Galois Group: 18T23

• Label: 18T23
• Order: \(54\)
• n: \(18\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_3\times C_3:S_3$

Group action invariants

Degree $n$ :		$18$
Transitive number $t$ :		$23$
Group :		$C_3\times C_3:S_3$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,18,7,6,14,10)(2,17,8,5,15,12)(3,16,9,4,13,11), (1,8,13)(2,9,14)(3,7,15)(4,5,6)(10,11,12)(16,17,18), (1,9,15)(2,7,13)(3,8,14)(4,6,5)(10,12,11)(16,18,17)
$|\Aut(F/K)|$:		$9$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$
3:	$C_3$
6:	$S_3$ x 4, $C_6$
18:	$S_3\times C_3$ x 4, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$, $S_3\times C_3$ x 3

Degree 9: None

Low degree siblings

18T23 x 3

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

Group invariants

Order:		$54=2 \cdot 3^{3}$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[54, 13]

Character table:

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