Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $27$ | |
Group: | $C_2\times C_3:S_3.C_2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,13,18,6)(2,14,17,5)(3,4)(7,9,11,15)(8,10,12,16), (1,15,9,14)(2,16,10,13)(3,17,7,6)(4,18,8,5) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $36$: $C_3^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: None
Degree 9: $C_3^2:C_4$
Low degree siblings
12T40 x 2, 12T41 x 2, 18T27Siblings 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 $ | $1$ | $1$ | $()$ |
$ 4, 4, 4, 4, 1, 1 $ | $9$ | $4$ | $( 3,10,17,11)( 4, 9,18,12)( 5, 8,15,14)( 6, 7,16,13)$ |
$ 4, 4, 4, 4, 1, 1 $ | $9$ | $4$ | $( 3,11,17,10)( 4,12,18, 9)( 5,14,15, 8)( 6,13,16, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
$ 4, 4, 4, 4, 2 $ | $9$ | $4$ | $( 1, 2)( 3, 9,17,12)( 4,10,18,11)( 5, 7,15,13)( 6, 8,16,14)$ |
$ 4, 4, 4, 4, 2 $ | $9$ | $4$ | $( 1, 2)( 3,12,17, 9)( 4,11,18,10)( 5,13,15, 7)( 6,14,16, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,11)(10,12)$ |
$ 6, 6, 6 $ | $4$ | $6$ | $( 1, 3,18, 2, 4,17)( 5, 7, 9, 6, 8,10)(11,14,16,12,13,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 4,18)( 2, 3,17)( 5, 8, 9)( 6, 7,10)(11,13,16)(12,14,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,15)( 2, 6,16)( 3, 7,11)( 4, 8,12)( 9,14,18)(10,13,17)$ |
$ 6, 6, 6 $ | $4$ | $6$ | $( 1, 6,15, 2, 5,16)( 3, 8,11, 4, 7,12)( 9,13,18,10,14,17)$ |
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [72, 45] |
Character table: |
2 3 3 3 3 3 3 3 3 1 1 1 1 3 2 . . . 2 . . . 2 2 2 2 1a 4a 4b 2a 2b 4c 4d 2c 6a 3a 3b 6b 2P 1a 2a 2a 1a 1a 2a 2a 1a 3a 3a 3b 3b 3P 1a 4b 4a 2a 2b 4d 4c 2c 2b 1a 1a 2b 5P 1a 4a 4b 2a 2b 4c 4d 2c 6a 3a 3b 6b X.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 X.3 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 X.5 1 A -A -1 -1 -A A 1 -1 1 1 -1 X.6 1 -A A -1 -1 A -A 1 -1 1 1 -1 X.7 1 A -A -1 1 A -A -1 1 1 1 1 X.8 1 -A A -1 1 -A A -1 1 1 1 1 X.9 4 . . . -4 . . . -1 1 -2 2 X.10 4 . . . -4 . . . 2 -2 1 -1 X.11 4 . . . 4 . . . -2 -2 1 1 X.12 4 . . . 4 . . . 1 1 -2 -2 A = -E(4) = -Sqrt(-1) = -i |