Galois group: 12T14

• Label: 12T14
• Degree: $12$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $D_4 \times C_3$

Group invariants

Abstract group:		$D_4 \times C_3$
Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$2$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$14$
CHM label:		$D(4)[x]C(3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,7)(3,9)(5,11)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$

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
$8$:	$D_{4}$
$12$:	$C_6\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $D_{4}$

Degree 6: $C_6$

Low degree siblings

12T14, 24T15

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	$2^{6}$	$1$	$2$	$6$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B	$2^{3},1^{6}$	$2$	$2$	$3$	$( 1, 7)( 3, 9)( 5,11)$
2C	$2^{6}$	$2$	$2$	$6$	$( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$
3A1	$3^{4}$	$1$	$3$	$8$	$( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3A-1	$3^{4}$	$1$	$3$	$8$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
4A	$4^{3}$	$2$	$4$	$9$	$( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
6A1	$6^{2}$	$1$	$6$	$10$	$( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6A-1	$6^{2}$	$1$	$6$	$10$	$( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
6B1	$6,3^{2}$	$2$	$6$	$9$	$( 1,11, 9, 7, 5, 3)( 2, 6,10)( 4, 8,12)$
6B-1	$6,3^{2}$	$2$	$6$	$9$	$( 1, 9, 5)( 2, 4, 6, 8,10,12)( 3,11, 7)$
6C1	$6^{2}$	$2$	$6$	$10$	$( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
6C-1	$6^{2}$	$2$	$6$	$10$	$( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$
12A1	$12$	$2$	$12$	$11$	$( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
12A-1	$12$	$2$	$12$	$11$	$( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$:     $1/3$

Character table

		1A	2A	2B	2C	3A1	3A-1	4A	6A1	6A-1	6B1	6B-1	6C1	6C-1	12A1	12A-1
	Size	1	1	2	2	1	1	2	1	1	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	3A-1	3A1	2A	3A-1	3A1	3A1	3A-1	3A1	3A-1	6A1	6A-1
	3 P	1A	2A	2B	2C	1A	1A	4A	2A	2A	2B	2B	2C	2C	4A	4A
	Type
24.10.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.10.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
24.10.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
24.10.1d	R	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
24.10.1e1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
24.10.1e2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
24.10.1f1	C	$1$	$1$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
24.10.1f2	C	$1$	$1$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
24.10.1g1	C	$1$	$1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
24.10.1g2	C	$1$	$1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
24.10.1h1	C	$1$	$1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
24.10.1h2	C	$1$	$1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
24.10.2a	R	$2$	$-2$	$0$	$0$	$2$	$2$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
24.10.2b1	C	$2$	$-2$	$0$	$0$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$0$	$-2{\zeta }_3^{-1}$	$-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
24.10.2b2	C	$2$	$-2$	$0$	$0$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$0$	$-2{\zeta }_3$	$-2{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed