Galois group: 12T2

• Label: 12T2
• Degree: $12$
• Order: $12$
• Cyclic: no
• Abelian: yes
• Solvable: yes
• Transitivity: $1$
• Primitive: no
• $p$-group: no
• Group: $C_6\times C_2$

Group invariants

Abstract group:		$C_6\times C_2$
Order:		$12=2^{2} \cdot 3$
Cyclic:		no
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$2$
CHM label:		$E(4)[x]C(3)=6x2$
Parity:		$1$
Transitivity:		1
Primitive:		no
$\card{\Aut(F/K)}$:		$12$
Generators:		$(1,10)(2,5)(3,12)(4,7)(6,9)(8,11)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$C_6$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3

Low degree siblings

There are no siblings 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	$2^{6}$	$1$	$2$	$6$	$( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$
2B	$2^{6}$	$1$	$2$	$6$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2C	$2^{6}$	$1$	$2$	$6$	$( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
3A1	$3^{4}$	$1$	$3$	$8$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3A-1	$3^{4}$	$1$	$3$	$8$	$( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
6A1	$6^{2}$	$1$	$6$	$10$	$( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$
6A-1	$6^{2}$	$1$	$6$	$10$	$( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
6B1	$6^{2}$	$1$	$6$	$10$	$( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
6B-1	$6^{2}$	$1$	$6$	$10$	$( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6C1	$6^{2}$	$1$	$6$	$10$	$( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$
6C-1	$6^{2}$	$1$	$6$	$10$	$( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	3A1	3A-1	6A1	6A-1	6B1	6B-1	6C1	6C-1
	Size	1	1	1	1	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	2A	2A	2B	2B	2C	2C
	Type
12.5.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
12.5.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
12.5.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
12.5.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
12.5.1e1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
12.5.1e2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
12.5.1f1	C	$1$	$-1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
12.5.1f2	C	$1$	$-1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
12.5.1g1	C	$1$	$-1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
12.5.1g2	C	$1$	$-1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
12.5.1h1	C	$1$	$1$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
12.5.1h2	C	$1$	$1$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$

Regular extensions

Data not computed