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Properties

Label	27T11
Degree	$27$
Order	$54$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(27, 11);

Group action invariants

Degree $n$:		$27$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$11$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_3^2:C_6$
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:		(1,18,4,22,25,20)(2,16,5,23,26,21)(3,17,6,24,27,19)(7,8,9)(10,15,12,14,11,13), (1,27)(2,25)(3,26)(7,19)(8,20)(9,21)(10,17)(11,18)(12,16)(13,24)(14,22)(15,23)	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: $S_3\times C_3$, $C_3^2 : C_6$, $C_3^2 : S_3 $

Low degree siblings

9T11, 9T13, 18T20, 18T21, 18T22
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 $	$9$	$2$	$( 4,26)( 5,27)( 6,25)( 7,23)( 8,24)( 9,22)(10,21)(11,19)(12,20)(13,16)(14,17) (15,18)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$3$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7,12,14)( 8,10,15)( 9,11,13)(16,22,19)(17,23,20) (18,24,21)(25,26,27)$
	$ 6, 6, 6, 6, 3 $	$9$	$6$	$( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,20,14,23,12,17)( 8,21,15,24,10,18) ( 9,19,13,22,11,16)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$3$	$3$	$( 1, 3, 2)( 4, 6, 5)( 7,14,12)( 8,15,10)( 9,13,11)(16,19,22)(17,20,23) (18,21,24)(25,27,26)$
	$ 6, 6, 6, 6, 3 $	$9$	$6$	$( 1, 3, 2)( 4,25, 5,26, 6,27)( 7,17,12,23,14,20)( 8,18,10,24,15,21) ( 9,16,11,22,13,19)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24) (17,21,22)(18,19,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$6$	$3$	$( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,10,19)( 5,11,20)( 6,12,21)(13,22,25) (14,23,26)(15,24,27)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$6$	$3$	$( 1, 8,19)( 2, 9,20)( 3, 7,21)( 4,11,22)( 5,12,23)( 6,10,24)(13,18,27) (14,16,25)(15,17,26)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$6$	$3$	$( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,12,16)( 5,10,17)( 6,11,18)(13,20,26) (14,21,27)(15,19,25)$

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

		1A	2A	3A	3B1	3B-1	3C	3D1	3D-1	6A1	6A-1
	Size	1	9	2	3	3	6	6	6	9	9
	2 P	1A	1A	3A	3B-1	3B1	3C	3D-1	3D1	3B1	3B-1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	2A	2A
	Type
54.5.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
54.5.1b	R	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
54.5.1c1	C	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$
54.5.1c2	C	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$
54.5.1d1	C	$1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
54.5.1d2	C	$1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
54.5.2a	R	$2$	$0$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$
54.5.2b1	C	$2$	$0$	$2$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$0$	$0$
54.5.2b2	C	$2$	$0$	$2$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$0$	$0$
54.5.6a	R	$6$	$0$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);