)( 2,29,17, 5,31,19, 8,34,23,11,37,25, 15)( 3,30,18, 6,32,20, 9,35,24,12,38,26,13)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $117$ | $3$ | $( 1,36,16)( 2,34,17)( 3,35,18)( 7,14,33)( 8,15,31)( 9,13,32)(10,22,21) (11,23,19)(12,24,20)(25,29,37)(26,30,38)(27,28,39)$ | |
| $ 9, 9, 9, 3, 3, 3, 3 $ | $234$ | $9$ | $( 1,36,18, 3,35,17, 2,34,16)( 4, 5, 6)( 7,13,32)( 8,14,33)( 9,15,31) (10,23,19,12,22,21,11,24,20)(25,30,39,26,28,37,27,29,38)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $117$ | $3$ | $( 1,16,36)( 2,17,34)( 3,18,35)( 7,33,14)( 8,31,15)( 9,32,13)(10,21,22) (11,19,23)(12,20,24)(25,37,29)(26,38,30)(27,39,28)$ | |
| $ 9, 9, 9, 3, 3, 3, 3 $ | $234$ | $9$ | $( 1,18,35, 3,17,34, 2,16,36)( 4, 5, 6)( 7,33,13)( 8,31,14)( 9,32,15) (10,21,23,12,20,22,11,19,24)(25,38,28,26,39,29,27,37,30)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $(10,11,12)(13,14,15)(16,18,17)(22,23,24)(28,30,29)(31,33,32)(37,39,38)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $( 7, 9, 8)(10,11,12)(16,17,18)(19,21,20)(22,24,23)(25,26,27)(28,30,29) (31,32,33)(34,35,36)(37,38,39)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)$ | |
| $ 39 $ | $162$ | $39$ | $( 1,21,39,18,34,15,32,11,30, 8,25, 5,23, 3,20,38,17,36,14,31,10,29, 7,27, 4, 22, 2,19,37,16,35,13,33,12,28, 9,26, 6,24)$ | |
| $ 39 $ | $162$ | $39$ | $( 1,39,35,32,29,27,22,19,17,14,11, 9, 6, 3,38,34,31,28,26,24,21,16,13,10, 8, 5, 2,37,36,33,30,25,23,20,18,15,12, 7, 4)$ | |
| $ 39 $ | $162$ | $39$ | $( 1,36,28,24,16,12, 4,39,32,25,19,15, 9, 3,35,30,23,18,11, 6,38,31,27,21,14, 8, 2,34,29,22,17,10, 5,37,33,26,20,13, 7)$ | |
| $ 39 $ | $162$ | $39$ | $( 1,28,18, 5,31,21, 7,36,22,11,38,25,15, 3,30,17, 4,33,20, 9,35,24,10,37,27, 14, 2,29,16, 6,32,19, 8,34,23,12,39,26,13)$ | |
| $ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ | $234$ | $9$ | $( 1,36,16, 3,35,18, 2,34,17)( 7,14,31)( 8,15,32)( 9,13,33)(10,23,20,12,22,19, 11,24,21)(25,29,39,26,30,37,27,28,38)$ | |
| $ 9, 9, 9, 3, 3, 3, 3 $ | $234$ | $9$ | $( 1,36,18, 2,34,16, 3,35,17)( 4, 5, 6)( 7,13,33)( 8,14,31)( 9,15,32) (10,24,21,11,22,19,12,23,20)(25,30,38,27,29,37,26,28,39)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $234$ | $3$ | $( 1,36,17)( 2,34,18)( 3,35,16)( 4, 6, 5)( 7,15,32)( 8,13,33)( 9,14,31) (10,22,19)(11,23,20)(12,24,21)(25,28,37)(26,29,38)(27,30,39)$ | |
| $ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ | $234$ | $9$ | $( 1,16,35, 3,18,34, 2,17,36)( 7,33,13)( 8,31,14)( 9,32,15)(10,19,23,12,21,22, 11,20,24)(25,37,28,26,38,29,27,39,30)$ | |
| $ 9, 9, 9, 3, 3, 3, 3 $ | $234$ | $9$ | $( 1,18,34, 2,16,35, 3,17,36)( 4, 5, 6)( 7,33,15)( 8,31,13)( 9,32,14) (10,19,24,11,20,22,12,21,23)(25,38,30,27,37,29,26,39,28)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $234$ | $3$ | $( 1,17,36)( 2,18,34)( 3,16,35)( 4, 6, 5)( 7,33,14)( 8,31,15)( 9,32,13) (10,19,22)(11,20,23)(12,21,24)(25,39,29)(26,37,30)(27,38,28)$ | |
| $ 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 $ | $81$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(13,15)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32) (34,35)(37,38)$ | |
| $ 26, 13 $ | $243$ | $26$ | $( 1,21,39,16,36,14,33,10,28, 7,27, 4,22)( 2,20,37,18,34,13,31,12,29, 9,25, 6, 23, 3,19,38,17,35,15,32,11,30, 8,26, 5,24)$ | |
| $ 26, 13 $ | $243$ | $26$ | $( 1,39,36,33,28,27,22,21,16,14,10, 7, 4)( 2,38,34,32,29,26,23,20,17,13,11, 9, 5, 3,37,35,31,30,25,24,19,18,15,12, 8, 6)$ | |
| $ 26, 13 $ | $243$ | $26$ | $( 1,36,28,22,16,10, 4,39,33,27,21,14, 7)( 2,35,29,24,17,12, 5,38,31,26,19,13, 8, 3,34,30,23,18,11, 6,37,32,25,20,15, 9)$ | |
| $ 26, 13 $ | $243$ | $26$ | $( 1,28,16, 4,33,21, 7,36,22,10,39,27,14)( 2,30,17, 6,31,20, 8,35,23,12,37,26, 15, 3,29,18, 5,32,19, 9,34,24,11,38,25,13)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $1053$ | $6$ | $( 1,36,16)( 2,35,17, 3,34,18)( 5, 6)( 7,14,33)( 8,13,31, 9,15,32)(10,22,21) (11,24,19,12,23,20)(25,30,37,26,29,38)(27,28,39)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $1053$ | $6$ | $( 1,16,36)( 2,18,34, 3,17,35)( 5, 6)( 7,33,14)( 8,32,15, 9,31,13)(10,21,22) (11,20,23,12,19,24)(25,38,29,26,37,30)(27,39,28)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $6318=2 \cdot 3^{5} \cdot 13$ | 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: | 6318.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);