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Properties

Label	18T141
Degree	$18$
Order	$324$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^3:A_4$

Related objects

Number fields with this Galois group

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Show commands: Magma

magma: G := TransitiveGroup(18, 141);

Group action invariants

Degree $n$:		$18$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$141$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_3^3:A_4$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$2$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,15,8,17,11,6,3,13,10)(2,16,7,18,12,5,4,14,9), (1,4)(2,3)(7,9)(8,10)(11,12)(13,14)(15,16)(17,18)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$3$: $C_3$
$12$: $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4$
Degree 9: $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$

Low degree siblings

9T25, 12T132 x 2, 12T133, 18T141, 18T142, 18T143, 27T130, 27T131, 36T511, 36T512, 36T524, 36T546 x 2, 36T547
Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$6$	$3$	$(11,13,15)(12,14,16)$
	$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$12$	$3$	$( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$27$	$2$	$( 3,17)( 4,18)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$
	$ 6, 2, 2, 2, 2, 2, 1, 1 $	$54$	$6$	$( 3,17)( 4,18)( 5, 8, 9, 6, 7,10)(11,12)(13,16)(14,15)$
	$ 3, 3, 3, 3, 3, 3 $	$4$	$3$	$( 1, 3,17)( 2, 4,18)( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$
	$ 3, 3, 3, 3, 3, 3 $	$4$	$3$	$( 1, 3,17)( 2, 4,18)( 5, 7, 9)( 6, 8,10)(11,15,13)(12,16,14)$
	$ 3, 3, 3, 3, 3, 3 $	$36$	$3$	$( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$
	$ 9, 9 $	$36$	$9$	$( 1, 5,11, 3, 7,13,17, 9,15)( 2, 6,12, 4, 8,14,18,10,16)$
	$ 9, 9 $	$36$	$9$	$( 1, 5,11,17, 9,15, 3, 7,13)( 2, 6,12,18,10,16, 4, 8,14)$
	$ 3, 3, 3, 3, 3, 3 $	$36$	$3$	$( 1,11, 5)( 2,12, 6)( 3,13, 7)( 4,14, 8)( 9,17,15)(10,18,16)$
	$ 9, 9 $	$36$	$9$	$( 1,11, 7, 3,13, 9,17,15, 5)( 2,12, 8, 4,14,10,18,16, 6)$
	$ 9, 9 $	$36$	$9$	$( 1,11, 9,17,15, 7, 3,13, 5)( 2,12,10,18,16, 8, 4,14, 6)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$324=2^{2} \cdot 3^{4}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		324.160	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A1	3A-1	3B	3C	3D1	3D-1	6A	9A1	9A-1	9B1	9B-1
	Size	1	27	4	4	6	12	36	36	54	36	36	36	36
	2 P	1A	1A	3A-1	3A1	3B	3C	3D-1	3D1	3B	9A1	9B1	9B-1	9A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	2A	3A-1	3A-1	3A1	3A1
	Type
324.160.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
324.160.1b1	C	$1$	$1$	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
324.160.1b2	C	$1$	$1$	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
324.160.3a	R	$3$	$- 1$	$3$	$3$	$3$	$3$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$
324.160.4a1	C	$4$	$0$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$- 2$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$0$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
324.160.4a2	C	$4$	$0$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$- 2$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$0$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
324.160.4b1	C	$4$	$0$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$- 2$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$0$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$
324.160.4b2	C	$4$	$0$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$- 2$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$0$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$
324.160.4c1	C	$4$	$0$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$- 2$	$1$	$1$	$1$	$0$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$
324.160.4c2	C	$4$	$0$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$- 2$	$1$	$1$	$1$	$0$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$
324.160.6a	R	$6$	$2$	$- 3$	$- 3$	$3$	$0$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$
324.160.6b	R	$6$	$- 2$	$- 3$	$- 3$	$3$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$
324.160.12a	R	$12$	$0$	$3$	$3$	$0$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);