# Properties

 Label 27T42 Degree $27$ Order $162$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_9:C_3:S_3$

## Group action invariants

 Degree $n$: $27$ Transitive number $t$: $42$ Group: $C_9:C_3:S_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,12,3,10,2,11)(4,18,6,16,5,17)(7,14,9,15,8,13)(19,23)(20,22)(21,24)(25,26), (1,26,3,27,2,25)(4,23,6,24,5,22)(7,21,9,19,8,20)(11,12)(13,17)(14,16)(15,18), (1,16,19,2,17,20,3,18,21)(4,10,23,5,11,24,6,12,22)(7,15,27,8,13,25,9,14,26)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$

## Low degree siblings

27T68

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

## Group invariants

 Order: $162=2 \cdot 3^{4}$ Cyclic: no Abelian: no Solvable: yes GAP id: [162, 20]
 Character table:  2 1 1 1 1 1 1 . . . . . . . 3 4 3 3 1 1 1 4 3 3 3 2 2 2 1a 3a 3b 6a 6b 2a 3c 9a 9b 9c 3d 9d 9e 2P 1a 3b 3a 3a 3b 1a 3c 9c 9a 9b 3d 9d 9e 3P 1a 1a 1a 2a 2a 2a 1a 3c 3c 3c 1a 3c 3c 5P 1a 3b 3a 6b 6a 2a 3c 9b 9c 9a 3d 9d 9e 7P 1a 3a 3b 6a 6b 2a 3c 9c 9a 9b 3d 9d 9e X.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 X.3 2 2 2 . . . 2 2 2 2 -1 -1 -1 X.4 2 2 2 . . . 2 -1 -1 -1 2 -1 -1 X.5 2 2 2 . . . 2 -1 -1 -1 -1 -1 2 X.6 2 2 2 . . . 2 -1 -1 -1 -1 2 -1 X.7 3 A /A B /B -1 3 . . . . . . X.8 3 /A A /B B -1 3 . . . . . . X.9 3 A /A -B -/B 1 3 . . . . . . X.10 3 /A A -/B -B 1 3 . . . . . . X.11 6 . . . . . -3 C E D . . . X.12 6 . . . . . -3 D C E . . . X.13 6 . . . . . -3 E D C . . . A = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 B = -E(3) = (1-Sqrt(-3))/2 = -b3 C = -2*E(9)^2-E(9)^4-E(9)^5-2*E(9)^7 D = E(9)^2-E(9)^4-E(9)^5+E(9)^7 E = E(9)^2+2*E(9)^4+2*E(9)^5+E(9)^7