# Properties

 Label 27T7 Degree $27$ Order $54$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^3:C_2$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $7$ Group: $C_3^3:C_2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,21,10)(2,19,11)(3,20,12)(4,23,15)(5,24,13)(6,22,14)(7,27,17)(8,25,18)(9,26,16), (2,3)(4,26)(5,25)(6,27)(7,22)(8,24)(9,23)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16), (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), (1,27,6)(2,25,4)(3,26,5)(7,14,10)(8,15,11)(9,13,12)(16,24,20)(17,22,21)(18,23,19)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 13
$18$:  $C_3^2:C_2$ x 13

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$ x 13

Degree 9: $C_3^2:C_2$ 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $27$ $2$ $( 2, 3)( 4,26)( 5,25)( 6,27)( 7,22)( 8,24)( 9,23)(10,21)(11,20)(12,19)(13,18) (14,17)(15,16)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,11,13)( 8,12,14)( 9,10,15)(16,21,23) (17,19,24)(18,20,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,12,15)( 8,10,13)( 9,11,14)(16,19,22) (17,20,23)(18,21,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,10,14)( 8,11,15)( 9,12,13)(16,20,24) (17,21,22)(18,19,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 7,22)( 2, 8,23)( 3, 9,24)( 4,11,18)( 5,12,16)( 6,10,17)(13,20,26) (14,21,27)(15,19,25)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 8,24)( 2, 9,22)( 3, 7,23)( 4,12,17)( 5,10,18)( 6,11,16)(13,21,25) (14,19,26)(15,20,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 9,23)( 2, 7,24)( 3, 8,22)( 4,10,16)( 5,11,17)( 6,12,18)(13,19,27) (14,20,25)(15,21,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,10,21)( 2,11,19)( 3,12,20)( 4,15,23)( 5,13,24)( 6,14,22)( 7,17,27) ( 8,18,25)( 9,16,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,11,20)( 2,12,21)( 3,10,19)( 4,13,22)( 5,14,23)( 6,15,24)( 7,18,26) ( 8,16,27)( 9,17,25)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,12,19)( 2,10,20)( 3,11,21)( 4,14,24)( 5,15,22)( 6,13,23)( 7,16,25) ( 8,17,26)( 9,18,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,13,18)( 2,14,16)( 3,15,17)( 4, 7,20)( 5, 8,21)( 6, 9,19)(10,24,25) (11,22,26)(12,23,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,14,17)( 2,15,18)( 3,13,16)( 4, 8,19)( 5, 9,20)( 6, 7,21)(10,22,27) (11,23,25)(12,24,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,15,16)( 2,13,17)( 3,14,18)( 4, 9,21)( 5, 7,19)( 6, 8,20)(10,23,26) (11,24,27)(12,22,25)$

## Group invariants

 Order: $54=2 \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [54, 14]
 Character table:  2 1 1 . . . . . . . . . . . . . 3 3 . 3 3 3 3 3 3 3 3 3 3 3 3 3 1a 2a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 3k 3l 3m 2P 1a 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 3k 3l 3m 3P 1a 2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.3 2 . 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 X.4 2 . 2 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 X.5 2 . -1 2 -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 X.6 2 . -1 -1 -1 2 2 -1 -1 2 -1 -1 -1 2 -1 X.7 2 . -1 -1 2 -1 2 -1 -1 -1 -1 2 -1 -1 2 X.8 2 . -1 2 -1 -1 -1 -1 2 2 -1 -1 -1 -1 2 X.9 2 . -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 2 -1 X.10 2 . 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 X.11 2 . 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 X.12 2 . -1 -1 -1 2 -1 -1 2 -1 -1 2 2 -1 -1 X.13 2 . -1 -1 -1 2 -1 2 -1 -1 2 -1 -1 -1 2 X.14 2 . -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 2 -1 X.15 2 . -1 -1 2 -1 -1 2 -1 2 -1 -1 2 -1 -1