Galois group: 28T46

• Label: 28T46
• Degree: $28$
• Order: $336$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $\PGL(2,7)$

Group invariants

Abstract group:		$\PGL(2,7)$
Order:		$336=2^{4} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$28$
Transitive number $t$:		$46$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,27,23,7,14,9,21,18)(2,20,15,12,16,10,6,28)(3,5,25,22,17,13,8,26)(4,11,24,19)$, $(1,2,24,21,19,28,14,13)(3,25,16,12)(4,9,6,22,26,20,18,11)(5,15,8,7,23,27,17,10)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

8T43, 14T16, 16T713, 21T20, 24T707, 28T42, 42T81, 42T82, 42T83

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	$1^{28}$	$1$	$1$	$0$	$()$
2A	$2^{12},1^{4}$	$21$	$2$	$12$	$( 1,16)( 2,21)( 3,27)( 4, 8)( 5,26)( 7,24)( 9,17)(10,14)(11,18)(12,15)(19,25)(20,23)$
2B	$2^{12},1^{4}$	$28$	$2$	$12$	$( 1,16)( 2,17)( 3,18)( 5,14)( 6,13)( 7,15)( 9,21)(10,26)(11,27)(12,24)(19,23)(20,25)$
3A	$3^{9},1$	$56$	$3$	$18$	$( 1,12, 7)( 2,13,21)( 3,10,19)( 4, 8,22)( 5,11,20)( 6, 9,17)(14,27,25)(15,16,24)(18,26,23)$
4A	$4^{6},2^{2}$	$42$	$4$	$20$	$( 1, 8,16, 4)( 2,10,21,14)( 3,12,27,15)( 5, 9,26,17)( 6,13)( 7,11,24,18)(19,20,25,23)(22,28)$
6A	$6^{4},3,1$	$56$	$6$	$22$	$( 1,15,12,16, 7,24)( 2, 9,13,17,21, 6)( 3,23,10,18,19,26)( 4,22, 8)( 5,25,11,14,20,27)$
7A	$7^{4}$	$48$	$7$	$24$	$( 1, 5, 3, 7, 2, 4, 6)( 8,11,28,19,22,15,16)( 9,14,26,12,25,18,21)(10,20,23,17,24,13,27)$
8A1	$8^{3},4$	$42$	$8$	$24$	$( 1,20, 8,25,16,23, 4,19)( 2,17,10, 5,21, 9,14,26)( 3, 7,12,11,27,24,15,18)( 6,22,13,28)$
8A3	$8^{3},4$	$42$	$8$	$24$	$( 1,25, 4,20,16,19, 8,23)( 2, 5,14,17,21,26,10, 9)( 3,11,15, 7,27,18,12,24)( 6,28,13,22)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	2B	3A	4A	6A	7A	8A1	8A3
	Size	1	21	28	56	42	56	48	42	42
	2 P	1A	1A	1A	3A	2A	3A	7A	4A	4A
	3 P	1A	2A	2B	1A	4A	2B	7A	8A3	8A1
	7 P	1A	2A	2B	3A	4A	6A	1A	8A1	8A3
	Type
336.208.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
336.208.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
336.208.6a	R	$6$	$-2$	$0$	$0$	$2$	$0$	$-1$	$0$	$0$
336.208.6b1	R	$6$	$2$	$0$	$0$	$0$	$0$	$-1$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
336.208.6b2	R	$6$	$2$	$0$	$0$	$0$	$0$	$-1$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$
336.208.7a	R	$7$	$-1$	$1$	$1$	$-1$	$1$	$0$	$-1$	$-1$
336.208.7b	R	$7$	$-1$	$-1$	$1$	$-1$	$-1$	$0$	$1$	$1$
336.208.8a	R	$8$	$0$	$2$	$-1$	$0$	$-1$	$1$	$0$	$0$
336.208.8b	R	$8$	$0$	$-2$	$-1$	$0$	$1$	$1$	$0$	$0$

Regular extensions

Data not computed