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Magma
magma: G := TransitiveGroup(39, 39);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:(C_3\times S_3)$ | ||
| 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: | (1,30,11,32,7,27)(2,38,3,29,5,33)(4,28)(6,31,9,34,10,37)(8,35,12,39,13,36)(14,19,22)(15,16,18)(17,25,23)(20,26,21), (1,4,5,12,2,7,8,13,10,6,3,11,9)(14,32,16,37,21,27,18,33,24,34,22,31,23,35,19,39,15,30,17,29,25,36,26,38,20,28) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: None
Low degree siblings
26T27, 39T40Siblings 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 $ | $18$ | $13$ | $( 1, 6, 7, 4, 3, 8, 5,11,13,12, 9,10, 2)(14,18,23,17,20,21,22,15,26,16,24,19, 25)(27,30,37,39,32,35,28,31,38,34,36,33,29)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,23,20,22,26,24,25,18,17,21,15,16, 19)(27,37,32,28,38,36,29,30,39,35,31,34,33)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1, 3,13, 2, 4,11,10, 7, 5, 9, 6, 8,12)(14,20,26,25,17,15,19,23,22,24,18,21, 16)(27,32,38,29,39,31,33,37,28,36,30,35,34)$ |
| $ 13, 13, 13 $ | $18$ | $13$ | $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(14,26,17,19,22,18,16,20,25,15,23,24, 21)(27,38,39,33,28,30,34,32,29,31,37,36,35)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $13$ | $(14,18,23,17,20,21,22,15,26,16,24,19,25)(27,28,29,35,33,32,36,39,34,37,38,30, 31)$ |
| $ 13, 13, 13 $ | $9$ | $13$ | $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,17,22,16,25,23,21,26,19,18,20,15, 24)(27,38,39,33,28,30,34,32,29,31,37,36,35)$ |
| $ 13, 13, 13 $ | $9$ | $13$ | $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(14,16,21,18,24,22,23,19,15,17,25,26, 20)(27,30,37,39,32,35,28,31,38,34,36,33,29)$ |
| $ 13, 13, 13 $ | $9$ | $13$ | $( 1, 4, 5,12, 2, 7, 8,13,10, 6, 3,11, 9)(14,20,26,25,17,15,19,23,22,24,18,21, 16)(27,34,35,30,36,28,37,33,31,39,29,38,32)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 5, 2, 8,10, 3, 9, 4,12, 7,13, 6,11)(14,15,18,26,23,16,17,24,20,19,21,25, 22)(27,29,33,36,34,38,31,28,35,32,39,37,30)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,12, 8, 6, 9, 5, 7,10,11, 4, 2,13, 3)(14,24,15,20,18,19,26,21,23,25,16,22, 17)(27,37,32,28,38,36,29,30,39,35,31,34,33)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $13$ | $( 1,11, 6,13, 7,12, 4, 9, 3,10, 8, 2, 5)(14,26,17,19,22,18,16,20,25,15,23,24, 21)$ |
| $ 13, 13, 13 $ | $9$ | $13$ | $( 1, 5, 2, 8,10, 3, 9, 4,12, 7,13, 6,11)(14,26,17,19,22,18,16,20,25,15,23,24, 21)(27,35,36,37,31,29,32,34,30,28,33,39,38)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 2,10, 9,12,13,11, 5, 8, 3, 4, 7, 6)(14,18,23,17,20,21,22,15,26,16,24,19, 25)(27,33,34,31,35,39,30,29,36,38,28,32,37)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,10,12,11, 8, 4, 6, 2, 9,13, 5, 3, 7)(14,23,20,22,26,24,25,18,17,21,15,16, 19)(27,34,35,30,36,28,37,33,31,39,29,38,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $169$ | $3$ | $( 2, 9, 3)( 4,12, 6)( 5, 8, 7)(10,11,13)(15,26,19)(16,18,17)(20,25,24) (21,23,22)(28,35,37)(29,36,32)(30,39,34)(31,38,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $169$ | $3$ | $( 2, 3, 9)( 4, 6,12)( 5, 7, 8)(10,13,11)(15,19,26)(16,17,18)(20,24,25) (21,22,23)(28,37,35)(29,32,36)(30,34,39)(31,33,38)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $338$ | $3$ | $( 1,39,23)( 2,33,21)( 3,36,17)( 4,35,22)( 5,31,24)( 6,38,25)( 7,27,16) ( 8,37,14)( 9,30,19)(10,28,26)(11,29,15)(12,34,18)(13,32,20)$ |
| $ 39 $ | $78$ | $39$ | $( 1,34,17, 2,31,20,10,35,21, 9,39,22,12,30,15,13,29,26,11,36,16, 5,38,24, 8, 28,19, 3,32,25, 4,37,14, 7,27,18, 6,33,23)$ |
| $ 39 $ | $78$ | $39$ | $( 1,36,24, 5,34,20, 2,38,19, 8,31,21,10,28,25, 3,35,22, 9,32,14, 4,39,15,12, 37,18, 7,30,26,13,27,23, 6,29,16,11,33,17)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1,33,20)( 2,34,21)( 3,28,14)( 4,32,18)( 5,36,19)( 6,27,17)( 7,37,23) ( 8,38,25)( 9,35,15)(10,31,22)(11,29,24)(12,39,26)(13,30,16)$ |
| $ 39 $ | $78$ | $39$ | $( 1,27,21, 6,37,20, 7,32,17, 4,28,23, 3,38,18, 8,36,14, 5,29,25,11,30,19,13, 39,24,12,35,16, 9,31,26,10,34,15, 2,33,22)$ |
| $ 39 $ | $78$ | $39$ | $( 1,29,19,11,27,20, 6,30,24,13,37,17, 7,39,16,12,32,23, 4,35,26, 9,28,18, 3, 31,15,10,38,14, 8,34,22, 2,36,25, 5,33,21)$ |
| $ 39 $ | $78$ | $39$ | $( 1,30,26,13,36,18, 4,28,15,10,37,14, 5,33,22, 6,31,25,12,39,21, 3,29,19, 2, 38,20,11,32,24, 7,27,17, 9,34,16, 8,35,23)$ |
| $ 39 $ | $78$ | $39$ | $( 1,37,18, 3,27,20,13,33,15, 2,34,24, 4,31,14,11,35,17,10,39,22, 7,30,16, 5, 29,25, 9,36,23, 6,38,21, 8,28,26,12,32,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $26$ | $3$ | $( 1,39,15)( 2,36,17)( 3,30,24)( 4,38,22)( 5,27,21)( 6,34,19)( 7,37,23) ( 8,31,18)( 9,33,26)(10,32,25)(11,28,16)(12,35,20)(13,29,14)$ |
| $ 39 $ | $78$ | $39$ | $( 1,35,22, 8,34,14, 9,27,15, 7,32,18,11,38,26, 2,29,23, 3,39,16,12,31,17, 6, 33,24, 5,37,20,10,28,19, 4,36,21,13,30,25)$ |
| $ 39 $ | $78$ | $39$ | $( 1,32,14,12,28,24, 8,38,15, 6,36,20, 9,29,18, 5,30,19, 7,39,26,10,35,21,11, 31,23, 4,34,25, 2,33,16,13,27,22, 3,37,17)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $507$ | $6$ | $( 2, 3, 9)( 4, 6,12)( 5, 7, 8)(10,13,11)(14,32,25,28,16,29)(15,39) (17,33,26,30,24,36)(18,37,23,27,21,31)(19,38,22,35,20,34)$ |
| $ 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 $ | $39$ | $2$ | $(14,29)(15,34)(16,36)(17,31)(18,28)(19,33)(20,30)(21,38)(22,37)(23,27)(24,32) (25,35)(26,39)$ |
| $ 26, 13 $ | $117$ | $26$ | $( 1, 6, 7, 4, 3, 8, 5,11,13,12, 9,10, 2)(14,27,17,38,22,39,16,33,25,28,23,30, 21,34,26,32,19,29,18,31,20,37,15,36,24,35)$ |
| $ 26, 13 $ | $117$ | $26$ | $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,30,22,32,25,31,21,36,19,27,20,39, 24,28,17,34,16,29,23,37,26,35,18,38,15,33)$ |
| $ 26, 13 $ | $117$ | $26$ | $( 1, 3,13, 2, 4,11,10, 7, 5, 9, 6, 8,12)(14,39,25,34,19,37,24,38,16,30,26,31, 15,27,22,28,21,29,20,35,17,33,23,32,18,36)$ |
| $ 26, 13 $ | $117$ | $26$ | $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(14,31,19,28,16,35,15,32,21,39,17,37, 18,30,25,27,24,29,26,33,22,36,20,34,23,38)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $507$ | $6$ | $( 2, 9, 3)( 4,12, 6)( 5, 8, 7)(10,11,13)(14,36,18,35,24,29)(15,30,25,37,21,33) (16,32,20,39,19,31)(17,38,23,27,22,28)(26,34)$ |
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.e | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);