# Properties

 Label 27T37 Degree $27$ Order $162$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3\wr S_3$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $37$ Group: $C_3\wr S_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $9$ Generators: (1,4,25)(2,5,26)(3,6,27)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10,21)(2,11,19)(3,12,20)(4,13,24)(5,14,22)(6,15,23)(7,18,25)(8,16,26)(9,17,27), (7,18)(8,16)(9,17)(10,21)(11,19)(12,20)(13,24)(14,22)(15,23)

## 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: $C_3$, $S_3$

Degree 9: $S_3\times C_3$, $C_3 \wr S_3$ x 3

## Low degree siblings

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

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

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

## Group invariants

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