Show commands:
Magma
magma: G := TransitiveGroup(39, 3);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $S_3\times C_{13}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $13$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,13,25,38,10,23,36,9,21,33,6,18,30)(2,15,26,37,11,22,34,8,19,32,4,17,28,3,14,27,39,12,24,35,7,20,31,5,16,29), (1,15,25,37,10,22,36,8,21,32,6,17,30,3,13,27,38,12,23,35,9,20,33,5,18,29)(2,14,26,39,11,24,34,7,19,31,4,16,28) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $13$: $C_{13}$ $26$: $C_{26}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: $C_{13}$
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
| $ 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 $ | $3$ | $2$ | $( 2, 3)( 4, 5)( 7, 8)(11,12)(14,15)(16,17)(19,20)(22,24)(26,27)(28,29)(31,32) (34,35)(37,39)$ |
| $ 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 $ | $2$ | $39$ | $( 1, 4, 8,10,14,17,21,24,27,30,31,35,38, 2, 5, 9,11,15,18,19,22,25,28,32,36, 39, 3, 6, 7,12,13,16,20,23,26,29,33,34,37)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1, 4, 9,11,13,16,21,24,25,28,33,34,38, 2, 6, 7,10,14,18,19,23,26,30,31,36,39 )( 3, 5, 8,12,15,17,20,22,27,29,32,35,37)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1, 6, 9,10,13,18,21,23,25,30,33,36,38)( 2, 4, 7,11,14,16,19,24,26,28,31,34, 39)( 3, 5, 8,12,15,17,20,22,27,29,32,35,37)$ |
| $ 39 $ | $2$ | $39$ | $( 1, 7,15,21,26,32,38, 4,12,18,24,29,36, 2, 8,13,19,27,33,39, 5,10,16,22,30, 34, 3, 9,14,20,25,31,37, 6,11,17,23,28,35)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1, 7,13,19,25,31,38, 4,10,16,23,28,36, 2, 9,14,21,26,33,39, 6,11,18,24,30,34 )( 3, 8,15,20,27,32,37, 5,12,17,22,29,35)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1, 9,13,21,25,33,38, 6,10,18,23,30,36)( 2, 7,14,19,26,31,39, 4,11,16,24,28, 34)( 3, 8,15,20,27,32,37, 5,12,17,22,29,35)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,10,21,30,38, 9,18,25,36, 6,13,23,33)( 2,11,19,28,39, 7,16,26,34, 4,14,24, 31)( 3,12,20,29,37, 8,17,27,35, 5,15,22,32)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,10,21,30,38, 9,18,25,36, 6,13,23,33)( 2,12,19,29,39, 8,16,27,34, 5,14,22, 31, 3,11,20,28,37, 7,17,26,35, 4,15,24,32)$ |
| $ 39 $ | $2$ | $39$ | $( 1,11,20,30,39, 8,18,26,35, 6,14,22,33, 2,12,21,28,37, 9,16,27,36, 4,15,23, 31, 3,10,19,29,38, 7,17,25,34, 5,13,24,32)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,13,25,38,10,23,36, 9,21,33, 6,18,30)( 2,14,26,39,11,24,34, 7,19,31, 4,16, 28)( 3,15,27,37,12,22,35, 8,20,32, 5,17,29)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,13,25,38,10,23,36, 9,21,33, 6,18,30)( 2,15,26,37,11,22,34, 8,19,32, 4,17, 28, 3,14,27,39,12,24,35, 7,20,31, 5,16,29)$ |
| $ 39 $ | $2$ | $39$ | $( 1,14,27,38,11,22,36, 7,20,33, 4,17,30, 2,15,25,39,12,23,34, 8,21,31, 5,18, 28, 3,13,26,37,10,24,35, 9,19,32, 6,16,29)$ |
| $ 39 $ | $2$ | $39$ | $( 1,16,32, 9,24,37,13,28, 5,21,34,12,25, 2,17,33, 7,22,38,14,29, 6,19,35,10, 26, 3,18,31, 8,23,39,15,30, 4,20,36,11,27)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,16,33, 7,23,39,13,28, 6,19,36,11,25, 2,18,31, 9,24,38,14,30, 4,21,34,10,26 )( 3,17,32, 8,22,37,15,29, 5,20,35,12,27)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,18,33, 9,23,38,13,30, 6,21,36,10,25)( 2,16,31, 7,24,39,14,28, 4,19,34,11, 26)( 3,17,32, 8,22,37,15,29, 5,20,35,12,27)$ |
| $ 39 $ | $2$ | $39$ | $( 1,19,37,18,34,15,33,11,29, 9,26, 5,23, 2,20,38,16,35,13,31,12,30, 7,27, 6, 24, 3,21,39,17,36,14,32,10,28, 8,25, 4,22)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,19,38,16,36,14,33,11,30, 7,25, 4,23, 2,21,39,18,34,13,31,10,28, 9,26, 6,24 )( 3,20,37,17,35,15,32,12,29, 8,27, 5,22)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,21,38,18,36,13,33,10,30, 9,25, 6,23)( 2,19,39,16,34,14,31,11,28, 7,26, 4, 24)( 3,20,37,17,35,15,32,12,29, 8,27, 5,22)$ |
| $ 39 $ | $2$ | $39$ | $( 1,22, 4,25, 8,28,10,32,14,36,17,39,21, 3,24, 6,27, 7,30,12,31,13,35,16,38, 20, 2,23, 5,26, 9,29,11,33,15,34,18,37,19)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,22, 6,27, 9,29,10,32,13,35,18,37,21, 3,23, 5,25, 8,30,12,33,15,36,17,38,20 )( 2,24, 4,26, 7,28,11,31,14,34,16,39,19)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,23, 6,25, 9,30,10,33,13,36,18,38,21)( 2,24, 4,26, 7,28,11,31,14,34,16,39, 19)( 3,22, 5,27, 8,29,12,32,15,35,17,37,20)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,25,10,36,21, 6,30,13,38,23, 9,33,18)( 2,26,11,34,19, 4,28,14,39,24, 7,31, 16)( 3,27,12,35,20, 5,29,15,37,22, 8,32,17)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,25,10,36,21, 6,30,13,38,23, 9,33,18)( 2,27,11,35,19, 5,28,15,39,22, 7,32, 16, 3,26,12,34,20, 4,29,14,37,24, 8,31,17)$ |
| $ 39 $ | $2$ | $39$ | $( 1,26,12,36,19, 5,30,14,37,23, 7,32,18, 2,27,10,34,20, 6,28,15,38,24, 8,33, 16, 3,25,11,35,21, 4,29,13,39,22, 9,31,17)$ |
| $ 39 $ | $2$ | $39$ | $( 1,28,17, 6,31,20, 9,34,22,10,39,27,13, 2,29,18, 4,32,21, 7,35,23,11,37,25, 14, 3,30,16, 5,33,19, 8,36,24,12,38,26,15)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,28,18, 4,33,19, 9,34,23,11,38,26,13, 2,30,16, 6,31,21, 7,36,24,10,39,25,14 )( 3,29,17, 5,32,20, 8,35,22,12,37,27,15)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,30,18, 6,33,21, 9,36,23,10,38,25,13)( 2,28,16, 4,31,19, 7,34,24,11,39,26, 14)( 3,29,17, 5,32,20, 8,35,22,12,37,27,15)$ |
| $ 39 $ | $2$ | $39$ | $( 1,31,22,13, 4,35,25,16, 8,38,28,20,10, 2,32,23,14, 5,36,26,17, 9,39,29,21, 11, 3,33,24,15, 6,34,27,18, 7,37,30,19,12)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,31,23,14, 6,34,25,16, 9,39,30,19,10, 2,33,24,13, 4,36,26,18, 7,38,28,21,11 )( 3,32,22,15, 5,35,27,17, 8,37,29,20,12)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,33,23,13, 6,36,25,18, 9,38,30,21,10)( 2,31,24,14, 4,34,26,16, 7,39,28,19, 11)( 3,32,22,15, 5,35,27,17, 8,37,29,20,12)$ |
| $ 39 $ | $2$ | $39$ | $( 1,34,29,23,16,12, 6,39,32,25,19,15, 9, 2,35,30,24,17,10, 4,37,33,26,20,13, 7, 3,36,28,22,18,11, 5,38,31,27,21,14, 8)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,34,30,24,18,11, 6,39,33,26,21,14, 9, 2,36,28,23,16,10, 4,38,31,25,19,13, 7 )( 3,35,29,22,17,12, 5,37,32,27,20,15, 8)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,36,30,23,18,10, 6,38,33,25,21,13, 9)( 2,34,28,24,16,11, 4,39,31,26,19,14, 7)( 3,35,29,22,17,12, 5,37,32,27,20,15, 8)$ |
| $ 39 $ | $2$ | $39$ | $( 1,37,34,33,29,26,23,20,16,13,12, 7, 6, 3,39,36,32,28,25,22,19,18,15,11, 9, 5, 2,38,35,31,30,27,24,21,17,14,10, 8, 4)$ |
| $ 26, 13 $ | $3$ | $26$ | $( 1,37,36,32,30,27,23,20,18,15,10, 8, 6, 3,38,35,33,29,25,22,21,17,13,12, 9, 5 )( 2,39,34,31,28,26,24,19,16,14,11, 7, 4)$ |
| $ 13, 13, 13 $ | $1$ | $13$ | $( 1,38,36,33,30,25,23,21,18,13,10, 9, 6)( 2,39,34,31,28,26,24,19,16,14,11, 7, 4)( 3,37,35,32,29,27,22,20,17,15,12, 8, 5)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $78=2 \cdot 3 \cdot 13$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 78.3 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);