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 invariants

Abstract group:		$C_3^2:C_4$
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 $t$:		$10$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{6}$	$1$	$1$	$0$	$()$
2A	$2^{2},1^{2}$	$9$	$2$	$2$	$(1,3)(2,4)$
3A	$3,1^{3}$	$4$	$3$	$2$	$(1,3,5)$
3B	$3^{2}$	$4$	$3$	$4$	$(1,3,5)(2,4,6)$
4A1	$4,2$	$9$	$4$	$4$	$(1,2,3,4)(5,6)$
4A-1	$4,2$	$9$	$4$	$4$	$(1,4,3,2)(5,6)$

Malle's constant $a(G)$:     $1/2$

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	$4$	$0$	$1$	$-2$	$0$	$0$

Regular extensions

$f_{ 1 } =$

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