# Properties

 Label 27T27 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 $t$: $27$ 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), (1,12,19)(2,10,20)(3,11,21)(4,15,22)(5,13,23)(6,14,24)(7,17,26)(8,18,27)(9,16,25)

## 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: $C_3$

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

## Low degree siblings

9T17 x 3, 27T19, 27T21, 27T27 x 2

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

## Group invariants

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