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Properties

Label	12T6
Degree	$12$
Order	$24$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$A_4\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(12, 6);

Group action invariants

Degree $n$:		$12$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$6$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$A_4\times C_2$
CHM label:		$A_{4}(12)x2$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12)	magma: Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$6$: $C_6$
$12$: $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: $A_4$, $A_4\times C_2$

Low degree siblings

6T6, 8T13, 12T7, 24T9
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	Index	Representative
1A	$1^{12}$	$1$	$1$	$0$	$()$
2A	$2^{6}$	$1$	$2$	$6$	$( 1, 7)( 2,11)( 3,12)( 4,10)( 5, 8)( 6, 9)$
2B	$2^{6}$	$3$	$2$	$6$	$( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
2C	$2^{4},1^{4}$	$3$	$2$	$4$	$( 1,10)( 3, 9)( 4, 7)( 6,12)$
3A1	$3^{4}$	$4$	$3$	$8$	$( 1, 9, 5)( 2, 4, 3)( 6, 8, 7)(10,12,11)$
3A-1	$3^{4}$	$4$	$3$	$8$	$( 1, 5, 9)( 2, 3, 4)( 6, 7, 8)(10,11,12)$
6A1	$6^{2}$	$4$	$6$	$10$	$( 1, 8, 9, 7, 5, 6)( 2,12, 4,11, 3,10)$
6A-1	$6^{2}$	$4$	$6$	$10$	$( 1, 6, 5, 7, 9, 8)( 2,10, 3,11, 4,12)$

Malle's constant $a(G)$: $1/4$

magma: ConjugacyClasses(G);

Group invariants

Order:		$24=2^{3} \cdot 3$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		24.13	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	3A1	3A-1	6A1	6A-1
	Size	1	1	3	3	4	4	4	4
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	2A	2A
	Type
24.13.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.13.1b	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$
24.13.1c1	C	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
24.13.1c2	C	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
24.13.1d1	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
24.13.1d2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
24.13.3a	R	$3$	$3$	$- 1$	$- 1$	$0$	$0$	$0$	$0$
24.13.3b	R	$3$	$- 3$	$- 1$	$1$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);