Galois group: 6T1

• Label: 6T1
• Degree: $6$
• Order: $6$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_6$

Group invariants

Abstract group:		$C_6$
Order:		$6=2 \cdot 3$
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Low degree siblings

There are no siblings 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	$2^{3}$	$1$	$2$	$3$	$(1,4)(2,5)(3,6)$
3A1	$3^{2}$	$1$	$3$	$4$	$(1,3,5)(2,4,6)$
3A-1	$3^{2}$	$1$	$3$	$4$	$(1,5,3)(2,6,4)$
6A1	$6$	$1$	$6$	$5$	$(1,2,3,4,5,6)$
6A-1	$6$	$1$	$6$	$5$	$(1,6,5,4,3,2)$

Malle's constant $a(G)$:     $1/3$

Character table

		1A	2A	3A1	3A-1	6A1	6A-1
	Size	1	1	1	1	1	1
	2 P	1A	1A	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	1A	1A	2A	2A
	Type
6.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
6.2.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$
6.2.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
6.2.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
6.2.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
6.2.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$

Regular extensions

$f_{ 1 } =$

$x^{6} + 2 t x^{5} + \left(-5 t - 15\right) x^{4} + 20 x^{3} + 5 t x^{2} + \left(-2 t - 6\right) x + 1$