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Magma
magma: G := TransitiveGroup(39, 36);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^4:C_{26}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,16,33,8,22,37,14,29,5,21,36,12,26)(2,18,31,7,23,39,15,28,6,20,34,11,27,3,17,32,9,24,38,13,30,4,19,35,10,25), (1,28,17,6,32,19,7,35,24,12,37,27,13)(2,30,18,5,33,21,8,34,22,11,38,26,14,3,29,16,4,31,20,9,36,23,10,39,25,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $13$: $C_{13}$ $26$: $C_{26}$ $78$: 39T3 $702$: 27T219 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 13: $C_{13}$
Low degree siblings
39T36Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $( 1, 3, 2)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,20,21)(25,27,26) (28,29,30)(31,33,32)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,33,22,14, 5,36,26,16, 8,37,29,21,12)( 2,31,23,15, 6,34,27,17, 9,38,30,19, 10)( 3,32,24,13, 4,35,25,18, 7,39,28,20,11)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,22, 5,26, 8,29,12,33,14,36,16,37,21)( 2,23, 6,27, 9,30,10,31,15,34,17,38, 19)( 3,24, 4,25, 7,28,11,32,13,35,18,39,20)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1, 5, 8,12,14,16,21,22,26,29,33,36,37)( 2, 6, 9,10,15,17,19,23,27,30,31,34, 38)( 3, 4, 7,11,13,18,20,24,25,28,32,35,39)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1, 8,14,21,26,33,37, 5,12,16,22,29,36)( 2, 9,15,19,27,31,38, 6,10,17,23,30, 34)( 3, 7,13,20,25,32,39, 4,11,18,24,28,35)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,14,26,37,12,22,36, 8,21,33, 5,16,29)( 2,15,27,38,10,23,34, 9,19,31, 6,17, 30)( 3,13,25,39,11,24,35, 7,20,32, 4,18,28)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,26,12,36,21, 5,29,14,37,22, 8,33,16)( 2,27,10,34,19, 6,30,15,38,23, 9,31, 17)( 3,25,11,35,20, 4,28,13,39,24, 7,32,18)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,12,21,29,37, 8,16,26,36, 5,14,22,33)( 2,10,19,30,38, 9,17,27,34, 6,15,23, 31)( 3,11,20,28,39, 7,18,25,35, 4,13,24,32)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,21,37,16,36,14,33,12,29, 8,26, 5,22)( 2,19,38,17,34,15,31,10,30, 9,27, 6, 23)( 3,20,39,18,35,13,32,11,28, 7,25, 4,24)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,37,36,33,29,26,22,21,16,14,12, 8, 5)( 2,38,34,31,30,27,23,19,17,15,10, 9, 6)( 3,39,35,32,28,25,24,20,18,13,11, 7, 4)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,36,29,22,16,12, 5,37,33,26,21,14, 8)( 2,34,30,23,17,10, 6,38,31,27,19,15, 9)( 3,35,28,24,18,11, 4,39,32,25,20,13, 7)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,29,16, 5,33,21, 8,36,22,12,37,26,14)( 2,30,17, 6,31,19, 9,34,23,10,38,27, 15)( 3,28,18, 4,32,20, 7,35,24,11,39,25,13)$ |
| $ 13, 13, 13 $ | $27$ | $13$ | $( 1,16,33, 8,22,37,14,29, 5,21,36,12,26)( 2,17,31, 9,23,38,15,30, 6,19,34,10, 27)( 3,18,32, 7,24,39,13,28, 4,20,35,11,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $(10,12,11)(13,15,14)(16,17,18)(22,24,23)(28,29,30)(31,32,33)(37,38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $( 1, 3, 2)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,24,23) (25,27,26)(28,30,29)(37,38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$ |
| $ 39 $ | $54$ | $39$ | $( 1,33,23,14, 4,35,25,18, 8,37,30,20,11, 2,31,24,15, 5,36,26,16, 9,38,28,21, 12, 3,32,22,13, 6,34,27,17, 7,39,29,19,10)$ |
| $ 39 $ | $54$ | $39$ | $( 1,22, 4,25, 7,28,12,32,14,35,18,37,19, 2,23, 5,26, 8,29,10,33,15,36,16,38, 20, 3,24, 6,27, 9,30,11,31,13,34,17,39,21)$ |
| $ 39 $ | $54$ | $39$ | $( 1, 5, 8,12,13,17,20,24,27,30,32,36,37, 2, 6, 9,10,14,18,21,22,25,28,33,34, 38, 3, 4, 7,11,15,16,19,23,26,29,31,35,39)$ |
| $ 39 $ | $54$ | $39$ | $( 1, 8,14,20,25,32,37, 6,10,16,23,29,34, 2, 9,15,21,26,33,38, 4,11,17,24,30, 35, 3, 7,13,19,27,31,39, 5,12,18,22,28,36)$ |
| $ 39 $ | $54$ | $39$ | $( 1,14,25,39,12,24,34, 9,19,31, 4,18,29, 2,15,26,37,10,22,35, 7,20,32, 5,16, 30, 3,13,27,38,11,23,36, 8,21,33, 6,17,28)$ |
| $ 39 $ | $54$ | $39$ | $( 1,26,12,35,20, 4,28,14,39,22, 7,32,16, 2,27,10,36,21, 5,29,15,37,23, 8,33, 17, 3,25,11,34,19, 6,30,13,38,24, 9,31,18)$ |
| $ 39 $ | $54$ | $39$ | $( 1,12,20,28,37, 9,17,25,35, 4,13,23,33, 2,10,21,29,38, 7,18,26,36, 5,14,24, 31, 3,11,19,30,39, 8,16,27,34, 6,15,22,32)$ |
| $ 39 $ | $54$ | $39$ | $( 1,21,37,17,35,13,31,11,30, 7,25, 4,24, 2,19,38,18,36,14,32,12,28, 8,26, 5, 22, 3,20,39,16,34,15,33,10,29, 9,27, 6,23)$ |
| $ 39 $ | $54$ | $39$ | $( 1,37,34,31,28,26,22,20,18,14,11, 9, 6, 2,38,35,32,29,27,23,21,16,15,12, 7, 4, 3,39,36,33,30,25,24,19,17,13,10, 8, 5)$ |
| $ 39 $ | $54$ | $39$ | $( 1,36,29,23,16,10, 5,37,31,25,20,13, 9, 2,34,30,24,17,11, 6,38,32,26,21,14, 7, 3,35,28,22,18,12, 4,39,33,27,19,15, 8)$ |
| $ 39 $ | $54$ | $39$ | $( 1,29,17, 4,32,21, 8,36,22,11,38,25,13, 2,30,18, 5,33,19, 9,34,23,12,39,26, 14, 3,28,16, 6,31,20, 7,35,24,10,37,27,15)$ |
| $ 39 $ | $54$ | $39$ | $( 1,16,31, 7,24,38,13,30, 4,20,35,11,27, 2,17,32, 8,22,39,14,28, 5,21,36,12, 25, 3,18,33, 9,23,37,15,29, 6,19,34,10,26)$ |
| $ 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 $ | $81$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(13,15)(16,17)(19,20)(22,24)(25,26)(28,30)(32,33) (34,35)(38,39)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,33,24,14, 5,34,25,16, 9,37,29,21,12, 3,31,23,15, 4,35,27,17, 8,38,28,19,11 )( 2,32,22,13, 6,36,26,18, 7,39,30,20,10)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,22, 4,25, 8,30,11,33,13,34,18,39,19, 3,23, 6,26, 7,28,10,31,15,35,17,37,21 )( 2,24, 5,27, 9,29,12,32,14,36,16,38,20)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1, 5, 9,12,13,17,21,22,25,29,33,35,38, 3, 6, 8,10,15,18,20,23,27,30,32,36,37 )( 2, 4, 7,11,14,16,19,24,26,28,31,34,39)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1, 8,15,20,27,31,38, 4,11,16,23,30,35, 2, 7,13,19,25,33,39, 6,12,18,24,29,36 )( 3, 9,14,21,26,32,37, 5,10,17,22,28,34)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,14,26,39,10,23,34, 7,20,31, 6,16,30, 3,15,25,37,12,24,36, 8,19,32, 5,17,29 )( 2,13,27,38,11,22,35, 9,21,33, 4,18,28)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,26,11,36,21, 5,30,13,38,24, 8,31,17)( 2,25,12,35,19, 4,28,15,39,23, 9,33, 18, 3,27,10,34,20, 6,29,14,37,22, 7,32,16)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,12,20,30,39, 9,16,27,34, 4,13,23,31, 2,11,21,29,37, 8,17,26,35, 6,14,22,32 )( 3,10,19,28,38, 7,18,25,36, 5,15,24,33)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,21,37,16,34,13,31,10,30, 7,25, 5,23, 2,20,38,18,35,15,32,12,28, 9,26, 4,24 )( 3,19,39,17,36,14,33,11,29, 8,27, 6,22)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,37,36,33,28,27,23,19,18,13,10, 9, 5, 2,39,34,32,29,26,24,21,16,15,11, 8, 6 )( 3,38,35,31,30,25,22,20,17,14,12, 7, 4)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,36,29,22,18,11, 5,38,32,26,20,15, 7, 3,34,28,23,17,12, 4,39,31,27,19,13, 9 )( 2,35,30,24,16,10, 6,37,33,25,21,14, 8)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,29,16, 6,33,20, 9,36,22,11,37,26,13, 2,28,17, 5,31,19, 7,35,23,10,38,25,14 )( 3,30,18, 4,32,21, 8,34,24,12,39,27,15)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,16,31, 9,22,39,15,28, 6,21,36,12,25)( 2,18,32, 8,23,38,13,30, 4,20,34,11, 26, 3,17,33, 7,24,37,14,29, 5,19,35,10,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2106=2 \cdot 3^{4} \cdot 13$ | 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: | 2106.b | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);