Galois group: 8T13

• Label: 8T13
• Degree: $8$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $A_4\times C_2$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$13$
Group:		$A_4\times C_2$
CHM label:		$E(8):3=A(4)[x]2$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$
$12$:	$A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $A_4$

Low degree siblings

6T6, 12T6, 12T7, 24T9

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 $	$1$	$1$	$()$
	$ 3, 3, 1, 1 $	$4$	$3$	$(2,3,8)(4,7,5)$
	$ 3, 3, 1, 1 $	$4$	$3$	$(2,8,3)(4,5,7)$
	$ 2, 2, 2, 2 $	$3$	$2$	$(1,2)(3,8)(4,7)(5,6)$
	$ 6, 2 $	$4$	$6$	$(1,4,8,6,3,7)(2,5)$
	$ 6, 2 $	$4$	$6$	$(1,4,2,6,3,5)(7,8)$
	$ 2, 2, 2, 2 $	$3$	$2$	$(1,4)(2,7)(3,6)(5,8)$
	$ 2, 2, 2, 2 $	$1$	$2$	$(1,6)(2,5)(3,4)(7,8)$

Group invariants

Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		24.13
Character table:

		1A	2A	2B	2C	3A1	3A-1	6A1	6A-1
	Size	1	1	3	3	4	4	4	4
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	2A	2A
	Type
24.13.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.13.1b	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
24.13.1c1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
24.13.1c2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
24.13.1d1	C	$1$	$-1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
24.13.1d2	C	$1$	$-1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
24.13.3a	R	$3$	$3$	$-1$	$-1$	$0$	$0$	$0$	$0$
24.13.3b	R	$3$	$-3$	$-1$	$1$	$0$	$0$	$0$	$0$