# Properties

 Label 27T10 Order $$54$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3:D_9$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $D_{9}$ x 3, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$ x 4

Degree 9: $D_{9}$ x 3, $C_3^2:C_2$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $54=2 \cdot 3^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [54, 7]
 Character table: 2 1 1 . . . . . . . . . . . . . 3 3 . 3 3 3 3 3 3 3 3 3 3 3 3 3 1a 2a 3a 9a 9b 9c 9d 9e 9f 3b 3c 3d 9g 9h 9i 2P 1a 1a 3a 9d 9f 9e 9g 9i 9h 3b 3c 3d 9a 9b 9c 3P 1a 2a 1a 3c 3c 3c 3c 3c 3c 1a 1a 1a 3c 3c 3c 5P 1a 2a 3a 9g 9h 9i 9a 9c 9b 3b 3c 3d 9d 9f 9e 7P 1a 2a 3a 9d 9f 9e 9g 9i 9h 3b 3c 3d 9a 9b 9c X.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 X.3 2 . 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 X.4 2 . -1 2 -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 X.5 2 . -1 -1 -1 2 -1 2 -1 -1 2 -1 -1 -1 2 X.6 2 . -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 2 -1 X.7 2 . -1 A B C B A C -1 -1 2 C A B X.8 2 . -1 B C A C B A -1 -1 2 A B C X.9 2 . -1 C A B A C B -1 -1 2 B C A X.10 2 . -1 A C B B C A 2 -1 -1 C B A X.11 2 . -1 B A C C A B 2 -1 -1 A C B X.12 2 . -1 C B A A B C 2 -1 -1 B A C X.13 2 . 2 A A A B B B -1 -1 -1 C C C X.14 2 . 2 B B B C C C -1 -1 -1 A A A X.15 2 . 2 C C C A A A -1 -1 -1 B B B A = E(9)^2+E(9)^7 B = E(9)^4+E(9)^5 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7