Galois group: 18T44

• Label: 18T44
• Degree: $18$
• Order: $108$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:C_{12}$

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$44$
Group:		$C_3^2:C_{12}$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,18,13,12,3,17,15,11,2,16,14,10)(4,8,5,9,6,7), (4,17,10)(5,18,11)(6,16,12)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$4$:	$C_4$
$6$:	$C_6$
$12$:	$C_{12}$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$, $C_3^2:C_4$

Degree 9: None

Low degree siblings

12T73 x 2, 18T44, 27T33, 36T81 x 2, 36T95 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
	$1^{18}$	$1$	$1$	$()$
	$2^{6},1^{6}$	$9$	$2$	$( 7,14)( 8,15)( 9,13)(10,17)(11,18)(12,16)$
	$3^{3},1^{9}$	$4$	$3$	$( 4,10,17)( 5,11,18)( 6,12,16)$
	$3^{6}$	$1$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
	$6^{2},3^{2}$	$9$	$6$	$( 1, 2, 3)( 4, 5, 6)( 7,15, 9,14, 8,13)(10,18,12,17,11,16)$
	$3^{6}$	$4$	$3$	$( 1, 2, 3)( 4,11,16)( 5,12,17)( 6,10,18)( 7, 8, 9)(13,14,15)$
	$3^{6}$	$1$	$3$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
	$6^{2},3^{2}$	$9$	$6$	$( 1, 3, 2)( 4, 6, 5)( 7,13, 8,14, 9,15)(10,16,11,17,12,18)$
	$3^{6}$	$4$	$3$	$( 1, 3, 2)( 4,12,18)( 5,10,16)( 6,11,17)( 7, 9, 8)(13,15,14)$
	$12,6$	$9$	$12$	$( 1, 4, 3, 6, 2, 5)( 7,12,13,18, 8,10,14,16, 9,11,15,17)$
	$12,6$	$9$	$12$	$( 1, 4, 3, 6, 2, 5)( 7,16,13,11, 8,17,14,12, 9,18,15,10)$
	$12,6$	$9$	$12$	$( 1, 5, 2, 6, 3, 4)( 7,10,15,18, 9,12,14,17, 8,11,13,16)$
	$12,6$	$9$	$12$	$( 1, 5, 2, 6, 3, 4)( 7,17,15,11, 9,16,14,10, 8,18,13,12)$
	$4^{3},2^{3}$	$9$	$4$	$( 1, 6)( 2, 4)( 3, 5)( 7,11,14,18)( 8,12,15,16)( 9,10,13,17)$
	$4^{3},2^{3}$	$9$	$4$	$( 1, 6)( 2, 4)( 3, 5)( 7,18,14,11)( 8,16,15,12)( 9,17,13,10)$
	$3^{6}$	$4$	$3$	$( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,12,18)( 5,10,16)( 6,11,17)$
	$3^{6}$	$4$	$3$	$( 1, 8,15)( 2, 9,13)( 3, 7,14)( 4,10,17)( 5,11,18)( 6,12,16)$
	$3^{6}$	$4$	$3$	$( 1, 9,14)( 2, 7,15)( 3, 8,13)( 4,11,16)( 5,12,17)( 6,10,18)$

Group invariants

Order:		$108=2^{2} \cdot 3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		108.36
Character table:

		1A	2A	3A1	3A-1	3B	3C	3D1	3D-1	3E1	3E-1	4A1	4A-1	6A1	6A-1	12A1	12A-1	12A5	12A-5
	Size	1	9	1	1	4	4	4	4	4	4	9	9	9	9	9	9	9	9
	2 P	1A	1A	3A-1	3A1	3B	3E1	3C	3D1	3E-1	3D-1	2A	2A	3A1	3A-1	6A-1	6A-1	6A1	6A1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	1A	1A	4A-1	4A1	2A	2A	4A1	4A-1	4A-1	4A1
	Type
108.36.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.36.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
108.36.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
108.36.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
108.36.1d1	C	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$	$i$	$-i$
108.36.1d2	C	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$i$	$-i$	$-1$	$-1$	$-i$	$i$	$-i$	$i$
108.36.1e1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
108.36.1e2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
108.36.1f1	C	$1$	$-1$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	$1$	$1$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$-{\zeta }_{12}^3$	${\zeta }_{12}^3$	$-{\zeta }_{12}^4$	${\zeta }_{12}^2$	$-{\zeta }_{12}^5$	${\zeta }_{12}$	$-{\zeta }_{12}$	${\zeta }_{12}^5$
108.36.1f2	C	$1$	$-1$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$1$	$1$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	${\zeta }_{12}^3$	$-{\zeta }_{12}^3$	${\zeta }_{12}^2$	$-{\zeta }_{12}^4$	${\zeta }_{12}$	$-{\zeta }_{12}^5$	${\zeta }_{12}^5$	$-{\zeta }_{12}$
108.36.1f3	C	$1$	$-1$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	$1$	$1$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	${\zeta }_{12}^3$	$-{\zeta }_{12}^3$	$-{\zeta }_{12}^4$	${\zeta }_{12}^2$	${\zeta }_{12}^5$	$-{\zeta }_{12}$	${\zeta }_{12}$	$-{\zeta }_{12}^5$
108.36.1f4	C	$1$	$-1$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$1$	$1$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	$-{\zeta }_{12}^3$	${\zeta }_{12}^3$	${\zeta }_{12}^2$	$-{\zeta }_{12}^4$	$-{\zeta }_{12}$	${\zeta }_{12}^5$	$-{\zeta }_{12}^5$	${\zeta }_{12}$
108.36.4a	R	$4$	$0$	$4$	$4$	$-2$	$1$	$-2$	$-2$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
108.36.4b	R	$4$	$0$	$4$	$4$	$1$	$-2$	$1$	$1$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
108.36.4c1	C	$4$	$0$	$4{\zeta }_3^{-1}$	$4{\zeta }_3$	$-2$	$1$	$-2{\zeta }_3^{-1}$	$-2{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
108.36.4c2	C	$4$	$0$	$4{\zeta }_3$	$4{\zeta }_3^{-1}$	$-2$	$1$	$-2{\zeta }_3$	$-2{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
108.36.4d1	C	$4$	$0$	$4{\zeta }_3^{-1}$	$4{\zeta }_3$	$1$	$-2$	${\zeta }_3^{-1}$	${\zeta }_3$	$-2{\zeta }_3$	$-2{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
108.36.4d2	C	$4$	$0$	$4{\zeta }_3$	$4{\zeta }_3^{-1}$	$1$	$-2$	${\zeta }_3$	${\zeta }_3^{-1}$	$-2{\zeta }_3^{-1}$	$-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$