Galois group: 12T18

• Label: 12T18
• Degree: $12$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_6\times S_3$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$18$
Group:		$C_6\times S_3$
CHM label:		$[3^{2}]E(4)$
Parity:		$1$
Primitive:		no
Nilpotency class:		$-1$ (not nilpotent)
$\card{\Aut(F/K)}$:		$6$
Generators:		(1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,6,10)(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)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$S_3$, $C_6$ x 3
$12$:	$D_{6}$, $C_6\times C_2$
$18$:	$S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3\times C_3$

Low degree siblings

18T6 x 2, 36T6

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Label:		36.12

Character table:

2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 2 3 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 2 2 2 1a 3a 3b 2a 6a 6b 6c 6d 6e 6f 6g 2b 3c 3d 2c 6h 3e 6i 2P 1a 3b 3a 1a 3c 3e 3c 3a 3d 3c 3e 1a 3e 3d 1a 3b 3c 3e 3P 1a 1a 1a 2a 2a 2a 2c 2c 2c 2b 2b 2b 1a 1a 2c 2c 1a 2c 5P 1a 3b 3a 2a 6b 6a 6i 6h 6e 6g 6f 2b 3e 3d 2c 6d 3c 6c X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 X.3 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 X.5 1 A /A -1 -A -/A -A -/A -1 A /A 1 /A 1 -1 -A A -/A X.6 1 /A A -1 -/A -A -/A -A -1 /A A 1 A 1 -1 -/A /A -A X.7 1 A /A -1 -A -/A A /A 1 -A -/A -1 /A 1 1 A A /A X.8 1 /A A -1 -/A -A /A A 1 -/A -A -1 A 1 1 /A /A A X.9 1 A /A 1 A /A -A -/A -1 -A -/A -1 /A 1 -1 -A A -/A X.10 1 /A A 1 /A A -/A -A -1 -/A -A -1 A 1 -1 -/A /A -A X.11 1 A /A 1 A /A A /A 1 A /A 1 /A 1 1 A A /A X.12 1 /A A 1 /A A /A A 1 /A A 1 A 1 1 /A /A A X.13 2 -1 -1 . . . -2 1 1 . . . 2 -1 -2 1 2 -2 X.14 2 -1 -1 . . . 2 -1 -1 . . . 2 -1 2 -1 2 2 X.15 2 -/A -A . . . B A 1 . . . -/B -1 -2 /A -B /B X.16 2 -A -/A . . . /B /A 1 . . . -B -1 -2 A -/B B X.17 2 -/A -A . . . -B -A -1 . . . -/B -1 2 -/A -B -/B X.18 2 -A -/A . . . -/B -/A -1 . . . -B -1 2 -A -/B -B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -2*E(3) = 1-Sqrt(-3) = 1-i3