Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $4$ | |
Group: | $C_3^2 : C_2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $18$ | |
Generators: | (1,8,14)(2,7,13)(3,10,16)(4,9,15)(5,11,17)(6,12,18), (1,4,17)(2,3,18)(5,8,9)(6,7,10)(11,14,15)(12,13,16), (1,3)(2,4)(5,12)(6,11)(7,15)(8,16)(9,13)(10,14)(17,18) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ x 4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 4
Degree 6: $S_3$ x 4
Degree 9: $C_3^2:C_2$
Low degree siblings
9T5Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5,15)( 2, 6,16)( 3, 7,12)( 4, 8,11)( 9,14,17)(10,13,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,14)( 2, 7,13)( 3,10,16)( 4, 9,15)( 5,11,17)( 6,12,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,11)( 2,10,12)( 3, 6,13)( 4, 5,14)( 7,16,18)( 8,15,17)$ |
Group invariants
Order: | $18=2 \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [18, 4] |
Character table: |
2 1 1 . . . . 3 2 . 2 2 2 2 1a 2a 3a 3b 3c 3d 2P 1a 1a 3a 3b 3c 3d 3P 1a 2a 1a 1a 1a 1a X.1 1 1 1 1 1 1 X.2 1 -1 1 1 1 1 X.3 2 . 2 -1 -1 -1 X.4 2 . -1 2 -1 -1 X.5 2 . -1 -1 -1 2 X.6 2 . -1 -1 2 -1 |