# Properties

 Label 18T22 Order $$54$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $He_3:C_2$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $22$ Group : $He_3:C_2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,10,8,18,14,5)(2,12,9,17,15,4)(3,11,7,16,13,6), (1,18,9,6,13,12)(2,17,7,5,14,11)(3,16,8,4,15,10) $|\Aut(F/K)|$: $3$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 9: None

## Low degree siblings

9T11, 9T13, 18T20, 18T21

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

## Group invariants

 Order: $54=2 \cdot 3^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [54, 5]
 Character table:  2 1 . . . 1 1 1 1 . 1 3 3 2 2 3 1 1 1 2 2 2 1a 3a 3b 3c 2a 6a 6b 3d 3e 3f 2P 1a 3b 3a 3c 1a 3d 3f 3f 3e 3d 3P 1a 1a 1a 1a 2a 2a 2a 1a 1a 1a 5P 1a 3b 3a 3c 2a 6b 6a 3f 3e 3d X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 1 1 1 X.3 1 A /A 1 -1 -A -/A /A 1 A X.4 1 /A A 1 -1 -/A -A A 1 /A X.5 1 A /A 1 1 A /A /A 1 A X.6 1 /A A 1 1 /A A A 1 /A X.7 2 -1 -1 2 . . . 2 -1 2 X.8 2 -A -/A 2 . . . B -1 /B X.9 2 -/A -A 2 . . . /B -1 B X.10 6 . . -3 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3