Group action invariants
Degree $n$: | $27$ | |
Transitive number $t$: | $30$ | |
Group: | $S_3\times D_9$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,19,10,3,21,12,2,20,11)(4,17,14,25,22,9,5,18,15,26,23,7,6,16,13,27,24,8), (1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 2 $12$: $D_{6}$ x 2 $18$: $D_{9}$ $36$: $S_3^2$, $D_{18}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$ x 2
Low degree siblings
18T50, 36T86Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 4,26)( 5,27)( 6,25)( 7,14)( 8,15)( 9,13)(16,22)(17,23)(18,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 2, 3)( 5, 6)( 7,18)( 8,17)( 9,16)(10,20)(11,19)(12,21)(13,22)(14,24)(15,23) (25,27)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $27$ | $2$ | $( 2, 3)( 4,26)( 5,25)( 6,27)( 7,24)( 8,23)( 9,22)(10,20)(11,19)(12,21)(13,16) (14,18)(15,17)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,15, 9,14, 8,13)(10,11,12)(16,23,18,22,17,24) (19,20,21)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,10,14)( 8,11,15)( 9,12,13)(16,21,22) (17,19,23)(18,20,24)$ |
$ 6, 6, 6, 6, 3 $ | $18$ | $6$ | $( 1, 4,26)( 2, 6,27, 3, 5,25)( 7,20,14,18,10,24)( 8,19,15,17,11,23) ( 9,21,13,16,12,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,11,13)( 8,12,14)( 9,10,15)(16,19,24) (17,20,22)(18,21,23)$ |
$ 9, 9, 9 $ | $4$ | $9$ | $( 1, 7,22, 2, 8,23, 3, 9,24)( 4,10,16, 5,11,17, 6,12,18)(13,20,26,14,21,27,15, 19,25)$ |
$ 18, 9 $ | $6$ | $18$ | $( 1, 7,21,27,11,17, 3, 9,20,26,10,16, 2, 8,19,25,12,18)( 4,14,22, 5,15,23, 6, 13,24)$ |
$ 9, 9, 9 $ | $4$ | $9$ | $( 1, 8,24, 2, 9,22, 3, 7,23)( 4,11,18, 5,12,16, 6,10,17)(13,21,25,14,19,26,15, 20,27)$ |
$ 18, 9 $ | $6$ | $18$ | $( 1, 8,20,27,12,16, 3, 7,19,26,11,18, 2, 9,21,25,10,17)( 4,15,24, 5,13,22, 6, 14,23)$ |
$ 9, 9, 9 $ | $4$ | $9$ | $( 1, 9,23, 2, 7,24, 3, 8,22)( 4,12,17, 5,10,18, 6,11,16)(13,19,27,14,20,25,15, 21,26)$ |
$ 18, 9 $ | $6$ | $18$ | $( 1, 9,19,27,10,18, 3, 8,21,26,12,17, 2, 7,20,25,11,16)( 4,13,23, 5,14,24, 6, 15,22)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1,10,21, 2,11,19, 3,12,20)( 4,14,22, 5,15,23, 6,13,24)( 7,16,27, 8,17,25, 9, 18,26)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1,11,20, 2,12,21, 3,10,19)( 4,15,24, 5,13,22, 6,14,23)( 7,17,26, 8,18,27, 9, 16,25)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1,12,19, 2,10,20, 3,11,21)( 4,13,23, 5,14,24, 6,15,22)( 7,18,25, 8,16,26, 9, 17,27)$ |
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [108, 16] |
Character table: |
2 2 2 2 2 1 1 1 1 . . 1 . 1 . 1 1 1 1 3 3 2 1 . 3 2 3 1 3 3 2 3 2 3 2 3 3 3 1a 2a 2b 2c 3a 6a 3b 6b 3c 9a 18a 9b 18b 9c 18c 9d 9e 9f 2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 9c 9f 9a 9d 9b 9e 9f 9d 9e 3P 1a 2a 2b 2c 1a 2a 1a 2b 1a 3a 6a 3a 6a 3a 6a 3a 3a 3a 5P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9b 18b 9c 18c 9a 18a 9e 9f 9d 7P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9c 18c 9a 18a 9b 18b 9f 9d 9e 11P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9c 18c 9a 18a 9b 18b 9f 9d 9e 13P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9b 18b 9c 18c 9a 18a 9e 9f 9d 17P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9a 18a 9b 18b 9c 18c 9d 9e 9f X.1 1 1 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 1 1 1 X.3 1 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 1 1 1 X.4 1 1 -1 -1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 X.5 2 -2 . . 2 -2 2 . 2 -1 1 -1 1 -1 1 -1 -1 -1 X.6 2 2 . . 2 2 2 . 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 X.7 2 . -2 . 2 . -1 1 -1 -1 . -1 . -1 . 2 2 2 X.8 2 . 2 . 2 . -1 -1 -1 -1 . -1 . -1 . 2 2 2 X.9 2 -2 . . -1 1 2 . -1 A -A B -B C -C A B C X.10 2 -2 . . -1 1 2 . -1 B -B C -C A -A B C A X.11 2 -2 . . -1 1 2 . -1 C -C A -A B -B C A B X.12 2 2 . . -1 -1 2 . -1 A A B B C C A B C X.13 2 2 . . -1 -1 2 . -1 B B C C A A B C A X.14 2 2 . . -1 -1 2 . -1 C C A A B B C A B X.15 4 . . . 4 . -2 . -2 1 . 1 . 1 . -2 -2 -2 X.16 4 . . . -2 . -2 . 1 -C . -A . -B . D F E X.17 4 . . . -2 . -2 . 1 -B . -C . -A . E D F X.18 4 . . . -2 . -2 . 1 -A . -B . -C . F E D A = -E(9)^2-E(9)^4-E(9)^5-E(9)^7 B = E(9)^4+E(9)^5 C = E(9)^2+E(9)^7 D = 2*E(9)^2+2*E(9)^7 E = 2*E(9)^4+2*E(9)^5 F = -2*E(9)^2-2*E(9)^4-2*E(9)^5-2*E(9)^7 |