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Magma
magma: G := TransitiveGroup(39, 38);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:(C_3\times C_6)$ | ||
| 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: | (1,9,3)(2,5,6)(4,10,12)(7,11,8)(14,24,23)(15,20,26)(17,25,19)(18,21,22)(27,33,35)(28,29,38)(30,34,31)(32,39,37), (1,32,16,3,37,14)(2,28,15)(4,33,26,13,36,17)(5,29,25,12,27,18)(6,38,24,11,31,19)(7,34,23,10,35,20)(8,30,22,9,39,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $18$: $C_6 \times C_3$ $78$: $C_{13}:C_6$ x 2 $234$: 39T13 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 13: None
Low degree siblings
39T38 x 3Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,19,24,16,21,26,18,23,15,20,25,17, 22)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20,26,19,25,18,24,17,23,16,22,15, 21)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24, 19)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,26,25,24,23,22,21,20,19,18,17,16, 15)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,17,20,23,26,16,19,22,25,15,18,21, 24)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,16,18,20,22,24,26,15,17,19,21,23, 25)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,15,16,17,18,19,20,21,22,23,24,25, 26)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,20,26,19,25,18,24,17,23,16,22,15, 21)(27,29,31,33,35,37,39,28,30,32,34,36,38)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23, 25)(27,29,31,33,35,37,39,28,30,32,34,36,38)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $169$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)(27,29,34)(28,38,37)(31,39,33)(32,35,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $169$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)(27,34,29)(28,37,38)(31,33,39)(32,36,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $13$ | $3$ | $( 1,16,37)( 2,15,28)( 3,14,32)( 4,26,36)( 5,25,27)( 6,24,31)( 7,23,35) ( 8,22,39)( 9,21,30)(10,20,34)(11,19,38)(12,18,29)(13,17,33)$ |
| $ 39 $ | $78$ | $39$ | $( 1,17,33, 3,15,28, 5,26,36, 7,24,31, 9,22,39,11,20,34,13,18,29, 2,16,37, 4, 14,32, 6,25,27, 8,23,35,10,21,30,12,19,38)$ |
| $ 39 $ | $78$ | $39$ | $( 1,18,29, 5,14,32, 9,23,35,13,19,38, 4,15,28, 8,24,31,12,20,34, 3,16,37, 7, 25,27,11,21,30, 2,17,33, 6,26,36,10,22,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $169$ | $3$ | $( 1,19,37)( 2,23,36)( 3,14,35)( 4,18,34)( 5,22,33)( 6,26,32)( 7,17,31) ( 8,21,30)( 9,25,29)(10,16,28)(11,20,27)(12,24,39)(13,15,38)$ |
| $ 39 $ | $78$ | $39$ | $( 1,20,29, 8,25,34, 2,17,39, 9,22,31, 3,14,36,10,19,28, 4,24,33,11,16,38, 5, 21,30,12,26,35, 6,18,27,13,23,32, 7,15,37)$ |
| $ 39 $ | $78$ | $39$ | $( 1,21,30, 2,18,27, 3,15,37, 4,25,34, 5,22,31, 6,19,28, 7,16,38, 8,26,35, 9, 23,32,10,20,29,11,17,39,12,14,36,13,24,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $13$ | $3$ | $( 1,19,28)( 2,16,38)( 3,26,35)( 4,23,32)( 5,20,29)( 6,17,39)( 7,14,36) ( 8,24,33)( 9,21,30)(10,18,27)(11,15,37)(12,25,34)(13,22,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $13$ | $3$ | $( 1,37,16)( 2,28,15)( 3,32,14)( 4,36,26)( 5,27,25)( 6,31,24)( 7,35,23) ( 8,39,22)( 9,30,21)(10,34,20)(11,38,19)(12,29,18)(13,33,17)$ |
| $ 39 $ | $78$ | $39$ | $( 1,37,17, 3,32,15, 5,27,26, 7,35,24, 9,30,22,11,38,20,13,33,18, 2,28,16, 4, 36,14, 6,31,25, 8,39,23,10,34,21,12,29,19)$ |
| $ 39 $ | $78$ | $39$ | $( 1,37,18, 5,27,14, 9,30,23,13,33,19, 4,36,15, 8,39,24,12,29,20, 3,32,16, 7, 35,25,11,38,21, 2,28,17, 6,31,26,10,34,22)$ |
| $ 39 $ | $78$ | $39$ | $( 1,28,23, 3,35,17, 5,29,24, 7,36,18, 9,30,25,11,37,19,13,31,26, 2,38,20, 4, 32,14, 6,39,21, 8,33,15,10,27,22,12,34,16)$ |
| $ 39 $ | $78$ | $39$ | $( 1,28,24,10,27,23, 6,39,22, 2,38,21,11,37,20, 7,36,19, 3,35,18,12,34,17, 8, 33,16, 4,32,15,13,31,14, 9,30,26, 5,29,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $13$ | $3$ | $( 1,28,19)( 2,38,16)( 3,35,26)( 4,32,23)( 5,29,20)( 6,39,17)( 7,36,14) ( 8,33,24)( 9,30,21)(10,27,18)(11,37,15)(12,34,25)(13,31,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $169$ | $3$ | $( 1,38,16)( 2,37,20)( 3,36,24)( 4,35,15)( 5,34,19)( 6,33,23)( 7,32,14) ( 8,31,18)( 9,30,22)(10,29,26)(11,28,17)(12,27,21)(13,39,25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)(27,33)(28,32)(29,31)(34,39)(35,38)(36,37)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1, 1 $ | $169$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21) (27,31,34,33,29,39)(28,35,37,32,38,36)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1, 1 $ | $169$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22) (27,39,29,33,34,31)(28,36,38,32,37,35)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,16,27,13,15,36)( 2,17,31,12,14,32)( 3,18,35,11,26,28)( 4,19,39,10,25,37) ( 5,20,30, 9,24,33)( 6,21,34, 8,23,29)( 7,22,38)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,19,33,11,18,36)( 2,15,32,10,22,37)( 3,24,31, 9,26,38)( 4,20,30, 8,17,39) ( 5,16,29, 7,21,27)( 6,25,28)(12,14,35,13,23,34)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,20,30,12,14,36)( 2,23,27,11,24,39)( 3,26,37,10,21,29)( 4,16,34, 9,18,32) ( 5,19,31, 8,15,35)( 6,22,28, 7,25,38)(13,17,33)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,37,26, 2,33,25)( 3,29,24,13,28,14)( 4,38,23,12,32,15)( 5,34,22,11,36,16) ( 6,30,21,10,27,17)( 7,39,20, 9,31,18)( 8,35,19)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,28,14, 6,30,25)( 2,31,24, 5,27,15)( 3,34,21, 4,37,18)( 7,33,22,13,38,17) ( 8,36,19,12,35,20)( 9,39,16,11,32,23)(10,29,26)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,38,15,10,34,25)( 2,39,19, 9,33,21)( 3,27,23, 8,32,17)( 4,28,14, 7,31,26) ( 5,29,18, 6,30,22)(11,35,16,13,37,24)(12,36,20)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $3042=2 \cdot 3^{2} \cdot 13^{2}$ | 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: | 3042.f | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);