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Magma
magma: G := TransitiveGroup(39, 27);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:S_3$ | ||
| 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: | (1,33,18)(2,28,20)(3,35,23)(4,27,21)(5,37,16)(6,39,19)(7,38,26)(8,36,25)(9,34,24)(10,30,15)(11,31,22)(12,32,14)(13,29,17), (1,15,3,22,13,21,2,20,4,17,11,23,10,18,7,14,5,25,9,19,6,24,8,16,12,26)(27,39)(28,36)(29,32)(30,37)(31,34)(33,35) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: None
Low degree siblings
26T14, 39T26Siblings 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 $ | $6$ | $13$ | $( 1,10,12,11, 8, 4, 6, 2, 9,13, 5, 3, 7)(14,16,21,18,24,22,23,19,15,17,25,26, 20)(27,29,33,36,34,38,31,28,35,32,39,37,30)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,12, 8, 6, 9, 5, 7,10,11, 4, 2,13, 3)(14,21,24,23,15,25,20,16,18,22,19,17, 26)(27,33,34,31,35,39,30,29,36,38,28,32,37)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 8, 9, 7,11, 2, 3,12, 6, 5,10, 4,13)(14,24,15,20,18,19,26,21,23,25,16,22, 17)(27,34,35,30,36,28,37,33,31,39,29,38,32)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 9,11, 3, 6,10,13, 8, 7, 2,12, 5, 4)(14,15,18,26,23,16,17,24,20,19,21,25, 22)(27,35,36,37,31,29,32,34,30,28,33,39,38)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,11, 6,13, 7,12, 4, 9, 3,10, 8, 2, 5)(14,18,23,17,20,21,22,15,26,16,24,19, 25)(27,36,31,32,30,33,38,35,37,29,34,28,39)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 6, 7, 4, 3, 8, 5,11,13,12, 9,10, 2)(14,23,20,22,26,24,25,18,17,21,15,16, 19)(27,31,30,38,37,34,39,36,32,33,35,29,28)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,20,26,25,17,15,19,23,22,24,18,21, 16)(27,30,37,39,32,35,28,31,38,34,36,33,29)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 3,13, 2, 4,11,10, 7, 5, 9, 6, 8,12)(14,26,17,19,22,18,16,20,25,15,23,24, 21)(27,37,32,28,38,36,29,30,39,35,31,34,33)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(14,17,22,16,25,23,21,26,19,18,20,15, 24)(27,32,38,29,39,31,33,37,28,36,30,35,34)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 4, 5,12, 2, 7, 8,13,10, 6, 3,11, 9)(14,22,25,21,19,20,24,17,16,23,26,18, 15)(27,38,39,33,28,30,34,32,29,31,37,36,35)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 5, 2, 8,10, 3, 9, 4,12, 7,13, 6,11)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,39,28,34,29,37,35,38,33,30,32,31,36)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 2,10, 9,12,13,11, 5, 8, 3, 4, 7, 6)(14,19,16,15,21,17,18,25,24,26,22,20, 23)(27,28,29,35,33,32,36,39,34,37,38,30,31)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,18,23,17,20,21,22,15,26,16,24,19,25)(27,28,29,35,33,32,36,39,34,37,38,30, 31)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,10,12,11, 8, 4, 6, 2, 9,13, 5, 3, 7)(14,24,15,20,18,19,26,21,23,25,16,22, 17)(27,35,36,37,31,29,32,34,30,28,33,39,38)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 8, 9, 7,11, 2, 3,12, 6, 5,10, 4,13)(14,19,16,15,21,17,18,25,24,26,22,20, 23)(27,37,32,28,38,36,29,30,39,35,31,34,33)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 9,11, 3, 6,10,13, 8, 7, 2,12, 5, 4)(14,26,17,19,22,18,16,20,25,15,23,24, 21)(27,33,34,31,35,39,30,29,36,38,28,32,37)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,11, 6,13, 7,12, 4, 9, 3,10, 8, 2, 5)(14,23,20,22,26,24,25,18,17,21,15,16, 19)(27,39,28,34,29,37,35,38,33,30,32,31,36)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $( 1, 6, 7, 4, 3, 8, 5,11,13,12, 9,10, 2)(14,17,22,16,25,23,21,26,19,18,20,15, 24)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 3,13, 2, 4,11,10, 7, 5, 9, 6, 8,12)(14,16,21,18,24,22,23,19,15,17,25,26, 20)(27,38,39,33,28,30,34,32,29,31,37,36,35)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $( 1, 5, 2, 8,10, 3, 9, 4,12, 7,13, 6,11)(27,34,35,30,36,28,37,33,31,39,29,38, 32)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,23,20,22,26,24,25,18,17,21,15,16,19)(27,29,33,36,34,38,31,28,35,32,39,37, 30)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,12, 8, 6, 9, 5, 7,10,11, 4, 2,13, 3)(14,15,18,26,23,16,17,24,20,19,21,25, 22)(27,36,31,32,30,33,38,35,37,29,34,28,39)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,11, 6,13, 7,12, 4, 9, 3,10, 8, 2, 5)(14,17,22,16,25,23,21,26,19,18,20,15, 24)(27,34,35,30,36,28,37,33,31,39,29,38,32)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,22,25,21,19,20,24,17,16,23,26,18, 15)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(14,21,24,23,15,25,20,16,18,22,19,17, 26)(27,39,28,34,29,37,35,38,33,30,32,31,36)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,20,26,25,17,15,19,23,22,24,18,21,16)(27,33,34,31,35,39,30,29,36,38,28,32, 37)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 8, 9, 7,11, 2, 3,12, 6, 5,10, 4,13)(14,18,23,17,20,21,22,15,26,16,24,19, 25)(27,31,30,38,37,34,39,36,32,33,35,29,28)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 4, 5,12, 2, 7, 8,13,10, 6, 3,11, 9)(14,24,15,20,18,19,26,21,23,25,16,22, 17)(27,28,29,35,33,32,36,39,34,37,38,30,31)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,26,17,19,22,18,16,20,25,15,23,24,21)(27,34,35,30,36,28,37,33,31,39,29,38, 32)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,17,22,16,25,23,21,26,19,18,20,15,24)(27,35,36,37,31,29,32,34,30,28,33,39, 38)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,22,25,21,19,20,24,17,16,23,26,18,15)(27,36,31,32,30,33,38,35,37,29,34,28, 39)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,25,19,24,16,26,15,22,21,20,17,23,18)(27,31,30,38,37,34,39,36,32,33,35,29, 28)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,19,16,15,21,17,18,25,24,26,22,20,23)(27,30,37,39,32,35,28,31,38,34,36,33, 29)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $13$ | $(14,16,21,18,24,22,23,19,15,17,25,26,20)(27,37,32,28,38,36,29,30,39,35,31,34, 33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $338$ | $3$ | $( 1,33,18)( 2,28,20)( 3,35,23)( 4,27,21)( 5,37,16)( 6,39,19)( 7,38,26) ( 8,36,25)( 9,34,24)(10,30,15)(11,31,22)(12,32,14)(13,29,17)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 2, 6)( 3,12)( 4, 9)( 5,11)( 7,10)( 8,13)(14,39,16,35,21,31,18,34,24,33,22, 27,23,37,19,32,15,28,17,38,25,36,26,29,20,30)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1,10)( 3,11)( 4,13)( 5, 8)( 6, 9)( 7,12)(14,37,15,35,18,38,26,33,23,30,16, 32,17,31,24,36,20,27,19,39,21,28,25,34,22,29)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1,12)( 2, 9)( 3, 8)( 4, 5)( 6,13)( 7,11)(14,30,21,35,24,34,23,27,15,32,25, 38,20,29,16,39,18,31,22,33,19,37,17,28,26,36)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 8)( 2, 5)( 3, 6)( 4, 7)( 9,13)(10,11)(14,29,18,35,23,33,17,32,20,36,21, 39,22,34,15,37,26,38,16,30,24,31,19,27,25,28)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 9)( 2,10)( 3, 5)( 4,11)( 6,12)( 7,13)(14,38,22,35,25,37,21,29,19,34,20, 28,24,39,17,27,16,36,23,31,26,32,18,30,15,33)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1,11)( 2,13)( 3, 4)( 5, 6)( 7, 8)(10,12)(14,27,17,35,22,36,16,37,25,31,23, 29,21,32,26,34,19,30,18,28,20,33,15,39,24,38)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 6)( 2, 7)( 3, 9)( 4,10)( 5,13)( 8,12)(14,36,24,35,15,27,20,38,18,39,19, 33,26,28,21,30,23,34,25,32,16,29,22,31,17,37)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 7)( 2, 4)( 3,10)( 5,12)( 8, 9)(11,13)(14,32,19,35,16,28,15,31,21,38,17, 34,18,36,25,33,24,29,26,27,22,30,20,37,23,39)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $2$ | $( 1, 3)( 2, 8)( 4, 6)( 5,10)( 9,11)(12,13)(14,35)(15,38)(16,31)(17,36)(18,33) (19,28)(20,39)(21,34)(22,37)(23,32)(24,27)(25,29)(26,30)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1,13)( 2,12)( 4, 8)( 5, 7)( 6,11)( 9,10)(14,31,20,35,26,39,25,30,17,29,15, 36,19,38,23,28,22,32,24,37,18,27,21,33,16,34)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 4)( 2, 3)( 5, 9)( 6, 7)( 8,10)(11,12)(14,33,25,35,19,29,24,28,16,27,26, 31,15,30,22,38,21,37,20,34,17,39,23,36,18,32)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 5)( 2,11)( 3, 7)( 6, 8)( 9,12)(10,13)(14,28,23,35,20,32,22,39,26,37,24, 30,25,27,18,29,17,33,21,36,15,34,16,38,19,31)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $26$ | $( 1, 2)( 3,13)( 4,12)( 6,10)( 7, 9)( 8,11)(14,34,26,35,17,30,19,36,22,28,18, 37,16,33,20,31,25,39,15,29,23,38,24,32,21,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1014=2 \cdot 3 \cdot 13^{2}$ | 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: | 1014.7 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);