Properties

Label 27T36
Order \(108\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_3^2$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $36$
Group :  $C_3\times S_3^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,12)(2,19,10)(3,20,11)(4,17,15,27,22,7)(5,18,13,25,23,8)(6,16,14,26,24,9), (1,26,6)(2,27,4)(3,25,5)(7,23,10,18,15,20)(8,24,11,16,13,21)(9,22,12,17,14,19)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$ x 2

Degree 9: $S_3\times C_3$ x 2, $S_3^2$

Low degree siblings

12T70, 18T43, 18T46 x 2, 36T80, 36T82 x 2, 36T92

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 38]
Character table: Data not available.