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Properties

Label	12T59
Degree	$12$
Order	$96$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_2^3:A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 59);

Group action invariants

Degree $n$:		$12$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$59$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_2^3:A_4$
CHM label:		$[2^{3}]A_{4}(6)$
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:		(2,8)(3,9)(4,10)(5,11), (1,12)(2,3)(6,7)(8,9), (1,12)(2,3)(4,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$6$: $C_6$
$12$: $A_4$
$24$: $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: $A_4$

Low degree siblings

8T33 x 2, 12T58 x 2, 12T59, 16T183, 24T181 x 2, 24T182 x 2, 24T183 x 2, 24T184 x 2, 24T185, 24T186, 32T389
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
	$1^{12}$	$1$	$1$	$()$
	$2^{3},1^{6}$	$4$	$2$	$(4,5)(6,7)(8,9)$
	$2^{4},1^{4}$	$3$	$2$	$( 2, 3)( 4, 5)( 8, 9)(10,11)$
	$2^{4},1^{4}$	$6$	$2$	$( 2, 8)( 3, 9)( 4,10)( 5,11)$
	$4^{2},2,1^{2}$	$12$	$4$	$( 2, 8, 3, 9)( 4,11, 5,10)( 6, 7)$
	$3^{4}$	$16$	$3$	$( 1, 2, 4)( 3, 5,12)( 6, 9,11)( 7, 8,10)$
	$6,3^{2}$	$16$	$6$	$( 1, 2, 4,12, 3, 5)( 6, 8,11)( 7, 9,10)$
	$3^{4}$	$16$	$3$	$( 1, 4, 2)( 3,12, 5)( 6,11, 9)( 7,10, 8)$
	$6,3^{2}$	$16$	$6$	$( 1, 4, 3,12, 5, 2)( 6,10, 8)( 7,11, 9)$
	$2^{6}$	$6$	$2$	$( 1, 6)( 2, 3)( 4,10)( 5,11)( 7,12)( 8, 9)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$96=2^{5} \cdot 3$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		96.70	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	3A1	3A-1	4A	6A1	6A-1
	Size	1	3	4	6	6	16	16	12	16	16
	2 P	1A	1A	1A	1A	1A	3A-1	3A1	2A	3A1	3A-1
	3 P	1A	2A	2B	2C	2D	1A	1A	4A	2B	2B
	Type
96.70.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.70.1b	R	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$
96.70.1c1	C	$1$	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$
96.70.1c2	C	$1$	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$
96.70.1d1	C	$1$	$1$	$- 1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
96.70.1d2	C	$1$	$1$	$- 1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
96.70.3a	R	$3$	$3$	$3$	$- 1$	$- 1$	$0$	$0$	$- 1$	$0$	$0$
96.70.3b	R	$3$	$3$	$- 3$	$- 1$	$- 1$	$0$	$0$	$1$	$0$	$0$
96.70.6a	R	$6$	$- 2$	$0$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$
96.70.6b	R	$6$	$- 2$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);