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Properties

Label	39T6
Degree	$39$
Order	$78$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{13}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(39, 6);

Group action invariants

Degree $n$:		$39$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$6$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_{13}:C_6$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$3$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,6,34,22,21,28)(2,4,35,23,19,29)(3,5,36,24,20,30)(7,27,11,18,37,14)(8,25,12,16,38,15)(9,26,10,17,39,13)(31,32,33), (1,30,26,35,9,12)(2,28,27,36,7,10)(3,29,25,34,8,11)(4,20,14,33,16,23)(5,21,15,31,17,24)(6,19,13,32,18,22)(37,38,39)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$6$: $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$
Degree 13: $C_{13}:C_6$

Low degree siblings

13T5, 26T6
Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:		78.1	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	13A1	13A2
	Size	1	13	13	13	13	13	6	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	13A2	13A1
	3 P	1A	2A	1A	1A	2A	2A	13A1	13A2
	13 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A
	Type
78.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
78.1.1b	R	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$
78.1.1c1	C	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
78.1.1c2	C	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
78.1.1d1	C	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$1$
78.1.1d2	C	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$1$
78.1.6a1	R	$6$	$0$	$0$	$0$	$0$	$0$	$ζ_{13}^{- 6} + ζ_{13}^{- 5} + ζ_{13}^{- 2} + ζ_{13}^{2} + ζ_{13}^{5} + ζ_{13}^{6}$	$- ζ_{13}^{- 6} - ζ_{13}^{- 5} - ζ_{13}^{- 2} - 1 - ζ_{13}^{2} - ζ_{13}^{5} - ζ_{13}^{6}$
78.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	$- ζ_{13}^{- 6} - ζ_{13}^{- 5} - ζ_{13}^{- 2} - 1 - ζ_{13}^{2} - ζ_{13}^{5} - ζ_{13}^{6}$	$ζ_{13}^{- 6} + ζ_{13}^{- 5} + ζ_{13}^{- 2} + ζ_{13}^{2} + ζ_{13}^{5} + ζ_{13}^{6}$

magma: CharacterTable(G);