Galois Group: 18T6

• Label: 18T6
• Order: \(36\)
• n: \(18\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $S_3 \times C_6$

Group action invariants

Degree $n$ :		$18$
Transitive number $t$ :		$6$
Group :		$S_3 \times C_6$
Parity:		$-1$
Primitive:		No
Generators:		(1,15,8,4,13,9)(2,16,7,3,14,10)(5,17,12)(6,18,11), (1,3,18,2,4,17)(5,8,10,6,7,9)(11,14,15,12,13,16)
$|\Aut(F/K)|$:		$6$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
3:	$C_3$
4:	$V_4$
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: $C_2$

Degree 3: $C_3$, $S_3$

Degree 6: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

Low degree siblings

12T18, 18T6

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, 1, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 3,17)( 4,18)( 5,10)( 6, 9)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3,18)( 4,17)( 5, 9)( 6,10)( 7, 8)(11,16)(12,15)(13,14)$
$ 6, 6, 6 $	$2$	$6$	$( 1, 3,18, 2, 4,17)( 5, 8,10, 6, 7, 9)(11,14,15,12,13,16)$
$ 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 4,18)( 2, 3,17)( 5, 7,10)( 6, 8, 9)(11,13,15)(12,14,16)$
$ 6, 6, 6 $	$2$	$6$	$( 1, 5,15, 2, 6,16)( 3, 8,12, 4, 7,11)( 9,14,18,10,13,17)$
$ 6, 6, 6 $	$3$	$6$	$( 1, 5,13,17, 8,12)( 2, 6,14,18, 7,11)( 3, 9,16, 4,10,15)$
$ 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 6,15)( 2, 5,16)( 3, 7,12)( 4, 8,11)( 9,13,18)(10,14,17)$
$ 6, 6, 3, 3 $	$3$	$6$	$( 1, 6,13,18, 8,11)( 2, 5,14,17, 7,12)( 3,10,16)( 4, 9,15)$
$ 6, 6, 6 $	$1$	$6$	$( 1, 7,13, 2, 8,14)( 3, 9,16, 4,10,15)( 5,11,17, 6,12,18)$
$ 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 8,13)( 2, 7,14)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)$
$ 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1,11, 9)( 2,12,10)( 3,14, 5)( 4,13, 6)( 7,17,16)( 8,18,15)$
$ 6, 6, 3, 3 $	$3$	$6$	$( 1,11, 8,18,13, 6)( 2,12, 7,17,14, 5)( 3,16,10)( 4,15, 9)$
$ 6, 6, 6 $	$2$	$6$	$( 1,12, 9, 2,11,10)( 3,13, 5, 4,14, 6)( 7,18,16, 8,17,15)$
$ 6, 6, 6 $	$3$	$6$	$( 1,12, 8,17,13, 5)( 2,11, 7,18,14, 6)( 3,15,10, 4,16, 9)$
$ 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,13, 8)( 2,14, 7)( 3,16,10)( 4,15, 9)( 5,17,12)( 6,18,11)$
$ 6, 6, 6 $	$1$	$6$	$( 1,14, 8, 2,13, 7)( 3,15,10, 4,16, 9)( 5,18,12, 6,17,11)$

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[36, 12]

Character table:

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