Properties

Label 27T47
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_3:C_9$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $47$
Group :  $C_3:S_3:C_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (7,10,13)(8,11,14)(9,12,15)(16,23,21)(17,24,19)(18,22,20), (1,8,22,2,9,23,3,7,24)(4,15,16,25,10,19,6,14,18,27,12,21,5,13,17,26,11,20)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
9:  $C_9$
18:  $S_3\times C_3$, $C_{18}$
54:  $C_3^2 : C_6$, $C_9\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_9$, $C_3^2 : S_3 $

Low degree siblings

18T82

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

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 4]
Character table: Data not available.