Properties

 Label 27T25 Order $$81$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_3.He_3$

Group action invariants

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

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

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

Group invariants

 Order: $81=3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [81, 10]
 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 9g 9h 9i 9j 9k 9l 2P 1a 3b 3a 3d 3c 9e 9d 9f 9c 9b 9a 9l 9k 9j 9i 9h 9g 3P 1a 1a 1a 1a 1a 3c 3c 3c 3d 3d 3d 3d 3d 3d 3c 3c 3c 5P 1a 3b 3a 3d 3c 9f 9e 9d 9b 9a 9c 9l 9k 9j 9i 9h 9g 7P 1a 3a 3b 3c 3d 9c 9a 9b 9e 9f 9d 9g 9h 9i 9j 9k 9l 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