# Properties

 Label 27T41 Order $$162$$ n $$27$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $He_3.S_3$

## Group action invariants

 Degree $n$ : $27$ Transitive number $t$ : $41$ Group : $He_3.S_3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,10,25,2,11,26,3,12,27)(4,15,19,5,13,20,6,14,21)(7,16,23,8,17,24,9,18,22), (1,20,14,8,24,10)(2,19,15,7,22,12)(3,21,13,9,23,11)(4,26,16)(5,25,17,6,27,18) $|\Aut(F/K)|$: $1$

## 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: $S_3\times C_3$

## Low degree siblings

27T72

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

## Group invariants

 Order: $162=2 \cdot 3^{4}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [162, 13]
 Character table:  2 1 . 1 . . . . . 1 1 . 1 1 3 4 3 1 4 3 3 3 2 1 2 2 1 2 1a 3a 2a 3b 9a 9b 9c 9d 6a 3c 9e 6b 3d 2P 1a 3a 1a 3b 9b 9c 9a 9e 3d 3d 9d 3c 3c 3P 1a 1a 2a 1a 3b 3b 3b 3b 2a 1a 3b 2a 1a 5P 1a 3a 2a 3b 9c 9a 9b 9e 6b 3d 9d 6a 3c 7P 1a 3a 2a 3b 9b 9c 9a 9d 6a 3c 9e 6b 3d 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 1 1 -1 1 1 1 1 D -D D /D -/D /D X.4 1 1 -1 1 1 1 1 /D -/D /D D -D D X.5 1 1 1 1 1 1 1 D D D /D /D /D X.6 1 1 1 1 1 1 1 /D /D /D D D D X.7 2 2 . 2 -1 -1 -1 -1 . 2 -1 . 2 X.8 2 2 . 2 -1 -1 -1 -D . E -/D . /E X.9 2 2 . 2 -1 -1 -1 -/D . /E -D . E X.10 6 -3 . 6 . . . . . . . . . X.11 6 . . -3 A B C . . . . . . X.12 6 . . -3 B C A . . . . . . X.13 6 . . -3 C A B . . . . . . A = -2*E(9)^2-E(9)^4-E(9)^5-2*E(9)^7 B = E(9)^2-E(9)^4-E(9)^5+E(9)^7 C = E(9)^2+2*E(9)^4+2*E(9)^5+E(9)^7 D = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 E = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3