Properties

Label 27T50
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\wr S_3$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $50$
Group :  $C_3\wr S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,2,10,3,12)(4,14,5,13,6,15)(7,27,9,25,8,26)(16,20,22,17,19,23)(18,21,24), (1,3)(4,6)(7,16,15,23,11,21)(8,18,13,22,12,20)(9,17,14,24,10,19)(25,27)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $C_3^2 : C_6$, $C_3 \wr S_3 $

Low degree siblings

9T20 x 3, 18T86 x 3, 27T37, 27T50 x 2, 27T70

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

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 10]
Character table: Data not available.