# Properties

 Label 27T21 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 $t$: $21$ Group: $C_3\wr C_3$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $3$ Generators: (4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26), (1,11,19)(2,12,20)(3,10,21)(4,15,22)(5,13,23)(6,14,24)(7,16,25)(8,17,26)(9,18,27)

## 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: $C_3$

Degree 9: $C_3^2:C_3$

## Low degree siblings

9T17 x 3, 27T19, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $9$ $3$ $( 4, 5, 6)( 7,15,10)( 8,13,11)( 9,14,12)(16,20,22)(17,21,23)(18,19,24) (25,27,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $9$ $3$ $( 4, 6, 5)( 7,10,15)( 8,11,13)( 9,12,14)(16,22,20)(17,23,21)(18,24,19) (25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22) (17,20,23)(18,21,24)$ $9, 9, 9$ $9$ $9$ $( 1, 7,18, 3, 9,17, 2, 8,16)( 4,10,19, 6,12,21, 5,11,20)(13,23,27,15,22,26,14, 24,25)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 7,22)( 2, 8,23)( 3, 9,24)( 4,11,16)( 5,12,17)( 6,10,18)(13,20,26) (14,21,27)(15,19,25)$ $9, 9, 9$ $9$ $9$ $( 1, 7,19, 3, 9,21, 2, 8,20)( 4,12,24, 6,11,23, 5,10,22)(13,16,26,15,18,25,14, 17,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 8,24)( 2, 9,22)( 3, 7,23)( 4,12,18)( 5,10,16)( 6,11,17)(13,21,25) (14,19,26)(15,20,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1, 9,23)( 2, 7,24)( 3, 8,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,19,27) (14,20,25)(15,21,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,16,15)( 2,17,13)( 3,18,14)( 4,19, 7)( 5,20, 8)( 6,21, 9)(10,27,24) (11,25,22)(12,26,23)$ $9, 9, 9$ $9$ $9$ $( 1,16, 8, 2,17, 9, 3,18, 7)( 4,20,11, 5,21,12, 6,19,10)(13,25,24,14,26,22,15, 27,23)$ $9, 9, 9$ $9$ $9$ $( 1,16,11, 2,17,12, 3,18,10)( 4,21,13, 5,19,14, 6,20,15)( 7,27,22, 8,25,23, 9, 26,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,17,14)( 2,18,15)( 3,16,13)( 4,20, 9)( 5,21, 7)( 6,19, 8)(10,25,23) (11,26,24)(12,27,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,18,13)( 2,16,14)( 3,17,15)( 4,21, 8)( 5,19, 9)( 6,20, 7)(10,26,22) (11,27,23)(12,25,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $3$ $3$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,22,19) (17,23,20)(18,24,21)$

## Group invariants

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