Properties

Label 27T39
Order \(162\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $He_3.C_6$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $39$
Group :  $He_3.C_6$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,22,2,15,23,3,13,24)(4,17,26,5,18,27,6,16,25)(7,10,19,8,11,20,9,12,21), (1,19,9,27,4,22,3,21,8,26,6,24,2,20,7,25,5,23)(10,13,18,11,14,16,12,15,17), (1,21,8,25,5,22,3,20,7,27,4,24,2,19,9,26,6,23)(10,14,17,11,15,18,12,13,16)
$|\Aut(F/K)|$:  $9$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$ x 4, $C_6$
18:  $S_3\times C_3$ x 4, $C_3^2:C_2$
54:  18T23

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$

Low degree siblings

27T39 x 3

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

Group invariants

Order:  $162=2 \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [162, 44]
Character table: Data not available.