Galois group: 42T37

• Label: 42T37
• Degree: $42$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\PSL(2,7)$

Group action invariants

Degree $n$:		$42$
Transitive number $t$:		$37$
Group:		$\PSL(2,7)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,10,30,22,14,40,31)(2,9,29,21,13,39,32)(3,11,25,19,16,41,36)(4,12,26,20,15,42,35)(5,7,28,23,17,37,33)(6,8,27,24,18,38,34), (1,28,33)(2,27,34)(3,30,35)(4,29,36)(5,25,31)(6,26,32)(7,10,11)(8,9,12)(13,41,21)(14,42,22)(15,40,19)(16,39,20)(17,38,24)(18,37,23)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 7: $\GL(3,2)$ x 2

Degree 14: None

Degree 21: $\PSL(2,7)$

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$21$	$2$	$( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,38)(14,37)(15,39)(16,40)(17,41)(18,42) (19,20)(21,24)(22,23)(25,32)(26,31)(27,34)(28,33)(29,35)(30,36)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $	$42$	$4$	$( 3, 5, 4, 6)( 7,19, 8,20)( 9,22,11,23)(10,21,12,24)(13,31,38,26)(14,32,37,25) (15,35,39,29)(16,36,40,30)(17,34,41,27)(18,33,42,28)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$56$	$3$	$( 1, 3, 5)( 2, 4, 6)( 7,37,31)( 8,38,32)( 9,41,35)(10,42,36)(11,40,33) (12,39,34)(13,26,19)(14,25,20)(15,30,22)(16,29,21)(17,28,24)(18,27,23)$
	$ 7, 7, 7, 7, 7, 7 $	$24$	$7$	$( 1, 9,26,33,42,21,14)( 2,10,25,34,41,22,13)( 3, 7,27,31,40,24,17) ( 4, 8,28,32,39,23,18)( 5,11,30,36,37,20,15)( 6,12,29,35,38,19,16)$
	$ 7, 7, 7, 7, 7, 7 $	$24$	$7$	$( 1,10,18,23,25,33,38)( 2, 9,17,24,26,34,37)( 3, 8,16,22,29,32,40) ( 4, 7,15,21,30,31,39)( 5,11,14,19,28,35,42)( 6,12,13,20,27,36,41)$

Group invariants

Order:		$168=2^{3} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		168.42
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	$8$	$0$	$-1$	$0$	$1$	$1$