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Properties

Label	27T5
Degree	$27$
Order	$27$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_9:C_3$

Related objects

Number fields with this Galois group

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Show commands: Magma

magma: G := TransitiveGroup(27, 5);

Group action invariants

Degree $n$:		$27$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$5$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_9:C_3$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$27$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(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), (1,12,20)(2,10,21)(3,11,19)(4,22,13)(5,23,14)(6,24,15)(7,8,9)(16,17,18)(25,26,27)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$3$: $C_3$ x 4
$9$: $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4
Degree 9: $C_3^2$, $C_9:C_3$

Low degree siblings

9T6
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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$3$	$3$	$( 1, 2, 3)( 4,15,23)( 5,13,24)( 6,14,22)( 7,25,16)( 8,26,17)( 9,27,18) (10,11,12)(19,20,21)$
	$ 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 $	$3$	$9$	$( 1, 4, 8,11,14,18,21,24,25)( 2, 5, 9,12,15,16,19,22,26)( 3, 6, 7,10,13,17,20, 23,27)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 5,17,11,15,27,21,22, 7)( 2, 6,18,12,13,25,19,23, 8)( 3, 4,16,10,14,26,20, 24, 9)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 6,26,11,13, 9,21,23,16)( 2, 4,27,12,14, 7,19,24,17)( 3, 5,25,10,15, 8,20, 22,18)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 7,22,21,27,15,11,17, 5)( 2, 8,23,19,25,13,12,18, 6)( 3, 9,24,20,26,14,10, 16, 4)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 8,14,21,25, 4,11,18,24)( 2, 9,15,19,26, 5,12,16,22)( 3, 7,13,20,27, 6,10, 17,23)$
	$ 9, 9, 9 $	$3$	$9$	$( 1, 9, 6,21,26,23,11,16,13)( 2, 7, 4,19,27,24,12,17,14)( 3, 8, 5,20,25,22,10, 18,15)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27) ( 8,18,25)( 9,16,26)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,21,11)( 2,19,12)( 3,20,10)( 4,24,14)( 5,22,15)( 6,23,13)( 7,27,17) ( 8,25,18)( 9,26,16)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$27=3^{3}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		$2$
Label:		27.4	magma: IdentifyGroup(G);
Character table:

		1A	3A1	3A-1	3B1	3B-1	9A1	9A-1	9B1	9B-1	9C1	9C-1
	Size	1	1	1	3	3	3	3	3	3	3	3
	3 P	1A	3A-1	3A1	3B-1	3B1	9A-1	9B-1	9A1	9C-1	9B1	9C1
	Type
27.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
27.4.1b1	C	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
27.4.1b2	C	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
27.4.1c1	C	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$
27.4.1c2	C	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$
27.4.1d1	C	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$
27.4.1d2	C	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$
27.4.1e1	C	$1$	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
27.4.1e2	C	$1$	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
27.4.3a1	C	$3$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
27.4.3a2	C	$3$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);