Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $38$ | |
Group: | $C_2^2:D_9$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,15)(2,16)(3,14)(4,13)(5,18)(6,17)(9,12)(10,11), (1,8,2,7)(3,12,4,11)(5,9,6,10)(13,14)(15,18)(16,17) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $18$: $D_{9}$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4$
Degree 9: $D_{9}$
Low degree siblings
18T39Siblings 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, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $18$ | $2$ | $( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$ |
$ 4, 4, 4, 2, 2, 2 $ | $18$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,17, 8,18)( 9,15,10,16)(11,14,12,13)$ |
$ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 6, 2, 4, 5)( 7, 9,12)( 8,10,11)(13,16,17,14,15,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4, 6)( 2, 3, 5)( 7, 9,12)( 8,10,11)(13,15,17)(14,16,18)$ |
$ 9, 9 $ | $8$ | $9$ | $( 1, 7,15, 6,12,13, 4, 9,17)( 2, 8,16, 5,11,14, 3,10,18)$ |
$ 9, 9 $ | $8$ | $9$ | $( 1, 9,13, 6, 7,17, 4,12,15)( 2,10,14, 5, 8,18, 3,11,16)$ |
$ 9, 9 $ | $8$ | $9$ | $( 1,11,18, 6,10,16, 4, 8,14)( 2,12,17, 5, 9,15, 3, 7,13)$ |
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [72, 15] |
Character table: |
2 3 3 2 2 2 2 . . . 3 2 1 . . 1 2 2 2 2 1a 2a 2b 4a 6a 3a 9a 9b 9c 2P 1a 1a 1a 2a 3a 3a 9b 9c 9a 3P 1a 2a 2b 4a 2a 1a 3a 3a 3a 5P 1a 2a 2b 4a 6a 3a 9c 9a 9b 7P 1a 2a 2b 4a 6a 3a 9b 9c 9a X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 X.3 2 2 . . 2 2 -1 -1 -1 X.4 2 2 . . -1 -1 A C B X.5 2 2 . . -1 -1 B A C X.6 2 2 . . -1 -1 C B A X.7 3 -1 -1 1 -1 3 . . . X.8 3 -1 1 -1 -1 3 . . . X.9 6 -2 . . 1 -3 . . . A = E(9)^4+E(9)^5 B = E(9)^2+E(9)^7 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7 |