Galois group: 9T32

• Label: 9T32
• Degree: $9$
• Order: $1512$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $\mathrm{P}\Gamma\mathrm{L}(2,8)$

Group action invariants

Degree $n$:		$9$
Transitive number $t$:		$32$
Group:		$\mathrm{P}\Gamma\mathrm{L}(2,8)$
CHM label:		$L(9):3=P|L(2,8)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(2,5)(3,6)(4,7)(8,9), (1,9)(2,3)(4,5)(6,7), (1,2,4,3,6,7,5), (2,4,6)(3,5,7)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

27T391, 28T165, 36T2342

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	$1^{9}$	$1$	$1$	$0$	$()$
2A	$2^{4},1$	$63$	$2$	$4$	$(1,4)(2,7)(3,6)(5,9)$
3A	$3^{3}$	$56$	$3$	$6$	$(1,2,3)(4,6,7)(5,9,8)$
3B1	$3^{2},1^{3}$	$84$	$3$	$4$	$(1,2,7)(4,8,9)$
3B-1	$3^{2},1^{3}$	$84$	$3$	$4$	$(1,7,2)(4,9,8)$
6A1	$6,2,1$	$252$	$6$	$6$	$(1,5,3,4,9,6)(2,7)$
6A-1	$6,2,1$	$252$	$6$	$6$	$(1,6,9,4,3,5)(2,7)$
7A	$7,1^{2}$	$216$	$7$	$6$	$(1,2,6,9,8,3,5)$
9A	$9$	$168$	$9$	$8$	$(1,7,5,3,6,8,2,4,9)$
9B1	$9$	$168$	$9$	$8$	$(1,3,8,4,6,5,9,7,2)$
9B-1	$9$	$168$	$9$	$8$	$(1,8,7,2,5,4,3,9,6)$

Malle's constant $a(G)$:     $1/4$

Group invariants

Order:		$1512=2^{3} \cdot 3^{3} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		1512.779
Character table:

		1A	2A	3A	3B1	3B-1	6A1	6A-1	7A	9A	9B1	9B-1
	Size	1	63	56	84	84	252	252	216	168	168	168
	2 P	1A	1A	3A	3B-1	3B1	3B1	3B-1	7A	9B1	9A	9B-1
	3 P	1A	2A	1A	1A	1A	2A	2A	7A	3A	3A	3A
	7 P	1A	2A	3A	3B1	3B-1	6A1	6A-1	1A	9B-1	9A	9B1
	Type
1512.779.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1512.779.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
1512.779.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
1512.779.7a	R	$7$	$-1$	$-2$	$1$	$1$	$-1$	$-1$	$0$	$1$	$1$	$1$
1512.779.7b1	C	$7$	$-1$	$-2$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
1512.779.7b2	C	$7$	$-1$	$-2$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
1512.779.8a	R	$8$	$0$	$-1$	$2$	$2$	$0$	$0$	$1$	$-1$	$-1$	$-1$
1512.779.8b1	C	$8$	$0$	$-1$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$0$	$0$	$1$	$-1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
1512.779.8b2	C	$8$	$0$	$-1$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$0$	$0$	$1$	$-1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
1512.779.21a	R	$21$	$-3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
1512.779.27a	R	$27$	$3$	$0$	$0$	$0$	$0$	$0$	$-1$	$0$	$0$	$0$