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Magma
magma: G := TransitiveGroup(39, 45);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:C_3:C_{12}$ | ||
| 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: | (1,35,18)(2,34,22)(3,33,26)(4,32,17)(5,31,21)(6,30,25)(7,29,16)(8,28,20)(9,27,24)(10,39,15)(11,38,19)(12,37,23)(13,36,14), (1,33,12,34,5,31,13,27,2,39,9,29)(3,32,6,37,10,35,11,28,8,36,4,38)(7,30)(14,15,22,19,24,20,18,17,23,26,21,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $S_3$, $C_6$ $12$: $C_{12}$, $C_3 : C_4$ $18$: $S_3\times C_3$ $36$: $C_3\times (C_3 : C_4)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: None
Low degree siblings
26T41, 39T45Siblings 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, 13 $ | $36$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,31,35,39,30,34,38,29,33,37,28,32,36)$ |
| $ 13, 13, 13 $ | $36$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,35,30,38,33,28,36,31,39,34,29,37,32)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $13$ | $(14,18,22,26,17,21,25,16,20,24,15,19,23)(27,28,29,30,31,32,33,34,35,36,37,38, 39)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,16,18,20,22,24,26,15,17,19,21,23, 25)(27,32,37,29,34,39,31,36,28,33,38,30,35)$ |
| $ 13, 13, 13 $ | $36$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,15,16,17,18,19,20,21,22,23,24,25, 26)(27,34,28,35,29,36,30,37,31,38,32,39,33)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,20,26,19,25,18,24,17,23,16,22,15, 21)(27,33,39,32,38,31,37,30,36,29,35,28,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $338$ | $3$ | $( 1,35,18)( 2,34,22)( 3,33,26)( 4,32,17)( 5,31,21)( 6,30,25)( 7,29,16) ( 8,28,20)( 9,27,24)(10,39,15)(11,38,19)(12,37,23)(13,36,14)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $169$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)(27,29,34)(28,38,37)(31,39,33)(32,35,36)$ |
| $ 39 $ | $156$ | $39$ | $( 1,35,14,13,31,15,12,27,16,11,36,17,10,32,18, 9,28,19, 8,37,20, 7,33,21, 6, 29,22, 5,38,23, 4,34,24, 3,30,25, 2,39,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1,39,24)( 2,30,23)( 3,34,22)( 4,38,21)( 5,29,20)( 6,33,19)( 7,37,18) ( 8,28,17)( 9,32,16)(10,36,15)(11,27,14)(12,31,26)(13,35,25)$ |
| $ 39 $ | $156$ | $39$ | $( 1,28,16, 8,30,22, 2,32,15, 9,34,21, 3,36,14,10,38,20, 4,27,26,11,29,19, 5, 31,25,12,33,18, 6,35,24,13,37,17, 7,39,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $169$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)(27,34,29)(28,37,38)(31,33,39)(32,36,35)$ |
| $ 39 $ | $156$ | $39$ | $( 1,35,17,12,28,23,10,34,16, 8,27,22, 6,33,15, 4,39,21, 2,32,14,13,38,20,11, 31,26, 9,37,19, 7,30,25, 5,36,18, 3,29,24)$ |
| $ 39 $ | $156$ | $39$ | $( 1,39,19, 5,27,20, 9,28,21,13,29,22, 4,30,23, 8,31,24,12,32,25, 3,33,26, 7, 34,14,11,35,15, 2,36,16, 6,37,17,10,38,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1,32,22)( 2,29,19)( 3,39,16)( 4,36,26)( 5,33,23)( 6,30,20)( 7,27,17) ( 8,37,14)( 9,34,24)(10,31,21)(11,28,18)(12,38,15)(13,35,25)$ |
| $ 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 $ | $338$ | $6$ | $( 1,35,20, 2,38,17)( 3,28,14,13,32,23)( 4,31,24,12,29,26)( 5,34,21,11,39,16) ( 6,37,18,10,36,19)( 7,27,15, 9,33,22)( 8,30,25)$ |
| $ 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, 3 $ | $338$ | $6$ | $( 1,35,19, 2,36,23)( 3,37,14,13,34,15)( 4,38,18,12,33,24)( 5,39,22,11,32,20) ( 6,27,26,10,31,16)( 7,28,17, 9,30,25)( 8,29,21)$ |
| $ 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, 3 $ | $338$ | $6$ | $( 1,35,23, 9,29,15)( 2,31,22, 8,33,16)( 3,27,21, 7,37,17)( 4,36,20, 6,28,18) ( 5,32,19)(10,38,14,13,39,24)(11,34,26,12,30,25)$ |
| $ 12, 12, 12, 2, 1 $ | $507$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(14,30)(15,38,24,32,23,37,26,35,17,28,18, 36)(16,33,21,34,19,31,25,27,20,39,22,29)$ |
| $ 12, 12, 12, 2, 1 $ | $507$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(14,30)(15,37,18,32,17,38,26,36,23,28,24, 35)(16,31,22,34,20,33,25,29,19,39,21,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $507$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(14,30)(15,28,26,32)(16,39,25,34) (17,37,24,36)(18,35,23,38)(19,33,22,27)(20,31,21,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $507$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(14,30)(15,32,26,28)(16,34,25,39) (17,36,24,37)(18,38,23,35)(19,27,22,33)(20,29,21,31)$ |
| $ 12, 12, 12, 2, 1 $ | $507$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(14,30)(15,35,24,28,23,36,26,38,17,32,18, 37)(16,27,21,39,19,29,25,33,20,34,22,31)$ |
| $ 12, 12, 12, 2, 1 $ | $507$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(14,30)(15,36,18,28,17,35,26,37,23,32,24, 38)(16,29,22,39,20,27,25,31,19,34,21,33)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $6084=2^{2} \cdot 3^{2} \cdot 13^{2}$ | 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: | 6084.j | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);