Galois group: 8T12

• Label: 8T12
• Degree: $8$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\SL(2,3)$

Group invariants

Abstract group:		$\SL(2,3)$
Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$12$
CHM label:		$2A_{4}(8)=[2]A(4)=SL(2,3)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,3,5,7)(2,4,6,8)$, $(1,3,8)(4,5,7)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$
$12$:	$A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Low degree siblings

24T7

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	$1^{8}$	$1$	$1$	$0$	$()$
2A	$2^{4}$	$1$	$2$	$4$	$(1,5)(2,6)(3,7)(4,8)$
3A1	$3^{2},1^{2}$	$4$	$3$	$4$	$(2,8,7)(3,6,4)$
3A-1	$3^{2},1^{2}$	$4$	$3$	$4$	$(2,7,8)(3,4,6)$
4A	$4^{2}$	$6$	$4$	$6$	$(1,7,5,3)(2,8,6,4)$
6A1	$6,2$	$4$	$6$	$6$	$(1,5)(2,3,8,6,7,4)$
6A-1	$6,2$	$4$	$6$	$6$	$(1,8,2,5,4,6)(3,7)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	3A1	3A-1	4A	6A1	6A-1
	Size	1	1	4	4	6	4	4
	2 P	1A	1A	3A-1	3A1	2A	3A1	3A-1
	3 P	1A	2A	1A	1A	4A	2A	2A
	Type
24.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.3.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
24.3.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
24.3.2a	S	$2$	$-2$	$-1$	$-1$	$0$	$1$	$1$
24.3.2b1	C	$2$	$-2$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	${\zeta }_3^{-1}$	${\zeta }_3$
24.3.2b2	C	$2$	$-2$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	${\zeta }_3$	${\zeta }_3^{-1}$
24.3.3a	R	$3$	$3$	$0$	$0$	$-1$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{8} + \left(t^{2} + t + 15\right) x^{6} + \left(6 t^{2} + 54\right) x^{4} + \left(5 t^{2} - 3 t + 27\right) x^{2} + t^{2}$