Galois group: 28T70

• Label: 28T70
• Degree: $28$
• Order: $504$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• pp-group: no
• Group: $\PSL(2,8)$

Group invariants

Abstract group:		$\PSL(2,8)$
Order:		$504=2^{3} \cdot 3^{2} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$28$
Transitive number $t$:		$70$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,21,27,15,25,10,24,7,20)(2,19,8,26,17,16,6,5,3)(9,28,18,13,11,22,14,23,12)$, $(1,28,21,4,25,18,10)(2,26,19,17,12,7,11)(3,24,20,16,13,14,9)(5,6,23,15,8,27,22)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

9T27, 36T712

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	$2^{12},1^{4}$	$63$	$2$	$12$	$( 1, 5)( 2, 4)( 6, 7)( 8,21)(10,11)(13,22)(14,24)(15,26)(16,28)(17,23)(18,25)(20,27)$
3A	$3^{9},1$	$56$	$3$	$18$	$( 1,20,17)( 2,24,10)( 3,16, 7)( 4, 6, 5)( 8,23,22)( 9,15,28)(11,21,26)(12,19,25)(13,18,27)$
7A1	$7^{4}$	$72$	$7$	$24$	$( 1,19,22,14,23,26,10)( 2,21, 9,13,25,17, 7)( 3,20,24,16, 6, 4, 5)( 8,12,27,18,28,15,11)$
7A2	$7^{4}$	$72$	$7$	$24$	$( 1,22,23,10,19,14,26)( 2, 9,25, 7,21,13,17)( 3,24, 6, 5,20,16, 4)( 8,27,28,11,12,18,15)$
7A3	$7^{4}$	$72$	$7$	$24$	$( 1,14,10,22,26,19,23)( 2,13, 7, 9,17,21,25)( 3,16, 5,24, 4,20, 6)( 8,18,11,27,15,12,28)$
9A1	$9^{3},1$	$56$	$9$	$24$	$( 1, 2,23,20,24,22,17,10, 8)( 3,28,18,16, 9,27, 7,15,13)( 4,11,19, 6,21,25, 5,26,12)$
9A2	$9^{3},1$	$56$	$9$	$24$	$( 1,23,24,17, 8, 2,20,22,10)( 3,18, 9, 7,13,28,16,27,15)( 4,19,21, 5,12,11, 6,25,26)$
9A4	$9^{3},1$	$56$	$9$	$24$	$( 1,24, 8,20,10,23,17, 2,22)( 3, 9,13,16,15,18, 7,28,27)( 4,21,12, 6,26,19, 5,11,25)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	3A	7A1	7A2	7A3	9A1	9A2	9A4
	Size	1	63	56	72	72	72	56	56	56
	2 P	1A	1A	3A	7A2	7A3	7A1	9A2	9A4	9A1
	3 P	1A	2A	1A	7A3	7A1	7A2	3A	3A	3A
	7 P	1A	2A	3A	1A	1A	1A	9A2	9A4	9A1
	Type
504.156.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
504.156.7a	R	$7$	$-1$	$-2$	$0$	$0$	$0$	$1$	$1$	$1$
504.156.7b1	R	$7$	$-1$	$1$	$0$	$0$	$0$	$-{\zeta }_9^{-1}-{\zeta }_9$	$-{\zeta }_9^{-2}-{\zeta }_9^2$	$-{\zeta }_9^{-4}-{\zeta }_9^4$
504.156.7b2	R	$7$	$-1$	$1$	$0$	$0$	$0$	$-{\zeta }_9^{-2}-{\zeta }_9^2$	$-{\zeta }_9^{-4}-{\zeta }_9^4$	$-{\zeta }_9^{-1}-{\zeta }_9$
504.156.7b3	R	$7$	$-1$	$1$	$0$	$0$	$0$	$-{\zeta }_9^{-4}-{\zeta }_9^4$	$-{\zeta }_9^{-1}-{\zeta }_9$	$-{\zeta }_9^{-2}-{\zeta }_9^2$
504.156.8a	R	$8$	$0$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
504.156.9a1	R	$9$	$1$	$0$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	$0$	$0$	$0$
504.156.9a2	R	$9$	$1$	$0$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	${\zeta }_7^{-1}+{\zeta }_7$	$0$	$0$	$0$
504.156.9a3	R	$9$	$1$	$0$	${\zeta }_7^{-1}+{\zeta }_7$	${\zeta }_7^{-2}+{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7^3$	$0$	$0$	$0$

Regular extensions

Data not computed