LMFDB - Galois group: 39T46

Show commands: Magma

magma: G := TransitiveGroup(39, 46);

Group action invariants

Degree $n$:	$39$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$46$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_{13}^2:(C_6\times S_3)$
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:	(1,31,9,38,7,33)(2,27,12,39,3,36)(4,32,5,28,8,29)(6,37,11,30,13,35)(10,34)(14,22,15,26,18,25)(16,17,21,24,23,19), (1,14,3,21,9,16)(2,24,6,25,5,15)(4,18,12,20,10,26)(7,22,8,19,11,23)(13,17)(27,31,28)(29,36,34)(30,32,37)(35,38,39)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $S_3$, $C_6$ x 3
$12$: $D_{6}$, $C_6\times C_2$
$18$: $S_3\times C_3$
$36$: $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$S_3$, $C_6$ x 3
$12$:	$D_{6}$, $C_6\times C_2$
$18$:	$S_3\times C_3$
$36$:	$C_6\times S_3$

Subfields

Degree 3: $S_3$
Degree 13: None

Low degree siblings

26T40, 39T46
Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$18$	$13$	$(14,16,18,20,22,24,26,15,17,19,21,23,25)(27,34,28,35,29,36,30,37,31,38,32,39, 33)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$18$	$13$	$(14,18,22,26,17,21,25,16,20,24,15,19,23)(27,28,29,30,31,32,33,34,35,36,37,38, 39)$
$ 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, 13 $	$18$	$13$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,18,22,26,17,21,25,16,20,24,15,19, 23)(27,39,38,37,36,35,34,33,32,31,30,29,28)$
$ 13, 13, 13 $	$18$	$13$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21, 24)(27,29,31,33,35,37,39,28,30,32,34,36,38)$
$ 13, 13, 13 $	$36$	$13$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,26,25,24,23,22,21,20,19,18,17,16, 15)(27,28,29,30,31,32,33,34,35,36,37,38,39)$
$ 13, 13, 13 $	$36$	$13$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,31,35,39,30,34,38,29,33,37,28,32,36)$
$ 13, 13, 13 $	$12$	$13$	$( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,18,22,26,17,21,25,16,20,24,15,19, 23)(27,37,34,31,28,38,35,32,29,39,36,33,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $	$169$	$3$	$( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)(27,34,29)(28,37,38)(31,33,39)(32,36,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $	$169$	$3$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)(27,29,34)(28,38,37)(31,39,33)(32,35,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$338$	$3$	$( 1,26,27)( 2,17,39)( 3,21,38)( 4,25,37)( 5,16,36)( 6,20,35)( 7,24,34) ( 8,15,33)( 9,19,32)(10,23,31)(11,14,30)(12,18,29)(13,22,28)$
$ 39 $	$156$	$39$	$( 1,24,29, 8,17,31, 2,23,33, 9,16,35, 3,22,37,10,15,39, 4,21,28,11,14,30, 5, 20,32,12,26,34, 6,19,36,13,25,38, 7,18,27)$
$ 39 $	$156$	$39$	$( 1,26,28,11,16,29, 8,19,30, 5,22,31, 2,25,32,12,15,33, 9,18,34, 6,21,35, 3, 24,36,13,14,37,10,17,38, 7,20,39, 4,23,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$26$	$3$	$( 1,15,27)( 2,14,31)( 3,26,35)( 4,25,39)( 5,24,30)( 6,23,34)( 7,22,38) ( 8,21,29)( 9,20,33)(10,19,37)(11,18,28)(12,17,32)(13,16,36)$
$ 39 $	$156$	$39$	$( 1,18,34, 3,25,28, 5,19,35, 7,26,29, 9,20,36,11,14,30,13,21,37, 2,15,31, 4, 22,38, 6,16,32, 8,23,39,10,17,33,12,24,27)$
$ 39 $	$156$	$39$	$( 1,20,30,13,23,33,12,26,36,11,16,39,10,19,29, 9,22,32, 8,25,35, 7,15,38, 6, 18,28, 5,21,31, 4,24,34, 3,14,37, 2,17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$26$	$3$	$( 1,15,27)( 2,25,37)( 3,22,34)( 4,19,31)( 5,16,28)( 6,26,38)( 7,23,35) ( 8,20,32)( 9,17,29)(10,14,39)(11,24,36)(12,21,33)(13,18,30)$
$ 6, 6, 6, 6, 6, 6, 2, 1 $	$507$	$6$	$( 1,29, 4,30,13,33)( 2,38, 7,31, 9,36)( 3,34,10,32, 5,39)( 6,35) ( 8,27,12,37,11,28)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $	$234$	$26$	$( 1,27, 8,34, 2,28, 9,35, 3,29,10,36, 4,30,11,37, 5,31,12,38, 6,32,13,39, 7,33 )(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $	$234$	$26$	$( 1,34, 2,35, 3,36, 4,37, 5,38, 6,39, 7,27, 8,28, 9,29,10,30,11,31,12,32,13,33 )(14,16)(17,26)(18,25)(19,24)(20,23)(21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$39$	$2$	$( 1,33)( 2,34)( 3,35)( 4,36)( 5,37)( 6,38)( 7,39)( 8,27)( 9,28)(10,29)(11,30) (12,31)(13,32)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$
$ 6, 6, 6, 6, 6, 6, 2, 1 $	$507$	$6$	$( 1,34, 4,30, 5,33)( 2,37,13,31, 8,29)( 3,27, 9,32,11,38)( 6,36,10,35, 7,39) (12,28)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 6, 6, 6, 6, 6, 6, 1, 1, 1 $	$169$	$6$	$( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22) (27,39,29,33,34,31)(28,36,38,32,37,35)$
$ 6, 6, 6, 6, 6, 6, 1, 1, 1 $	$169$	$6$	$( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21) (27,31,34,33,29,39)(28,35,37,32,38,36)$
$ 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 $	$338$	$6$	$( 1,26,39,12,24,31)( 2,14,30,11,23,27)( 3,15,34,10,22,36)( 4,16,38, 9,21,32) ( 5,17,29, 8,20,28)( 6,18,33, 7,19,37)(13,25,35)$
$ 6, 6, 6, 6, 6, 6, 3 $	$338$	$6$	$( 1,24,33, 6,26,31)( 2,14,30, 5,23,34)( 3,17,27, 4,20,37)( 7,16,28,13,21,36) ( 8,19,38,12,18,39)( 9,22,35,11,15,29)(10,25,32)$
$ 6, 6, 6, 6, 6, 6, 3 $	$338$	$6$	$( 1,18,31)( 2,14,30,13,22,32)( 3,23,29,12,26,33)( 4,19,28,11,17,34) ( 5,15,27,10,21,35)( 6,24,39, 9,25,36)( 7,20,38, 8,16,37)$
$ 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 $	$39$	$2$	$( 1,29)( 2,28)( 3,27)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32) (12,31)(13,30)$
$ 26, 13 $	$234$	$26$	$( 1,36, 7,30,13,37, 6,31,12,38, 5,32,11,39, 4,33,10,27, 3,34, 9,28, 2,35, 8,29 )(14,16,18,20,22,24,26,15,17,19,21,23,25)$
$ 26, 13 $	$234$	$26$	$( 1,30,13,31,12,32,11,33,10,34, 9,35, 8,36, 7,37, 6,38, 5,39, 4,27, 3,28, 2,29 )(14,18,22,26,17,21,25,16,20,24,15,19,23)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $	$507$	$6$	$( 1,27, 7,35, 9,29)( 2,37, 3,34,12,33)( 4,31, 8,32, 5,28)( 6,38,13,30,11,36) (10,39)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $	$507$	$6$	$( 1,34, 8,36, 3,29)( 2,38,11,35,12,39)( 4,33)( 5,37, 7,32,13,30) ( 6,28,10,31, 9,27)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$6084=2^{2} \cdot 3^{2} \cdot 13^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	6084.i	magma: IdentifyGroup(G);

Character table: not available.

magma: CharacterTable(G);

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Group action invariants

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Group invariants

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	39T46
Degree	$39$
Order	$6084$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{13}^2:(C_6\times S_3)$