Galois group: 18T73

• Label: 18T73
• Degree: $18$
• Order: $144$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_9:C_2$

Group invariants

Abstract group:		$F_9:C_2$
Order:		$144=2^{4} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$73$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,14,17,11,3,16)(2,13,18,12,4,15)(5,7,10)(6,8,9)$, $(1,2)(3,5,17,15)(4,6,18,16)(7,9,13,11)(8,10,14,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$
$16$:	$QD_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $(C_3^2:C_8):C_2$

Low degree siblings

9T19, 12T84, 18T68, 18T71, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175

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	$1^{18}$	$1$	$1$	$0$	$()$
2A	$2^{8},1^{2}$	$9$	$2$	$8$	$( 1,17)( 2,18)( 5,13)( 6,14)( 7,12)( 8,11)( 9,16)(10,15)$
2B	$2^{6},1^{6}$	$12$	$2$	$6$	$( 1,11)( 2,12)( 3,14)( 4,13)(15,18)(16,17)$
3A	$3^{6}$	$8$	$3$	$12$	$( 1, 3,17)( 2, 4,18)( 5, 7,10)( 6, 8, 9)(11,14,16)(12,13,15)$
4A	$4^{4},1^{2}$	$18$	$4$	$12$	$( 1,14,17, 6)( 2,13,18, 5)( 7,10,12,15)( 8, 9,11,16)$
4B	$4^{4},2$	$36$	$4$	$13$	$( 1,10, 3,13)( 2, 9, 4,14)( 5, 8,12,16)( 6, 7,11,15)(17,18)$
6A	$6^{2},3^{2}$	$24$	$6$	$14$	$( 1,16, 3,11,17,14)( 2,15, 4,12,18,13)( 5,10, 7)( 6, 9, 8)$
8A1	$8^{2},2$	$18$	$8$	$15$	$( 1,10,14,12,17,15, 6, 7)( 2, 9,13,11,18,16, 5, 8)( 3, 4)$
8A-1	$8^{2},2$	$18$	$8$	$15$	$( 1, 7, 6,15,17,12,14,10)( 2, 8, 5,16,18,11,13, 9)( 3, 4)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	3A	4A	4B	6A	8A1	8A-1
	Size	1	9	12	8	18	36	24	18	18
	2 P	1A	1A	1A	3A	2A	2A	3A	4A	4A
	3 P	1A	2A	2B	1A	4A	4B	2B	8A1	8A-1
	Type
144.182.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
144.182.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
144.182.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
144.182.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
144.182.2a	R	$2$	$2$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$
144.182.2b1	C	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
144.182.2b2	C	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$
144.182.8a	R	$8$	$0$	$2$	$-1$	$0$	$0$	$-1$	$0$	$0$
144.182.8b	R	$8$	$0$	$-2$	$-1$	$0$	$0$	$1$	$0$	$0$

Regular extensions

Data not computed