Group action invariants
Degree $n$: | $27$ | |
Transitive number $t$: | $38$ | |
Group: | $(C_3\times C_9):C_6$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,19,13,8,23,10)(2,21,14,7,24,12)(3,20,15,9,22,11)(4,25,17)(5,27,18,6,26,16), (1,9,4,3,8,6,2,7,5)(10,17,13,12,16,15,11,18,14)(19,27,24,21,26,23,20,25,22) |
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 9: $S_3\times C_3$
Low degree siblings
27T64Siblings 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 $ | $6$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,15,17,11,13,18,12,14,16)(19,22,25,20,23,26,21, 24,27)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,13,16,11,14,17,12,15,18)(19,23,27,20,24,25,21, 22,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,10,25)( 2,11,26)( 3,12,27)( 4,14,19)( 5,15,20)( 6,13,21)( 7,17,23) ( 8,18,24)( 9,16,22)$ |
$ 6, 6, 6, 6, 3 $ | $27$ | $6$ | $( 1,10,22, 7,14,20)( 2,12,23, 9,15,19)( 3,11,24, 8,13,21)( 4,17,27, 6,18,26) ( 5,16,25)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1,13,23)( 2,14,24)( 3,15,22)( 4,17,25)( 5,18,26)( 6,16,27)( 7,12,21) ( 8,10,19)( 9,11,20)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,19,17)( 2,20,18)( 3,21,16)( 4,24,10)( 5,22,11)( 6,23,12)( 7,25,13) ( 8,26,14)( 9,27,15)$ |
$ 6, 6, 6, 6, 3 $ | $27$ | $6$ | $( 1,19,13, 8,23,10)( 2,21,14, 7,24,12)( 3,20,15, 9,22,11)( 4,25,17) ( 5,27,18, 6,26,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $9$ | $3$ | $( 1,22,14)( 2,23,15)( 3,24,13)( 4,27,18)( 5,25,16)( 6,26,17)( 7,20,10) ( 8,21,11)( 9,19,12)$ |
Group invariants
Order: | $162=2 \cdot 3^{4}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [162, 15] |
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 3c 6a 3d 3e 6b 3f 2P 1a 3a 1a 3b 9c 9a 9b 3e 3f 3f 3c 3d 3d 3P 1a 1a 2a 1a 3b 3b 3b 1a 2a 1a 1a 2a 1a 5P 1a 3a 2a 3b 9b 9c 9a 3e 6b 3f 3c 6a 3d 7P 1a 3a 2a 3b 9c 9a 9b 3c 6a 3d 3e 6b 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 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 C B . . . . . . X.12 6 . . -3 B A C . . . . . . X.13 6 . . -3 C B A . . . . . . 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 |