Galois group: 36T175

• Label: 36T175
• 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 $t$:		$175$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,30,27,14,34,20,11,23)(2,29,28,13,33,19,12,24)(3,31,26,15,36,18,10,22)(4,32,25,16,35,17,9,21)(5,7,6,8)$, $(1,14,32,19)(2,13,31,20)(3,16,30,18)(4,15,29,17)(5,23,28,35)(6,24,27,36)(7,21,25,33)(8,22,26,34)(9,11)(10,12)$

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: 12T84

Degree 18: 18T73

Low degree siblings

9T19, 12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172

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

Malle's constant $a(G)$:     $1/15$

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