Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $21$ | |
Group: | $He_3:C_2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $6$ | |
Generators: | (1,5,11)(2,6,12)(3,7,13)(4,8,14)(9,15,17)(10,16,18), (1,3)(2,4)(5,12)(6,11)(7,15)(8,16)(9,13)(10,14)(17,18) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 9: $C_3^2 : C_6$
Low degree siblings
9T11, 9T13, 18T20, 18T22Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 5, 9, 8)( 6,10, 7)(11,14,15)(12,13,16)$ |
$ 6, 6, 2, 2, 2 $ | $9$ | $6$ | $( 1, 2)( 3,17)( 4,18)( 5,12, 8,16, 9,13)( 6,11, 7,15,10,14)$ |
$ 6, 6, 2, 2, 2 $ | $9$ | $6$ | $( 1, 2)( 3,17)( 4,18)( 5,13, 9,16, 8,12)( 6,14,10,15, 7,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 5,14)( 2, 6,13)( 3, 7,16)( 4, 8,15)( 9,11,17)(10,12,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 5,15)( 2, 6,16)( 3, 7,12)( 4, 8,11)( 9,14,17)(10,13,18)$ |
Group invariants
Order: | $54=2 \cdot 3^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [54, 5] |
Character table: |
2 1 1 1 1 1 1 . . . . 3 3 2 2 1 1 1 3 2 2 2 1a 3a 3b 6a 6b 2a 3c 3d 3e 3f 2P 1a 3b 3a 3a 3b 1a 3c 3e 3d 3f 3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 1a 5P 1a 3b 3a 6b 6a 2a 3c 3e 3d 3f X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 1 X.3 1 A /A -/A -A -1 1 A /A 1 X.4 1 /A A -A -/A -1 1 /A A 1 X.5 1 A /A /A A 1 1 A /A 1 X.6 1 /A A A /A 1 1 /A A 1 X.7 2 2 2 . . . 2 -1 -1 -1 X.8 2 B /B . . . 2 -/A -A -1 X.9 2 /B B . . . 2 -A -/A -1 X.10 6 . . . . . -3 . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |