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Magma
magma: G := TransitiveGroup(39, 34);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:D_6$ | ||
| 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: | (1,34,7,28,13,35,6,29,12,36,5,30,11,37,4,31,10,38,3,32,9,39,2,33,8,27)(14,16,18,20,22,24,26,15,17,19,21,23,25), (1,23,27,10,26,31)(2,19,39,9,17,32)(3,15,38,8,21,33)(4,24,37,7,25,34)(5,20,36,6,16,35)(11,22,30,13,14,28)(12,18,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: None
Low degree siblings
26T23, 39T34Siblings 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 $ | $6$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,22,26,17,21,25,16,20,24,15,19, 23)(27,34,28,35,29,36,30,37,31,38,32,39,33)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24, 19)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21, 24)(27,29,31,33,35,37,39,28,30,32,34,36,38)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,20,26,19,25,18,24,17,23,16,22,15, 21)(27,31,35,39,30,34,38,29,33,37,28,32,36)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,26,25,24,23,22,21,20,19,18,17,16, 15)(27,35,30,38,33,28,36,31,39,34,29,37,32)$ |
| $ 13, 13, 13 $ | $6$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,30,33,36,39,29,32,35,38,28,31,34,37)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $13$ | $(14,20,26,19,25,18,24,17,23,16,22,15,21)(27,35,30,38,33,28,36,31,39,34,29,37, 32)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,24,21,18,15,25,22,19,16,26,23,20, 17)(27,29,31,33,35,37,39,28,30,32,34,36,38)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25, 26)(27,36,32,28,37,33,29,38,34,30,39,35,31)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,37,34,31,28,38,35,32,29,39,36,33,30)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,19,24,16,21,26,18,23,15,20,25,17, 22)(27,30,33,36,39,29,32,35,38,28,31,34,37)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(27,31,35,39,30,34,38,29,33,37,28,32, 36)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $13$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)(27,30,33,36,39,29,32,35,38,28,31,34, 37)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21, 24)(27,37,34,31,28,38,35,32,29,39,36,33,30)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,21,15,22,16,23,17,24,18,25,19,26, 20)(27,31,35,39,30,34,38,29,33,37,28,32,36)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $13$ | $(14,25,23,21,19,17,15,26,24,22,20,18,16)(27,33,39,32,38,31,37,30,36,29,35,28, 34)$ |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,26,19,25,18,24,17,23,16,22,15, 21)(27,34,28,35,29,36,30,37,31,38,32,39,33)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $13$ | $(14,23,19,15,24,20,16,25,21,17,26,22,18)(27,39,38,37,36,35,34,33,32,31,30,29, 28)$ |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $13$ | $(14,19,24,16,21,26,18,23,15,20,25,17,22)(27,38,36,34,32,30,28,39,37,35,33,31, 29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $338$ | $3$ | $( 1,28,15)( 2,27,19)( 3,39,23)( 4,38,14)( 5,37,18)( 6,36,22)( 7,35,26) ( 8,34,17)( 9,33,21)(10,32,25)(11,31,16)(12,30,20)(13,29,24)$ |
| $ 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 $ | $39$ | $2$ | $(14,38)(15,28)(16,31)(17,34)(18,37)(19,27)(20,30)(21,33)(22,36)(23,39)(24,29) (25,32)(26,35)$ |
| $ 26, 13 $ | $78$ | $26$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,32,16,38,18,31,20,37,22,30,24,36, 26,29,15,35,17,28,19,34,21,27,23,33,25,39)$ |
| $ 26, 13 $ | $78$ | $26$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,39,18,38,22,37,26,36,17,35,21,34, 25,33,16,32,20,31,24,30,15,29,19,28,23,27)$ |
| $ 26, 13 $ | $78$ | $26$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,27,22,38,17,36,25,34,20,32,15,30, 23,28,18,39,26,37,21,35,16,33,24,31,19,29)$ |
| $ 26, 13 $ | $78$ | $26$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,29,17,38,20,34,23,30,26,39,16,35, 19,31,22,27,25,36,15,32,18,28,21,37,24,33)$ |
| $ 26, 13 $ | $78$ | $26$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,33,20,38,26,30,19,35,25,27,18,32, 24,37,17,29,23,34,16,39,22,31,15,36,21,28)$ |
| $ 26, 13 $ | $78$ | $26$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,28,26,38,25,35,24,32,23,29,22,39, 21,36,20,33,19,30,18,27,17,37,16,34,15,31)$ |
| $ 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 $ | $338$ | $6$ | $( 1,28,25,11,38,26)( 2,29,16,10,37,22)( 3,30,20, 9,36,18)( 4,31,24, 8,35,14) ( 5,32,15, 7,34,23)( 6,33,19)(12,39,17,13,27,21)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $78$ | $26$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,38,26,28,25,31,24,34,23,37,22, 27,21,30,20,33,19,36,18,39,17,29,16,32,15,35)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $78$ | $26$ | $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,32,19,30,24,28,16,39,21,37,26, 35,18,33,23,31,15,29,20,27,25,38,17,36,22,34)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $78$ | $26$ | $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,39,25,32,23,38,21,31,19,37,17, 30,15,36,26,29,24,35,22,28,20,34,18,27,16,33)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $78$ | $26$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(14,27,24,36,21,32,18,28,15,37,25, 33,22,29,19,38,16,34,26,30,23,39,20,35,17,31)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $78$ | $26$ | $( 1, 6)( 2, 5)( 3, 4)( 7,13)( 8,12)( 9,11)(14,33,18,34,22,35,26,36,17,37,21, 38,25,39,16,27,20,28,24,29,15,30,19,31,23,32)$ |
| $ 26, 2, 2, 2, 2, 2, 2, 1 $ | $78$ | $26$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,31,20,39,26,34,19,29,25,37,18, 32,24,27,17,35,23,30,16,38,22,33,15,28,21,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $39$ | $2$ | $( 1, 2)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(14,37)(15,34)(16,31)(17,28)(18,38) (19,35)(20,32)(21,29)(22,39)(23,36)(24,33)(25,30)(26,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2028=2^{2} \cdot 3 \cdot 13^{2}$ | 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: | 2028.s | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);