# Properties

 Label 27T14 Order $$54$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_9:C_3$

## Group action invariants

 Degree $n$ : $27$ Transitive number $t$ : $14$ Group : $D_9:C_3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,27,5,21,17,22,11,7,15)(2,25,6,19,18,23,12,8,13)(3,26,4,20,16,24,10,9,14), (1,27,19,8,10,16)(2,25,20,9,11,17)(3,26,21,7,12,18)(4,6,5)(13,22,14,23,15,24) $|\Aut(F/K)|$: $3$

## 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 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$, $(C_9:C_3):C_2$

## Low degree siblings

9T10, 18T18

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $9$ $2$ $( 4,25)( 5,26)( 6,27)( 7,23)( 8,24)( 9,22)(10,20)(11,21)(12,19)(13,17)(14,18) (15,16)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 2, 3)( 4,15,23)( 5,13,24)( 6,14,22)( 7,25,16)( 8,26,17)( 9,27,18) (10,11,12)(19,20,21)$ $6, 6, 6, 6, 3$ $9$ $6$ $( 1, 2, 3)( 4,16,23,25,15, 7)( 5,17,24,26,13, 8)( 6,18,22,27,14, 9) (10,21,12,20,11,19)$ $6, 6, 6, 6, 3$ $9$ $6$ $( 1, 3, 2)( 4, 7,15,25,23,16)( 5, 8,13,26,24,17)( 6, 9,14,27,22,18) (10,19,11,20,12,21)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 3, 2)( 4,23,15)( 5,24,13)( 6,22,14)( 7,16,25)( 8,17,26)( 9,18,27) (10,12,11)(19,21,20)$ $9, 9, 9$ $6$ $9$ $( 1, 4, 8,11,14,18,21,24,25)( 2, 5, 9,12,15,16,19,22,26)( 3, 6, 7,10,13,17,20, 23,27)$ $9, 9, 9$ $6$ $9$ $( 1, 5,17,11,15,27,21,22, 7)( 2, 6,18,12,13,25,19,23, 8)( 3, 4,16,10,14,26,20, 24, 9)$ $9, 9, 9$ $6$ $9$ $( 1, 6,26,11,13, 9,21,23,16)( 2, 4,27,12,14, 7,19,24,17)( 3, 5,25,10,15, 8,20, 22,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27) ( 8,18,25)( 9,16,26)$

## 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 1 2 1 1 2 2 2 2 3 1a 2a 3a 6a 6b 3b 9a 9b 9c 3c 2P 1a 1a 3b 3b 3a 3a 9a 9c 9b 3c 3P 1a 2a 1a 2a 2a 1a 3c 3c 3c 1a 5P 1a 2a 3b 6b 6a 3a 9a 9c 9b 3c 7P 1a 2a 3a 6a 6b 3b 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 -1 A -A -/A /A 1 A /A 1 X.4 1 -1 /A -/A -A A 1 /A A 1 X.5 1 1 A A /A /A 1 A /A 1 X.6 1 1 /A /A A A 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