Galois group: 8T23

• Label: 8T23
• Degree: $8$
• Order: $48$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\textrm{GL(2,3)}$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$23$
Group:		$\textrm{GL(2,3)}$
CHM label:		$2S_{4}(8)=GL(2,3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,2,3,4,5,6,7,8), (1,3,8)(4,5,7)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$
$24$:	$S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Low degree siblings

8T23, 16T66, 24T22

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 1, 1 $	$12$	$2$	$(2,3)(4,8)(6,7)$
	$ 3, 3, 1, 1 $	$8$	$3$	$(2,7,8)(3,4,6)$
	$ 8 $	$6$	$8$	$(1,2,3,4,5,6,7,8)$
	$ 4, 4 $	$6$	$4$	$(1,2,5,6)(3,8,7,4)$
	$ 6, 2 $	$8$	$6$	$(1,2,7,5,6,3)(4,8)$
	$ 8 $	$6$	$8$	$(1,2,8,3,5,6,4,7)$
	$ 2, 2, 2, 2 $	$1$	$2$	$(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:		$48=2^{4} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		48.29
Character table:

		1A	2A	2B	3A	4A	6A	8A1	8A-1
	Size	1	1	12	8	6	8	6	6
	2 P	1A	1A	1A	3A	2A	3A	4A	4A
	3 P	1A	2A	2B	1A	4A	2A	8A1	8A-1
	Type
48.29.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.29.1b	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
48.29.2a	R	$2$	$2$	$0$	$-1$	$2$	$-1$	$0$	$0$
48.29.2b1	C	$2$	$-2$	$0$	$-1$	$0$	$1$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
48.29.2b2	C	$2$	$-2$	$0$	$-1$	$0$	$1$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$
48.29.3a	R	$3$	$3$	$1$	$0$	$-1$	$0$	$-1$	$-1$
48.29.3b	R	$3$	$3$	$-1$	$0$	$-1$	$0$	$1$	$1$
48.29.4a	R	$4$	$-4$	$0$	$1$	$0$	$-1$	$0$	$0$