Galois group: 6T9

• Label: 6T9
• Degree: $6$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_3^2$

Group invariants

Abstract group:		$S_3^2$
Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$9$
CHM label:		$F_{18}(6):2 = [1/2.S(3)^{2}]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,4)(2,5)(3,6)$, $(2,4,6)$, $(1,5)(2,4)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$ x 2
$12$:	$D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

9T8, 12T16, 18T9, 18T11 x 2, 36T13

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	$2^{3}$	$3$	$2$	$3$	$(1,6)(2,5)(3,4)$
2B	$2^{3}$	$3$	$2$	$3$	$(1,2)(3,4)(5,6)$
2C	$2^{2},1^{2}$	$9$	$2$	$2$	$(1,5)(4,6)$
3A	$3^{2}$	$2$	$3$	$4$	$(1,5,3)(2,4,6)$
3B	$3^{2}$	$2$	$3$	$4$	$(1,3,5)(2,4,6)$
3C	$3,1^{3}$	$4$	$3$	$2$	$(1,5,3)$
6A	$6$	$6$	$6$	$5$	$(1,4,5,6,3,2)$
6B	$6$	$6$	$6$	$5$	$(1,6,3,2,5,4)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	3A	3B	3C	6A	6B
	Size	1	3	3	9	2	2	4	6	6
	2 P	1A	1A	1A	1A	3A	3B	3C	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	1A	2A	2B
	Type
36.10.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.10.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
36.10.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$
36.10.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$
36.10.2a	R	$2$	$0$	$2$	$0$	$2$	$-1$	$-1$	$0$	$-1$
36.10.2b	R	$2$	$2$	$0$	$0$	$-1$	$2$	$-1$	$-1$	$0$
36.10.2c	R	$2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$1$	$0$
36.10.2d	R	$2$	$0$	$-2$	$0$	$2$	$-1$	$-1$	$0$	$1$
36.10.4a	R	$4$	$0$	$0$	$0$	$-2$	$-2$	$1$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{6} + 6 x^{4} + t x^{3} + 9 x^{2} + 3 t x + 4$