Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $19$ | |
| Group : | $C_3\times D_9$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,5,7,2,6,8,3,4,9)(10,14,18,12,13,17,11,15,16), (1,15,2,13,3,14)(4,12,5,10,6,11)(7,18,8,16,9,17) | |
| $|\Aut(F/K)|$: | $9$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$, $D_{9}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 9: None
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,13,16,12,15,18,11,14,17)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,14,18,12,13,17,11,15,16)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,15,17,12,14,16,11,13,18)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,13,16,12,15,18,11,14,17)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,14,18,12,13,17,11,15,16)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,15,17,12,14,16,11,13,18)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,13,16,12,15,18,11,14,17)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,14,18,12,13,17,11,15,16)$ |
| $ 9, 9 $ | $2$ | $9$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,15,17,12,14,16,11,13,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1,10)( 2,11)( 3,12)( 4,17)( 5,18)( 6,16)( 7,13)( 8,14)( 9,15)$ |
| $ 6, 6, 6 $ | $9$ | $6$ | $( 1,10, 2,11, 3,12)( 4,17, 5,18, 6,16)( 7,13, 8,14, 9,15)$ |
| $ 6, 6, 6 $ | $9$ | $6$ | $( 1,10, 3,12, 2,11)( 4,17, 6,16, 5,18)( 7,13, 9,15, 8,14)$ |
Group invariants
| Order: | $54=2 \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [54, 3] |
| Character table: |
2 1 . . 1 . 1 . . . . . . . . . 1 1 1
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1
1a 3a 3b 3c 3d 3e 9a 9b 9c 9d 9e 9f 9g 9h 9i 2a 6a 6b
2P 1a 3b 3a 3e 3d 3c 9h 9e 9b 9g 9d 9a 9i 9f 9c 1a 3c 3e
3P 1a 1a 1a 1a 1a 1a 3d 3d 3d 3d 3d 3d 3d 3d 3d 2a 2a 2a
5P 1a 3b 3a 3e 3d 3c 9f 9c 9i 9e 9b 9h 9d 9a 9g 2a 6b 6a
7P 1a 3a 3b 3c 3d 3e 9h 9i 9g 9b 9c 9a 9e 9f 9d 2a 6a 6b
X.1 1 1 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 -1 -1 -1
X.3 1 A /A /A 1 A 1 A /A A /A 1 /A 1 A -1 -A -/A
X.4 1 /A A A 1 /A 1 /A A /A A 1 A 1 /A -1 -/A -A
X.5 1 A /A /A 1 A 1 A /A A /A 1 /A 1 A 1 A /A
X.6 1 /A A A 1 /A 1 /A A /A A 1 A 1 /A 1 /A A
X.7 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 . . .
X.8 2 B /B /B 2 B -1 -/A -A -/A -A -1 -A -1 -/A . . .
X.9 2 /B B B 2 /B -1 -A -/A -A -/A -1 -/A -1 -A . . .
X.10 2 -1 -1 2 -1 2 C E D D C E E D C . . .
X.11 2 -1 -1 2 -1 2 D C E E D C C E D . . .
X.12 2 -1 -1 2 -1 2 E D C C E D D C E . . .
X.13 2 -/A -A /B -1 B C F /H H /G E /F D G . . .
X.14 2 -/A -A /B -1 B D G /F F /H C /G E H . . .
X.15 2 -/A -A /B -1 B E H /G G /F D /H C F . . .
X.16 2 -A -/A B -1 /B C /F H /H G E F D /G . . .
X.17 2 -A -/A B -1 /B D /G F /F H C G E /H . . .
X.18 2 -A -/A B -1 /B E /H G /G F D H C /F . . .
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
= -1+Sqrt(-3) = 2b3
C = E(9)^2+E(9)^7
D = E(9)^4+E(9)^5
E = -E(9)^2-E(9)^4-E(9)^5-E(9)^7
F = E(9)^2+E(9)^4
G = -E(9)^4+E(9)^5-E(9)^7
H = -E(9)^2-E(9)^5+E(9)^7
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