Galois group: 12T73

• Label: 12T73
• Degree: $12$
• Order: $108$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:C_{12}$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$73$
Group:		$C_3^2:C_{12}$
CHM label:		$1/2[3^{3}:2]4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,8,7,2)(3,6,9,12)(4,11,10,5), (2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)

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: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T73, 18T44 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 3, 3, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$( 3, 7,11)( 4,12, 8)$
	$ 3, 3, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$( 3,11, 7)( 4, 8,12)$
	$ 3, 3, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$( 2, 6,10)( 4,12, 8)$
	$ 3, 3, 3, 1, 1, 1 $	$4$	$3$	$( 2, 6,10)( 3, 7,11)( 4, 8,12)$
	$ 3, 3, 3, 1, 1, 1 $	$4$	$3$	$( 2,10, 6)( 3,11, 7)( 4,12, 8)$
	$ 12 $	$9$	$12$	$( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$
	$ 12 $	$9$	$12$	$( 1, 2, 3, 8, 9, 6,11,12, 5,10, 7, 4)$
	$ 4, 4, 4 $	$9$	$4$	$( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$
	$ 2, 2, 2, 2, 2, 2 $	$9$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
	$ 6, 6 $	$9$	$6$	$( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
	$ 6, 6 $	$9$	$6$	$( 1, 3, 9,11, 5, 7)( 2, 4, 6, 8,10,12)$
	$ 12 $	$9$	$12$	$( 1, 4, 7,10, 5,12,11, 6, 9, 8, 3, 2)$
	$ 12 $	$9$	$12$	$( 1, 4, 3, 6, 9, 8,11,10, 5,12, 7, 2)$
	$ 4, 4, 4 $	$9$	$4$	$( 1, 4,11, 2)( 3,10, 5,12)( 6, 9, 8, 7)$
	$ 3, 3, 3, 3 $	$4$	$3$	$( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
	$ 3, 3, 3, 3 $	$1$	$3$	$( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
	$ 3, 3, 3, 3 $	$1$	$3$	$( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$

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	$4$	$0$	$4{\zeta }_3$	$4{\zeta }_3^{-1}$	$1$	$-2$	${\zeta }_3$	${\zeta }_3^{-1}$	$-2{\zeta }_3^{-1}$	$-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$