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Properties

Label	27T14
Degree	$27$
Order	$54$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_9:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(27, 14);

Group action invariants

Degree $n$:		$27$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$14$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_9:C_6$
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:		(1,27,5,21,17,22,11,7,15)(2,25,6,19,18,23,12,8,13)(3,26,4,20,16,24,10,9,14), (1,27,19,8,10,16)(2,25,20,9,11,17)(3,26,21,7,12,18)(4,6,5)(13,22,14,23,15,24)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$6$: $S_3$, $C_6$
$18$: $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$
Degree 9: $S_3\times C_3$, $(C_9:C_3):C_2$

Low degree siblings

9T10, 18T18
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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$9$	$2$	$( 4,25)( 5,26)( 6,27)( 7,23)( 8,24)( 9,22)(10,20)(11,21)(12,19)(13,17)(14,18) (15,16)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$3$	$3$	$( 1, 2, 3)( 4,15,23)( 5,13,24)( 6,14,22)( 7,25,16)( 8,26,17)( 9,27,18) (10,11,12)(19,20,21)$
	$ 6, 6, 6, 6, 3 $	$9$	$6$	$( 1, 2, 3)( 4,16,23,25,15, 7)( 5,17,24,26,13, 8)( 6,18,22,27,14, 9) (10,21,12,20,11,19)$
	$ 6, 6, 6, 6, 3 $	$9$	$6$	$( 1, 3, 2)( 4, 7,15,25,23,16)( 5, 8,13,26,24,17)( 6, 9,14,27,22,18) (10,19,11,20,12,21)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$3$	$3$	$( 1, 3, 2)( 4,23,15)( 5,24,13)( 6,22,14)( 7,16,25)( 8,17,26)( 9,18,27) (10,12,11)(19,21,20)$
	$ 9, 9, 9 $	$6$	$9$	$( 1, 4, 8,11,14,18,21,24,25)( 2, 5, 9,12,15,16,19,22,26)( 3, 6, 7,10,13,17,20, 23,27)$
	$ 9, 9, 9 $	$6$	$9$	$( 1, 5,17,11,15,27,21,22, 7)( 2, 6,18,12,13,25,19,23, 8)( 3, 4,16,10,14,26,20, 24, 9)$
	$ 9, 9, 9 $	$6$	$9$	$( 1, 6,26,11,13, 9,21,23,16)( 2, 4,27,12,14, 7,19,24,17)( 3, 5,25,10,15, 8,20, 22,18)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27) ( 8,18,25)( 9,16,26)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$54=2 \cdot 3^{3}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		54.6	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A	3B1	3B-1	6A1	6A-1	9A	9B1	9B-1
	Size	1	9	2	3	3	9	9	6	6	6
	2 P	1A	1A	3A	3B-1	3B1	3B1	3B-1	9A	9B-1	9B1
	3 P	1A	2A	1A	1A	1A	2A	2A	3A	3A	3A
	Type
54.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
54.6.1b	R	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$
54.6.1c1	C	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$
54.6.1c2	C	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$
54.6.1d1	C	$1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$
54.6.1d2	C	$1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$
54.6.2a	R	$2$	$0$	$2$	$2$	$2$	$0$	$0$	$- 1$	$- 1$	$- 1$
54.6.2b1	C	$2$	$0$	$2$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$0$	$0$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
54.6.2b2	C	$2$	$0$	$2$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$0$	$0$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
54.6.6a	R	$6$	$0$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);