# Properties

 Label 27T19 Degree $27$ Order $81$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_3\wr C_3$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $19$ Group: $C_3\wr C_3$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $9$ Generators: (1,4,25)(2,5,26)(3,6,27)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (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)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_3 \wr C_3$ x 3

## Low degree siblings

9T17 x 3, 27T21, 27T27 x 3

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

## Group invariants

 Order: $81=3^{4}$ Cyclic: no Abelian: no Solvable: yes GAP id: [81, 7]
 Character table:  3 4 3 3 3 3 3 3 3 3 4 2 2 2 2 2 2 4 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 9a 3j 9b 3k 9c 9d 3l 2P 1a 3b 3a 3e 3f 3c 3d 3h 3g 3l 9c 3k 9d 3j 9a 9b 3i 3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 3l 1a 3i 1a 3i 3l 1a 5P 1a 3b 3a 3e 3f 3c 3d 3h 3g 3l 9c 3k 9d 3j 9a 9b 3i 7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 9a 3j 9b 3k 9c 9d 3l X.1 1 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 B B B /B /B /B 1 X.3 1 1 1 1 1 1 1 1 1 1 /B /B /B B B B 1 X.4 1 1 1 B B /B /B B /B 1 1 B /B /B 1 B 1 X.5 1 1 1 /B /B B B /B B 1 1 /B B B 1 /B 1 X.6 1 1 1 B B /B /B B /B 1 B /B 1 B /B 1 1 X.7 1 1 1 /B /B B B /B B 1 /B B 1 /B B 1 1 X.8 1 1 1 B B /B /B B /B 1 /B 1 B 1 B /B 1 X.9 1 1 1 /B /B B B /B B 1 B 1 /B 1 /B B 1 X.10 3 A /A . . . . . . 3 . . . . . . 3 X.11 3 /A A . . . . . . 3 . . . . . . 3 X.12 3 . . C D /C -D -/C -C /A . . . . . . A X.13 3 . . /C -D C D -C -/C A . . . . . . /A X.14 3 . . -/C C -C /C D -D /A . . . . . . A X.15 3 . . -C /C -/C C -D D A . . . . . . /A X.16 3 . . D -/C -D -C C /C /A . . . . . . A X.17 3 . . -D -C D -/C /C C A . . . . . . /A A = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 C = -2*E(3)-E(3)^2 = (3-Sqrt(-3))/2 = 1-b3 D = E(3)-E(3)^2 = Sqrt(-3) = i3