Galois group: 6T10

• Label: 6T10
• Degree: $6$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:C_4$

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$10$
Group:		$C_3^2:C_4$
CHM label:		$F_{36}(6) = 1/2[S(3)^{2}]2$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,4,5,2)(3,6), (2,4,6), (1,5)(2,4)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

6T10, 9T9, 12T17 x 2, 18T10, 36T14

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, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 1, 1 $	$9$	$2$	$(3,5)(4,6)$
	$ 3, 1, 1, 1 $	$4$	$3$	$(2,4,6)$
	$ 4, 2 $	$9$	$4$	$(1,2)(3,4,5,6)$
	$ 4, 2 $	$9$	$4$	$(1,2)(3,6,5,4)$
	$ 3, 3 $	$4$	$3$	$(1,3,5)(2,4,6)$

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		36.9
Character table:

		1A	2A	3A	3B	4A1	4A-1
	Size	1	9	4	4	9	9
	2 P	1A	1A	3A	3B	2A	2A
	3 P	1A	2A	1A	1A	4A-1	4A1
	Type
36.9.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
36.9.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$
36.9.1c1	C	$1$	$-1$	$1$	$1$	$-i$	$i$
36.9.1c2	C	$1$	$-1$	$1$	$1$	$i$	$-i$
36.9.4a	R	$4$	$0$	$-2$	$1$	$0$	$0$
36.9.4b	R	$4$	$0$	$1$	$-2$	$0$	$0$