Galois group: 21T14

• Label: 21T14
• Degree: $21$
• Order: $168$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\PSL(2,7)$

Group action invariants

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

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

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

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 24T284, 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	$1^{21}$	$1$	$1$	$0$	$()$
2A	$2^{8},1^{5}$	$21$	$2$	$8$	$( 1,18)( 2,17)( 3,16)( 4, 9)( 5, 8)( 6, 7)(10,11)(19,20)$
3A	$3^{7}$	$56$	$3$	$14$	$( 1,12,13)( 2,11,14)( 3,10,15)( 4,21,16)( 5,19,18)( 6,20,17)( 7, 9, 8)$
4A	$4^{4},2^{2},1$	$42$	$4$	$14$	$( 1, 7,18, 6)( 2, 9,17, 4)( 3, 8,16, 5)(10,19,11,20)(12,21)(13,14)$
7A1	$7^{3}$	$24$	$7$	$18$	$( 1,14,11, 6, 7,16,21)( 2,13,12, 4, 8,18,20)( 3,15,10, 5, 9,17,19)$
7A-1	$7^{3}$	$24$	$7$	$18$	$( 1,16, 6,14,21, 7,11)( 2,18, 4,13,20, 8,12)( 3,17, 5,15,19, 9,10)$

Malle's constant $a(G)$:     $1/8$

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	$8$	$0$	$-1$	$0$	$1$	$1$