Properties

Label 27T18
Order \(81\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_3\times He_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 13
9:  $C_3^2$ x 13
27:  $C_3^2:C_3$ x 3, 27T4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

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

Low degree siblings

27T18 x 11

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:  $81=3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [81, 12]
Character table: Data not available.