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 invariants

Abstract group:		$\SOPlus(4,2)$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$36$
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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{18}$	$1$	$1$	$0$	$()$
2A	$2^{9}$	$6$	$2$	$9$	$( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,11)(10,12)(17,18)$
2B	$2^{9}$	$6$	$2$	$9$	$( 1, 8)( 2, 7)( 3,16)( 4,15)( 5,12)( 6,11)( 9,10)(13,14)(17,18)$
2C	$2^{8},1^{2}$	$9$	$2$	$8$	$( 1, 4)( 2, 3)( 5,11)( 6,12)( 7,16)( 8,15)( 9,13)(10,14)$
3A	$3^{6}$	$4$	$3$	$12$	$( 1,17, 4)( 2,18, 3)( 5,10, 8)( 6, 9, 7)(11,15,14)(12,16,13)$
3B	$3^{6}$	$4$	$3$	$12$	$( 1,13, 7)( 2,14, 8)( 3,15,10)( 4,16, 9)( 5,18,11)( 6,17,12)$
4A	$4^{4},1^{2}$	$18$	$4$	$12$	$( 1, 7, 4,16)( 2, 8, 3,15)( 5,14,11,10)( 6,13,12, 9)$
6A	$6^{3}$	$12$	$6$	$15$	$( 1,14,17,11, 4,15)( 2,13,18,12, 3,16)( 5, 7,10, 6, 8, 9)$
6B	$6^{3}$	$12$	$6$	$15$	$( 1,11,16, 8, 6, 3)( 2,12,15, 7, 5, 4)( 9,14,17,10,13,18)$

Malle's constant $a(G)$:     $1/8$

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	$4$	$0$	$-2$	$0$	$-2$	$1$	$0$	$0$	$1$

Regular extensions

Data not computed