# Properties

 Label 27T24 Degree $27$ Order $81$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $He_3:C_3$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $24$ Group: $He_3:C_3$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $9$ Generators: (1,4,7,2,5,8,3,6,9)(10,13,18,11,14,16,12,15,17)(19,23,25,20,24,26,21,22,27), (1,13,23)(2,14,24)(3,15,22)(4,16,25)(5,17,26)(6,18,27)(7,10,20)(8,11,21)(9,12,19)

## 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: $C_3^2$

## Low degree siblings

27T23 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $3$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $3$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ $9, 9, 9$ $3$ $9$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,13,18,11,14,16,12,15,17)(19,23,25,20,24,26,21, 22,27)$ $9, 9, 9$ $3$ $9$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,15,16,11,13,17,12,14,18)(19,24,27,20,22,25,21, 23,26)$ $9, 9, 9$ $3$ $9$ $( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,13,18,11,14,16,12,15,17)(19,22,26,20,23,27,21, 24,25)$ $9, 9, 9$ $3$ $9$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,16,13,12,18,15,11,17,14)(19,27,22,21,26,24,20, 25,23)$ $9, 9, 9$ $3$ $9$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,17,15,12,16,14,11,18,13)(19,26,23,21,25,22,20, 27,24)$ $9, 9, 9$ $3$ $9$ $( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,16,13,12,18,15,11,17,14)(19,26,23,21,25,22,20, 27,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,10,25)( 2,11,26)( 3,12,27)( 4,14,20)( 5,15,21)( 6,13,19)( 7,17,24) ( 8,18,22)( 9,16,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,13,22)( 2,14,23)( 3,15,24)( 4,16,27)( 5,17,25)( 6,18,26)( 7,10,19) ( 8,11,20)( 9,12,21)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,16,21)( 2,17,19)( 3,18,20)( 4,10,22)( 5,11,23)( 6,12,24)( 7,14,27) ( 8,15,25)( 9,13,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,19,16)( 2,20,17)( 3,21,18)( 4,23,10)( 5,24,11)( 6,22,12)( 7,25,14) ( 8,26,15)( 9,27,13)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,22,13)( 2,23,14)( 3,24,15)( 4,27,16)( 5,25,17)( 6,26,18)( 7,19,10) ( 8,20,11)( 9,21,12)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $9$ $3$ $( 1,25,10)( 2,26,11)( 3,27,12)( 4,20,14)( 5,21,15)( 6,19,13)( 7,24,17) ( 8,22,18)( 9,23,16)$

## Group invariants

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