Galois group: 18T8

• Label: 18T8
• Degree: $18$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $A_4 \times C_3$

Group invariants

Abstract group:		$A_4 \times C_3$
Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$3$:	$C_3$ x 4
$9$:	$C_3^2$
$12$:	$A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: $A_4$

Degree 9: $C_3^2$

Low degree siblings

12T20 x 3, 36T12

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	$2^{6},1^{6}$	$3$	$2$	$6$	$( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
3A1	$3^{6}$	$1$	$3$	$12$	$( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,16)(14,17,15)$
3A-1	$3^{6}$	$1$	$3$	$12$	$( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)$
3B1	$3^{6}$	$4$	$3$	$12$	$( 1,16,10)( 2,15, 9)( 3,18,11)( 4,17,12)( 5,13, 8)( 6,14, 7)$
3B-1	$3^{6}$	$4$	$3$	$12$	$( 1,10,16)( 2, 9,15)( 3,11,18)( 4,12,17)( 5, 8,13)( 6, 7,14)$
3C1	$3^{6}$	$4$	$3$	$12$	$( 1,13,11)( 2,14,12)( 3,16, 8)( 4,15, 7)( 5,18,10)( 6,17, 9)$
3C-1	$3^{6}$	$4$	$3$	$12$	$( 1,11,14)( 2,12,13)( 3, 8,15)( 4, 7,16)( 5,10,17)( 6, 9,18)$
3D1	$3^{6}$	$4$	$3$	$12$	$( 1,17, 7)( 2,18, 8)( 3,14, 9)( 4,13,10)( 5,15,12)( 6,16,11)$
3D-1	$3^{6}$	$4$	$3$	$12$	$( 1, 8,18)( 2, 7,17)( 3,10,13)( 4, 9,14)( 5,11,16)( 6,12,15)$
6A1	$6^{2},3^{2}$	$3$	$6$	$14$	$( 1, 4, 5, 2, 3, 6)( 7, 9,12)( 8,10,11)(13,15,18,14,16,17)$
6A-1	$6^{2},3^{2}$	$3$	$6$	$14$	$( 1, 5, 3)( 2, 6, 4)( 7,11, 9, 8,12,10)(13,17,16,14,18,15)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	6A1	6A-1
	Size	1	3	1	1	4	4	4	4	4	4	3	3
	2 P	1A	1A	3A-1	3A1	3B-1	3B1	3C-1	3C1	3D-1	3D1	3A1	3A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	1A	1A	2A	2A
	Type
36.11.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.11.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
36.11.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
36.11.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
36.11.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
36.11.1d1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
36.11.1d2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
36.11.1e1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
36.11.1e2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
36.11.3a	R	$3$	$-1$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
36.11.3b1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
36.11.3b2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$

Regular extensions

Data not computed