Galois group: 18T68

• Label: 18T68
• 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 $t$:		$68$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,9,17,16)(2,10,18,15)(5,12,13,8)(6,11,14,7)$, $(1,14,12,7,16,3,17,10)(2,13,11,8,15,4,18,9)(5,6)$

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, 18T71, 18T73, 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	$2^{8},1^{2}$	$9$	$2$	$8$	$( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,15)( 8,16)( 9,13)(10,14)$
2B	$2^{9}$	$12$	$2$	$9$	$( 1,10)( 2, 9)( 3, 5)( 4, 6)( 7,17)( 8,18)(11,12)(13,14)(15,16)$
3A	$3^{6}$	$8$	$3$	$12$	$( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$
4A	$4^{4},1^{2}$	$18$	$4$	$12$	$( 1,16, 4, 8)( 2,15, 3, 7)( 5, 9,12,13)( 6,10,11,14)$
4B	$4^{4},1^{2}$	$36$	$4$	$12$	$( 3,14,18, 7)( 4,13,17, 8)( 5, 9,16,12)( 6,10,15,11)$
6A	$6^{3}$	$24$	$6$	$15$	$( 1, 7, 4,10,17, 6)( 2, 8, 3, 9,18, 5)(11,16,14,12,15,13)$
8A1	$8^{2},2$	$18$	$8$	$15$	$( 1,10,16,11, 4,14, 8, 6)( 2, 9,15,12, 3,13, 7, 5)(17,18)$
8A-1	$8^{2},2$	$18$	$8$	$15$	$( 1, 6, 8,14, 4,11,16,10)( 2, 5, 7,13, 3,12,15, 9)(17,18)$

Malle's constant $a(G)$:     $1/8$

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