# Properties

 Label 27T6 Order $$54$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3^2:S_3$

## Group action invariants

 Degree $n$ : $27$ Transitive number $t$ : $6$ Group : $C_3^2:S_3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (2,3)(4,6)(7,16)(8,18)(9,17)(10,21)(11,20)(12,19)(13,23)(14,22)(15,24)(25,26), (1,10,21)(2,11,19)(3,12,20)(4,13,24)(5,14,22)(6,15,23)(7,18,25)(8,16,26)(9,17,27), (1,2,3)(4,5,6)(7,12,14)(8,10,15)(9,11,13)(16,22,19)(17,23,20)(18,24,21)(25,26,27) $|\Aut(F/K)|$: $3$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$, $(C_3^2:C_3):C_2$ x 4

## Low degree siblings

9T12 x 4, 18T24 x 4

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

## Group invariants

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