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Properties

Label	18T18
Degree	$18$
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(18, 18);

Group action invariants

Degree $n$:		$18$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$18$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_9:C_6$
Parity:		$-1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$6$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10), (1,15,7,9,13,3)(2,16,8,10,14,4)(5,6)(11,18)(12,17)	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 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 9: $(C_9:C_3):C_2$

Low degree siblings

9T10, 27T14
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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$3$	$3$	$( 3, 9,15)( 4,10,16)( 5,17,11)( 6,18,12)$
	$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$3$	$3$	$( 3,15, 9)( 4,16,10)( 5,11,17)( 6,12,18)$
	$ 6, 6, 2, 2, 2 $	$9$	$6$	$( 1, 2)( 3, 5, 9,17,15,11)( 4, 6,10,18,16,12)( 7,14)( 8,13)$
	$ 6, 6, 2, 2, 2 $	$9$	$6$	$( 1, 2)( 3,11,15,17, 9, 5)( 4,12,16,18,10, 6)( 7,14)( 8,13)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$9$	$2$	$( 1, 2)( 3,17)( 4,18)( 5,15)( 6,16)( 7,14)( 8,13)( 9,11)(10,12)$
	$ 9, 9 $	$6$	$9$	$( 1, 4, 5, 7,10,11,13,16,17)( 2, 3, 6, 8, 9,12,14,15,18)$
	$ 9, 9 $	$6$	$9$	$( 1, 4,11, 7,10,17,13,16, 5)( 2, 3,12, 8, 9,18,14,15, 6)$
	$ 9, 9 $	$6$	$9$	$( 1, 4,17, 7,10, 5,13,16,11)( 2, 3,18, 8, 9, 6,14,15,12)$
	$ 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,10,16)( 5,11,17)( 6,12,18)$

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);