Galois group: 8T36

• Label: 8T36
• Degree: $8$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_2^3:(C_7: C_3)$

Group invariants

Abstract group:		$C_2^3:(C_7: C_3)$
Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$36$
CHM label:		$E(8):F_{21}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,3)(2,8)(4,6)(5,7)$, $(1,2,6,3,4,5,7)$, $(1,8)(2,3)(4,5)(6,7)$, $(1,2,3)(4,6,5)$, $(1,5)(2,6)(3,7)(4,8)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$
$21$:	$C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

14T11, 24T283, 28T27, 42T26

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	$2^{4}$	$7$	$2$	$4$	$(1,8)(2,3)(4,5)(6,7)$
3A1	$3^{2},1^{2}$	$28$	$3$	$4$	$(2,6,4)(3,7,5)$
3A-1	$3^{2},1^{2}$	$28$	$3$	$4$	$(2,4,6)(3,5,7)$
6A1	$6,2$	$28$	$6$	$6$	$(1,8)(2,5,6,3,4,7)$
6A-1	$6,2$	$28$	$6$	$6$	$(1,8)(2,7,4,3,6,5)$
7A1	$7,1$	$24$	$7$	$6$	$(1,8,3,4,2,7,5)$
7A-1	$7,1$	$24$	$7$	$6$	$(1,5,7,2,4,3,8)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	3A1	3A-1	6A1	6A-1	7A1	7A-1
	Size	1	7	28	28	28	28	24	24
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	7A1	7A-1
	3 P	1A	2A	1A	1A	2A	2A	7A-1	7A1
	7 P	1A	2A	3A1	3A-1	6A1	6A-1	1A	1A
	Type
168.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
168.43.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
168.43.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
168.43.3a1	C	$3$	$3$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
168.43.3a2	C	$3$	$3$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
168.43.7a	R	$7$	$-1$	$1$	$1$	$-1$	$-1$	$0$	$0$
168.43.7b1	C	$7$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$
168.43.7b2	C	$7$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$9 x^{8} + 9 t x^{7} + \left(9 t^{2} + 108\right) x^{6} + \left(9 t^{3} + 108 t + 72\right) x^{5} + \left(9 t^{4} + 108 t^{2} + 72 t + 486\right) x^{4} + \left(9 t^{5} + 108 t^{3} + 72 t^{2} + 486 t + 504\right) x^{3} + \left(9 t^{6} + 108 t^{4} + 72 t^{3} + 486 t^{2} + 504 t + 1228\right) x^{2} + \left(9 t^{7} + 108 t^{5} + 72 t^{4} + 486 t^{3} + 504 t^{2} + 1228 t + 888\right) x + \left(9 t^{8} + 108 t^{6} + 72 t^{5} + 486 t^{4} + 504 t^{3} + 1228 t^{2} + 888 t + 1369\right)$