Galois group: 24T278

• Label: 24T278
• Degree: $24$
• 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 $n$:		$24$
Transitive number $t$:		$278$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,18)(2,17)(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(19,20)(21,23)(22,24)$, $(1,13,12,24,10,6,19,8)(2,14,11,23,9,5,20,7)(3,16,18,22)(4,15,17,21)$

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: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: None

Degree 8: $D_4$

Degree 12: 12T84

Low degree siblings

9T19, 12T84, 18T68, 18T71, 18T73, 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	$1^{24}$	$1$	$1$	$0$	$()$
2A	$2^{8},1^{8}$	$9$	$2$	$8$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,22)( 6,21)(15,24)(16,23)$
2B	$2^{12}$	$12$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,10)(11,19)(12,20)(13,23)(14,24)(15,22)(16,21)(17,18)$
3A	$3^{6},1^{6}$	$8$	$3$	$12$	$( 1,10,17)( 2, 9,18)( 5,22,14)( 6,21,13)( 7,16,23)( 8,15,24)$
4A	$4^{4},2^{4}$	$18$	$4$	$16$	$( 1, 4,10,12)( 2, 3, 9,11)( 5,16,22,23)( 6,15,21,24)( 7,14)( 8,13)(17,19)(18,20)$
4B	$4^{4},2^{4}$	$36$	$4$	$16$	$( 1,16)( 2,15)( 3,13,11,21)( 4,14,12,22)( 5,19)( 6,20)( 7,17,23,10)( 8,18,24, 9)$
6A	$6^{3},2^{3}$	$24$	$6$	$18$	$( 1,18,10, 2,17, 9)( 3, 4)( 5,24,22, 8,14,15)( 6,23,21, 7,13,16)(11,19)(12,20)$
8A1	$8^{2},4^{2}$	$18$	$8$	$20$	$( 1,21, 4,24,10, 6,12,15)( 2,22, 3,23, 9, 5,11,16)( 7,18,14,20)( 8,17,13,19)$
8A-1	$8^{2},4^{2}$	$18$	$8$	$20$	$( 1,15,12, 6,10,24, 4,21)( 2,16,11, 5, 9,23, 3,22)( 7,20,14,18)( 8,19,13,17)$

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