Galois group: 36T171

• Label: 36T171
• Degree: $36$
• Order: $144$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_9:C_2$

Group invariants

Abstract group:		$F_9:C_2$
Order:		$144=2^{4} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$171$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,33,25,31)(2,34,26,32)(3,35,27,30)(4,36,28,29)(5,20,22,11)(6,19,21,12)(7,17,23,9)(8,18,24,10)(13,14)$, $(1,6,10,14,31,21)(2,5,9,13,32,22)(3,7,12,16,29,23)(4,8,11,15,30,24)(17,36,26,19,34,27)(18,35,25,20,33,28)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$
$16$:	$QD_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Degree 9: $(C_3^2:C_8):C_2$

Degree 12: None

Degree 18: 18T68

Low degree siblings

9T19, 12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T172, 36T175

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

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	3A	4A	4B	6A	8A1	8A-1
	Size	1	9	12	8	18	36	24	18	18
	2 P	1A	1A	1A	3A	2A	2A	3A	4A	4A
	3 P	1A	2A	2B	1A	4A	4B	2B	8A1	8A-1
	Type
144.182.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
144.182.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
144.182.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
144.182.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
144.182.2a	R	$2$	$2$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$
144.182.2b1	C	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
144.182.2b2	C	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$
144.182.8a	R	$8$	$0$	$2$	$-1$	$0$	$0$	$-1$	$0$	$0$
144.182.8b	R	$8$	$0$	$-2$	$-1$	$0$	$0$	$1$	$0$	$0$

Regular extensions

Data not computed