Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $32$ | |
Group: | $S_3\times A_4$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,12,14)(2,11,13)(3,7,16)(4,8,15)(5,9,17)(6,10,18), (1,5)(2,6)(7,10)(8,9)(11,12)(13,15)(14,16) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $12$: $A_4$ $18$: $S_3\times C_3$ $24$: $A_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 6: $A_4\times C_2$
Degree 9: $S_3\times C_3$
Low degree siblings
12T43, 18T31Siblings 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, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 5)( 4, 6)( 7,11)( 8,12)( 9,10)(13,18)(14,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,11)( 8,12)( 9,10)(13,17)(14,18)(15,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,15,18)(14,16,17)$ |
$ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,16,18,14,15,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,16)( 6,11,15)$ |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1, 7,15, 6,10,14)( 2, 8,16, 5, 9,13)( 3,12,18, 4,11,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,15)( 2,10,16)( 3,12,18)( 4,11,17)( 5, 7,13)( 6, 8,14)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,13,12)( 2,14,11)( 3,15, 7)( 4,16, 8)( 5,18, 9)( 6,17,10)$ |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1,13,10, 6,16, 7)( 2,14, 9, 5,15, 8)( 3,18,11, 4,17,12)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,15, 9)( 2,16,10)( 3,18,12)( 4,17,11)( 5,13, 7)( 6,14, 8)$ |
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [72, 44] |
Character table: |
2 3 3 3 3 2 2 . 1 1 . 1 1 3 2 1 . 1 2 1 2 1 2 2 1 2 1a 2a 2b 2c 3a 6a 3b 6b 3c 3d 6c 3e 2P 1a 1a 1a 1a 3a 3a 3d 3e 3e 3b 3c 3c 3P 1a 2a 2b 2c 1a 2a 1a 2c 1a 1a 2c 1a 5P 1a 2a 2b 2c 3a 6a 3d 6c 3e 3b 6b 3c 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 A -A A /A -/A /A X.4 1 1 -1 -1 1 1 /A -/A /A A -A A X.5 1 1 1 1 1 1 A A A /A /A /A X.6 1 1 1 1 1 1 /A /A /A A A A X.7 2 2 . . -1 -1 -1 . 2 -1 . 2 X.8 2 2 . . -1 -1 -A . B -/A . /B X.9 2 2 . . -1 -1 -/A . /B -A . B X.10 3 -1 -1 3 3 -1 . . . . . . X.11 3 -1 1 -3 3 -1 . . . . . . X.12 6 -2 . . -3 1 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 |