Galois group: 36T712

• Label: 36T712
• Degree: $36$
• Order: $504$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-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 $n$:		$36$
Transitive number $t$:		$712$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,6,17,19,8,4)(3,10,23,12,28,25,13)(5,15,9,22,20,14,18)(7,16,31,30,29,21,32)(11,26,34,35,27,33,24)$, $(1,4)(2,8)(3,11)(5,14)(6,19)(7,21)(9,22)(10,24)(12,27)(13,26)(15,20)(16,29)(23,33)(25,34)(28,35)(30,31)$, $(1,3)(2,7)(4,13)(5,12)(6,18)(8,9)(10,14)(11,26)(16,23)(17,31)(20,32)(21,22)(24,27)(25,29)(33,34)(35,36)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: None

Low degree siblings

9T27, 28T70

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	$1^{36}$	$1$	$1$	$0$	$()$
2A	$2^{16},1^{4}$	$63$	$2$	$16$	$( 1, 9)( 3,23)( 4,11)( 5,21)( 6,31)( 7,26)( 8,24)(10,28)(12,36)(13,25)(14,17)(15,32)(18,33)(19,29)(20,35)(27,30)$
3A	$3^{12}$	$56$	$3$	$24$	$( 1,10,34)( 2, 3,26)( 4,16,20)( 5,17, 8)( 6,22,32)( 7,14, 9)(11,18,28)(12,35,21)(13,24,31)(15,23,33)(19,30,29)(25,36,27)$
7A1	$7^{5},1$	$72$	$7$	$30$	$( 1,34,31,30,19, 5,11)( 2,20,14,33,21, 4,26)( 6,17,29,24,15, 7,22)( 8,32, 9,16,35,27,18)(10,28,36,25,12,23,13)$
7A2	$7^{5},1$	$72$	$7$	$30$	$( 1,31,19,11,34,30, 5)( 2,14,21,26,20,33, 4)( 6,29,15,22,17,24, 7)( 8, 9,35,18,32,16,27)(10,36,12,13,28,25,23)$
7A3	$7^{5},1$	$72$	$7$	$30$	$( 1,30,11,31, 5,34,19)( 2,33,26,14, 4,20,21)( 6,24,22,29, 7,17,15)( 8,16,18, 9,27,32,35)(10,25,13,36,23,28,12)$
9A1	$9^{4}$	$56$	$9$	$32$	$( 1,29, 3,10,19,26,34,30, 2)( 4, 6,25,16,22,36,20,32,27)( 5,15,28,17,23,11, 8,33,18)( 7,13,21,14,24,12, 9,31,35)$
9A2	$9^{4}$	$56$	$9$	$32$	$( 1, 3,19,34, 2,29,10,26,30)( 4,25,22,20,27, 6,16,36,32)( 5,28,23, 8,18,15,17,11,33)( 7,21,24, 9,35,13,14,12,31)$
9A4	$9^{4}$	$56$	$9$	$32$	$( 1,19, 2,10,30, 3,34,29,26)( 4,22,27,16,32,25,20, 6,36)( 5,23,18,17,33,28, 8,15,11)( 7,24,35,14,31,21, 9,13,12)$

Malle's constant $a(G)$:     $1/16$

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