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

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