# Properties

 Label 27T4 Degree $27$ Order $27$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_3^3$

# Related objects

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $4$ Group: $C_3^3$ Parity: $1$ Primitive: no Nilpotency class: $1$ $|\Aut(F/K)|$: $27$ Generators: (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), (1,11,19)(2,12,20)(3,10,21)(4,15,22)(5,13,23)(6,14,24)(7,16,25)(8,17,26)(9,18,27), (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)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 13
$9$:  $C_3^2$ x 13

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$ x 13

Degree 9: $C_3^2$ x 13

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $27=3^{3}$ Cyclic: no Abelian: yes Solvable: yes GAP id: [27, 5]
 Character table: not available.