# Properties

 Label 27T44 Order $$162$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $He_3:C_3:C_2$

## Group action invariants

 Degree $n$ : $27$ Transitive number $t$ : $44$ Group : $He_3:C_3:C_2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,24,2,23,3,22)(4,20,5,19,6,21)(7,25,8,27,9,26)(10,18)(11,17)(12,16)(14,15), (1,10,3,11,2,12)(4,16,6,17,5,18)(7,15,9,13,8,14)(19,23)(20,22)(21,24)(25,26), (1,14)(2,13)(3,15)(4,10)(5,12)(6,11)(7,18)(8,17)(9,16)(19,27)(20,26)(21,25)(23,24) $|\Aut(F/K)|$: $1$

## 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

27T66 x 3

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

## Group invariants

 Order: $162=2 \cdot 3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [162, 21]
 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 2a 6a 6b 3c 9a 9b 9c 3d 3e 3f 2P 1a 3b 3a 1a 3a 3b 3c 9c 9a 9b 3d 3e 3f 3P 1a 1a 1a 2a 2a 2a 1a 3c 3c 3c 1a 1a 1a 5P 1a 3b 3a 2a 6b 6a 3c 9b 9c 9a 3d 3e 3f 7P 1a 3a 3b 2a 6a 6b 3c 9c 9a 9b 3d 3e 3f 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 -1 B /B 3 . . . . . . X.8 3 /A A -1 /B B 3 . . . . . . X.9 3 A /A 1 -B -/B 3 . . . . . . X.10 3 /A A 1 -/B -B 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