Galois group: 6T3

• Label: 6T3
• Degree: $6$
• Order: $12$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{6}$

Group invariants

Abstract group:		$D_{6}$
Order:		$12=2^{2} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$3$
CHM label:		$D(6) = S(3)[x]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,4)(2,3)(5,6)$, $(1,2,3,4,5,6)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Low degree siblings

6T3, 12T3

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	$1^{6}$	$1$	$1$	$0$	$()$
2A	$2^{3}$	$1$	$2$	$3$	$(1,4)(2,5)(3,6)$
2B	$2^{3}$	$3$	$2$	$3$	$(1,4)(2,3)(5,6)$
2C	$2^{2},1^{2}$	$3$	$2$	$2$	$(1,3)(4,6)$
3A	$3^{2}$	$2$	$3$	$4$	$(1,5,3)(2,6,4)$
6A	$6$	$2$	$6$	$5$	$(1,6,5,4,3,2)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	3A	6A
	Size	1	1	3	3	2	2
	2 P	1A	1A	1A	1A	3A	3A
	3 P	1A	2A	2B	2C	1A	2A
	Type
12.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
12.4.1b	R	$1$	$-1$	$-1$	$1$	$1$	$-1$
12.4.1c	R	$1$	$-1$	$1$	$-1$	$1$	$-1$
12.4.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$
12.4.2a	R	$2$	$2$	$0$	$0$	$-1$	$-1$
12.4.2b	R	$2$	$-2$	$0$	$0$	$-1$	$1$

Regular extensions

$f_{ 1 } =$

$x^{6} + 2 x^{4} + x^{2} + t$