# Properties

 Label 18T18 Order $$54$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_9:C_3$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $18$ Group : $D_9:C_3$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10), (1,15,7,9,13,3)(2,16,8,10,14,4)(5,6)(11,18)(12,17) $|\Aut(F/K)|$: $6$

## 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: $S_3$

Degree 6: $S_3$

Degree 9: $(C_9:C_3):C_2$

## Low degree siblings

9T10

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

## Group invariants

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