Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $37$ | |
Group: | $C_3:S_4$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,6,3,2,5,4)(7,12,10,8,11,9)(13,18,16)(14,17,15), (1,6)(2,5)(3,4)(7,16,8,15)(9,14,10,13)(11,18,12,17), (1,16)(2,15)(3,13)(4,14)(5,18)(6,17)(7,11)(8,12) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ x 4 $18$: $C_3^2:C_2$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$ x 4
Degree 6: $S_4$
Degree 9: $C_3^2:C_2$
Low degree siblings
12T44 x 3, 18T40Siblings 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 $ | $18$ | $2$ | $( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$ |
$ 4, 4, 4, 2, 2, 2 $ | $18$ | $4$ | $( 1, 2)( 3, 6)( 4, 5)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$ |
$ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,11, 8,10,12)(13,15,18,14,16,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,16,18)(14,15,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3,10,14)( 4, 9,13)( 5,11,15)( 6,12,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 9,15)( 2,10,16)( 3,12,17)( 4,11,18)( 5, 8,14)( 6, 7,13)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,11,14)( 2,12,13)( 3, 7,15)( 4, 8,16)( 5,10,17)( 6, 9,18)$ |
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [72, 43] |
Character table: |
2 3 3 2 2 2 2 . . . 3 2 1 . . 1 2 2 2 2 1a 2a 2b 4a 6a 3a 3b 3c 3d 2P 1a 1a 1a 2a 3a 3a 3b 3c 3d 3P 1a 2a 2b 4a 2a 1a 1a 1a 1a 5P 1a 2a 2b 4a 6a 3a 3b 3c 3d 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 2 -1 -1 X.5 2 2 . . -1 -1 -1 -1 2 X.6 2 2 . . -1 -1 -1 2 -1 X.7 3 -1 -1 1 -1 3 . . . X.8 3 -1 1 -1 -1 3 . . . X.9 6 -2 . . 1 -3 . . . |