Galois group: 6T5

• Label: 6T5
• Degree: $6$
• Order: $18$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_3\times C_3$

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$5$
Group:		$S_3\times C_3$
CHM label:		$F_{18}(6) = [3^{2}]2 = 3 wr 2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,4)(2,5)(3,6), (2,4,6)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Low degree siblings

9T4, 18T3

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

Group invariants

Order:		$18=2 \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		18.3
Character table:

		1A	2A	3A1	3A-1	3B	3C1	3C-1	6A1	6A-1
	Size	1	3	1	1	2	2	2	3	3
	2 P	1A	1A	3A-1	3A1	3B	3C-1	3C1	3A1	3A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	2A	2A
	Type
18.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
18.3.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
18.3.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
18.3.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
18.3.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
18.3.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
18.3.2a	R	$2$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$
18.3.2b1	C	$2$	$0$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$
18.3.2b2	C	$2$	$0$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$