Galois group: 36T49

• Label: 36T49
• Degree: $36$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_9$

Group invariants

Abstract group:		$F_9$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$49$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,3,2,4)(5,13,18,29,34,25,22,10)(6,14,17,30,33,26,21,9)(7,16,20,31,36,28,24,12)(8,15,19,32,35,27,23,11)$, $(1,22,27,6)(2,21,28,5)(3,24,25,8)(4,23,26,7)(9,35,19,29)(10,36,20,30)(11,34,18,31)(12,33,17,32)(13,14)(15,16)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$8$:	$C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Degree 9: $C_3^2:C_8$

Degree 12: 12T46

Degree 18: 18T28

Low degree siblings

9T15, 12T46, 18T28, 24T81

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

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	3A	4A1	4A-1	8A1	8A-1	8A3	8A-3
	Size	1	9	8	9	9	9	9	9	9
	2 P	1A	1A	3A	2A	2A	4A1	4A-1	4A-1	4A1
	3 P	1A	2A	1A	4A-1	4A1	8A3	8A-3	8A1	8A-1
	Type
72.39.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.39.1b	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
72.39.1c1	C	$1$	$1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
72.39.1c2	C	$1$	$1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$
72.39.1d1	C	$1$	$-1$	$1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$	${\zeta }_8$
72.39.1d2	C	$1$	$-1$	$1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$	$-{\zeta }_8^3$
72.39.1d3	C	$1$	$-1$	$1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$	$-{\zeta }_8$
72.39.1d4	C	$1$	$-1$	$1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$	${\zeta }_8^3$
72.39.8a	R	$8$	$0$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed