Galois group: 24T279

• Label: 24T279
• 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 $t$:		$279$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,9)(2,10)(3,20,11)(4,19,12)(5,15,14,24,22,8)(6,16,13,23,21,7)(17,18)$, $(1,5,3,23,17,21,19,8)(2,6,4,24,18,22,20,7)(9,13,12,15,10,14,11,16)$

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 4: $D_{4}$

Degree 6: None

Degree 8: $QD_{16}$

Degree 12: 12T84

Low degree siblings

9T19, 12T84, 18T68, 18T71, 18T73, 24T278, 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	$2^{12}$	$9$	$2$	$12$	$( 1, 2)( 3,19)( 4,20)( 5,13)( 6,14)( 7, 8)( 9,18)(10,17)(11,12)(15,23)(16,24)(21,22)$
2B	$2^{9},1^{6}$	$12$	$2$	$9$	$( 1, 9)( 2,10)( 5,24)( 6,23)( 7,13)( 8,14)(15,22)(16,21)(17,18)$
3A	$3^{6},1^{6}$	$8$	$3$	$12$	$( 3,11,20)( 4,12,19)( 5,14,22)( 6,13,21)( 7,16,23)( 8,15,24)$
4A	$4^{6}$	$18$	$4$	$18$	$( 1,11, 2,12)( 3, 9,19,18)( 4,10,20,17)( 5,23,13,15)( 6,24,14,16)( 7,21, 8,22)$
4B	$4^{6}$	$36$	$4$	$18$	$( 1,15, 2,16)( 3,21,12,14)( 4,22,11,13)( 5,20, 6,19)( 7,10,24,17)( 8, 9,23,18)$
6A	$6^{2},3^{2},2^{3}$	$24$	$6$	$17$	$( 1, 9)( 2,10)( 3,20,11)( 4,19,12)( 5,15,14,24,22, 8)( 6,16,13,23,21, 7)(17,18)$
8A1	$8^{3}$	$18$	$8$	$21$	$( 1,22,11, 7, 2,21,12, 8)( 3,16, 9, 6,19,24,18,14)( 4,15,10, 5,20,23,17,13)$
8A-1	$8^{3}$	$18$	$8$	$21$	$( 1, 8,12,21, 2, 7,11,22)( 3,14,18,24,19, 6, 9,16)( 4,13,17,23,20, 5,10,15)$

Malle's constant $a(G)$:     $1/9$

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