Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $30$ | |
Group: | $C_3\times S_4$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $6$ | |
Generators: | (1,3,5)(2,4,6)(7,18,11,15,9,13,8,17,12,16,10,14), (1,8,18)(2,7,17)(3,10,13)(4,9,14)(5,11,16)(6,12,15) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 6: $S_4$
Degree 9: $S_3\times C_3$
Low degree siblings
12T45, 18T33Siblings 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 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $6$ | $4$ | $( 7,15, 8,16)( 9,17,10,18)(11,13,12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,17)(10,18)(11,13)(12,14)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)$ |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,15,18,14,16,17)$ |
$ 12, 3, 3 $ | $6$ | $12$ | $( 1, 3, 5)( 2, 4, 6)( 7,17,11,16, 9,14, 8,18,12,15,10,13)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,17,12,15, 9,14)( 8,18,11,16,10,13)$ |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,11, 9, 8,12,10)(13,17,16,14,18,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,16)(14,17,15)$ |
$ 12, 3, 3 $ | $6$ | $12$ | $( 1, 5, 3)( 2, 6, 4)( 7,13,10,15,12,18, 8,14, 9,16,11,17)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,13, 9,16,12,18)( 8,14,10,15,11,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,15)( 6,11,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 9,16)( 2,10,15)( 3,12,18)( 4,11,17)( 5, 7,13)( 6, 8,14)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,11,13)( 2,12,14)( 3, 8,16)( 4, 7,15)( 5,10,18)( 6, 9,17)$ |
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [72, 42] |
Character table: |
2 3 3 2 2 3 3 2 2 3 3 2 2 . . . 3 2 1 1 1 2 1 1 1 1 2 1 1 2 2 2 1a 2a 4a 2b 3a 6a 12a 6b 6c 3b 12b 6d 3c 3d 3e 2P 1a 1a 2a 1a 3b 3b 6c 3b 3a 3a 6a 3a 3d 3c 3e 3P 1a 2a 4a 2b 1a 2a 4a 2b 2a 1a 4a 2b 1a 1a 1a 5P 1a 2a 4a 2b 3b 6c 12b 6d 6a 3a 12a 6b 3d 3c 3e 7P 1a 2a 4a 2b 3a 6a 12a 6b 6c 3b 12b 6d 3c 3d 3e 11P 1a 2a 4a 2b 3b 6c 12b 6d 6a 3a 12a 6b 3d 3c 3e X.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 X.3 1 1 -1 -1 A A -A -A /A /A -/A -/A A /A 1 X.4 1 1 -1 -1 /A /A -/A -/A A A -A -A /A A 1 X.5 1 1 1 1 A A A A /A /A /A /A A /A 1 X.6 1 1 1 1 /A /A /A /A A A A A /A A 1 X.7 2 2 . . 2 2 . . 2 2 . . -1 -1 -1 X.8 2 2 . . B B . . /B /B . . -A -/A -1 X.9 2 2 . . /B /B . . B B . . -/A -A -1 X.10 3 -1 -1 1 3 -1 -1 1 -1 3 -1 1 . . . X.11 3 -1 1 -1 3 -1 1 -1 -1 3 1 -1 . . . X.12 3 -1 -1 1 C -A -A A -/A /C -/A /A . . . X.13 3 -1 -1 1 /C -/A -/A /A -A C -A A . . . X.14 3 -1 1 -1 C -A A -A -/A /C /A -/A . . . X.15 3 -1 1 -1 /C -/A /A -/A -A C A -A . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 |