Galois group: 6T13

• Label: 6T13
• Degree: $6$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:D_4$

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$13$
Group:		$C_3^2:D_4$
CHM label:		$F_{36}(6):2 = [S(3)^{2}]2 = S(3) wr 2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(2,4), (1,4)(2,5)(3,6), (2,4,6)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

6T13, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2

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	$2^{3}$	$6$	$2$	$3$	$(1,2)(3,4)(5,6)$
2B	$2,1^{4}$	$6$	$2$	$1$	$(3,5)$
2C	$2^{2},1^{2}$	$9$	$2$	$2$	$(3,5)(4,6)$
3A	$3^{2}$	$4$	$3$	$4$	$(1,5,3)(2,6,4)$
3B	$3,1^{3}$	$4$	$3$	$2$	$(1,5,3)$
4A	$4,2$	$18$	$4$	$4$	$(1,2)(3,4,5,6)$
6A	$6$	$12$	$6$	$5$	$(1,2,5,6,3,4)$
6B	$3,2,1$	$12$	$6$	$3$	$(2,6,4)(3,5)$

Malle's constant $a(G)$:     $1$

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.40
Character table:

		1A	2A	2B	2C	3A	3B	4A	6A	6B
	Size	1	6	6	9	4	4	18	12	12
	2 P	1A	1A	1A	1A	3A	3B	2C	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	4A	2A	2B
	Type
72.40.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.40.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.40.1c	R	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
72.40.1d	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$
72.40.2a	R	$2$	$0$	$0$	$-2$	$2$	$2$	$0$	$0$	$0$
72.40.4a	R	$4$	$0$	$2$	$0$	$-2$	$1$	$0$	$0$	$-1$
72.40.4b	R	$4$	$2$	$0$	$0$	$1$	$-2$	$0$	$-1$	$0$
72.40.4c	R	$4$	$-2$	$0$	$0$	$1$	$-2$	$0$	$1$	$0$
72.40.4d	R	$4$	$0$	$-2$	$0$	$-2$	$1$	$0$	$0$	$1$