Galois group: 24T284

• Label: 24T284
• Degree: $24$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\PSL(2,7)$

Group invariants

Abstract group:		$\PSL(2,7)$
Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$284$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		$(1,8,24,21,17,10,13)(2,9,22,19,18,11,14)(3,7,23,20,16,12,15)$, $(1,7,12,13,22,5,18)(2,8,10,14,23,6,16)(3,9,11,15,24,4,17)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 8: $\PSL(2,7)$

Degree 12: None

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 28T32, 42T37, 42T38 x 2

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	$2^{12}$	$21$	$2$	$12$	$( 1, 9)( 2, 7)( 3, 8)( 4,21)( 5,19)( 6,20)(10,15)(11,13)(12,14)(16,23)(17,24)(18,22)$
3A	$3^{8}$	$56$	$3$	$16$	$( 1,17, 5)( 2,18, 6)( 3,16, 4)( 7,20,22)( 8,21,23)( 9,19,24)(10,12,11)(13,14,15)$
4A	$4^{6}$	$42$	$4$	$18$	$( 1,11, 9,13)( 2,12, 7,14)( 3,10, 8,15)( 4,16,21,23)( 5,17,19,24)( 6,18,20,22)$
7A1	$7^{3},1^{3}$	$24$	$7$	$18$	$( 1, 7,14,11,20,17, 6)( 2, 8,15,12,21,18, 4)( 3, 9,13,10,19,16, 5)$
7A-1	$7^{3},1^{3}$	$24$	$7$	$18$	$( 1, 6,17,20,11,14, 7)( 2, 4,18,21,12,15, 8)( 3, 5,16,19,10,13, 9)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	3A	4A	7A1	7A-1
	Size	1	21	56	42	24	24
	2 P	1A	1A	3A	2A	7A1	7A-1
	3 P	1A	2A	1A	4A	7A-1	7A1
	7 P	1A	2A	3A	4A	1A	1A
	Type
168.42.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
168.42.3a1	C	$3$	$-1$	$0$	$1$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
168.42.3a2	C	$3$	$-1$	$0$	$1$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
168.42.6a	R	$6$	$2$	$0$	$0$	$-1$	$-1$
168.42.7a	R	$7$	$-1$	$1$	$-1$	$0$	$0$
168.42.8a	R	$8$	$0$	$-1$	$0$	$1$	$1$

Regular extensions

Data not computed