Galois group: 18T36

• Label: 18T36
• Degree: $18$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\SOPlus(4,2)$

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$36$
Group:		$\SOPlus(4,2)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,10)(2,9)(3,6)(4,5)(7,18)(8,17)(11,12)(13,14)(15,16), (3,10,18,11)(4,9,17,12)(5,8,15,14)(6,7,16,13)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $S_3^2:C_2$

Low degree siblings

6T13 x 2, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 24T72 x 2, 36T53, 36T54 x 2

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	Representative
	$1^{18}$	$1$	$1$	$()$
	$4^{4},1^{2}$	$18$	$4$	$( 3,10,18,11)( 4, 9,17,12)( 5, 8,15,14)( 6, 7,16,13)$
	$2^{8},1^{2}$	$9$	$2$	$( 3,18)( 4,17)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
	$2^{9}$	$6$	$2$	$( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,11)(10,12)(17,18)$
	$2^{9}$	$6$	$2$	$( 1, 2)( 3, 9)( 4,10)( 5, 6)( 7,14)( 8,13)(11,17)(12,18)(15,16)$
	$6^{3}$	$12$	$6$	$( 1, 3, 6, 8,16,11)( 2, 4, 5, 7,15,12)( 9,18,13,10,17,14)$
	$6^{3}$	$12$	$6$	$( 1, 3,17, 2, 4,18)( 5,16,10,13, 8,12)( 6,15, 9,14, 7,11)$
	$3^{6}$	$4$	$3$	$( 1, 4,17)( 2, 3,18)( 5, 8,10)( 6, 7, 9)(11,14,15)(12,13,16)$
	$3^{6}$	$4$	$3$	$( 1, 6,16)( 2, 5,15)( 3, 8,11)( 4, 7,12)( 9,13,17)(10,14,18)$

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.40
Character table:

		1A	2A	2B	2C	3A	3B	4A	6A	6B
	Size	1	6	6	9	4	4	18	12	12
	2 P	1A	1A	1A	1A	3A	3B	2C	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	4A	2A	2B
	Type
72.40.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.40.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.40.1c	R	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
72.40.1d	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$
72.40.2a	R	$2$	$0$	$0$	$-2$	$2$	$2$	$0$	$0$	$0$
72.40.4a	R	$4$	$0$	$2$	$0$	$-2$	$1$	$0$	$0$	$-1$
72.40.4b	R	$4$	$2$	$0$	$0$	$1$	$-2$	$0$	$-1$	$0$
72.40.4c	R	$4$	$-2$	$0$	$0$	$1$	$-2$	$0$	$1$	$0$
72.40.4d	R	$4$	$0$	$-2$	$0$	$-2$	$1$	$0$	$0$	$1$