Group action invariants
| Degree $n$ : | $18$ | |
| Transitive number $t$ : | $2$ | |
| Group : | $C_6 \times C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $1$ | |
| Generators: | (1,7,13,2,8,14)(3,9,15,4,10,16)(5,11,17,6,12,18), (1,16,5,2,15,6)(3,11,8,4,12,7)(9,17,14,10,18,13) | |
| $|\Aut(F/K)|$: | $18$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ x 4 6: $C_6$ x 4 9: $C_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$ x 4
Degree 6: $C_6$ x 4
Degree 9: $C_3^2$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3,17)( 2, 4,18)( 5, 8,10)( 6, 7, 9)(11,14,16)(12,13,15)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 4,17, 2, 3,18)( 5, 7,10, 6, 8, 9)(11,13,16,12,14,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5,15)( 2, 6,16)( 3, 8,12)( 4, 7,11)( 9,14,18)(10,13,17)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 6,15, 2, 5,16)( 3, 7,12, 4, 8,11)( 9,13,18,10,14,17)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 7,13, 2, 8,14)( 3, 9,15, 4,10,16)( 5,11,17, 6,12,18)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,17)( 6,11,18)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1, 9,12, 2,10,11)( 3, 6,13, 4, 5,14)( 7,15,18, 8,16,17)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,10,12)( 2, 9,11)( 3, 5,13)( 4, 6,14)( 7,16,18)( 8,15,17)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,11,10, 2,12, 9)( 3,14, 5, 4,13, 6)( 7,17,16, 8,18,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,12,10)( 2,11, 9)( 3,13, 5)( 4,14, 6)( 7,18,16)( 8,17,15)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,13, 8)( 2,14, 7)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,14, 8, 2,13, 7)( 3,16,10, 4,15, 9)( 5,18,12, 6,17,11)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,15, 5)( 2,16, 6)( 3,12, 8)( 4,11, 7)( 9,18,14)(10,17,13)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,16, 5, 2,15, 6)( 3,11, 8, 4,12, 7)( 9,17,14,10,18,13)$ |
| $ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,17, 3)( 2,18, 4)( 5,10, 8)( 6, 9, 7)(11,16,14)(12,15,13)$ |
| $ 6, 6, 6 $ | $1$ | $6$ | $( 1,18, 3, 2,17, 4)( 5, 9, 8, 6,10, 7)(11,15,14,12,16,13)$ |
Group invariants
| Order: | $18=2 \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | Yes | |
| Solvable: | Yes | |
| GAP id: | [18, 5] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1a 2a 3a 6a 3b 6b 6c 3c 6d 3d 6e 3e 3f 6f 3g 6g 3h 6h
2P 1a 1a 3h 3h 3g 3g 3f 3f 3e 3e 3d 3d 3c 3c 3b 3b 3a 3a
3P 1a 2a 1a 2a 1a 2a 2a 1a 2a 1a 2a 1a 1a 2a 1a 2a 1a 2a
5P 1a 2a 3h 6h 3g 6g 6f 3f 6e 3e 6d 3d 3c 6c 3b 6b 3a 6a
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 -1 1 -1 A -A -A A -A A -/A /A /A -/A /A -/A 1 -1
X.4 1 -1 1 -1 /A -/A -/A /A -/A /A -A A A -A A -A 1 -1
X.5 1 -1 A -A 1 -1 -A A -/A /A -A A /A -/A 1 -1 /A -/A
X.6 1 -1 /A -/A 1 -1 -/A /A -A A -/A /A A -A 1 -1 A -A
X.7 1 -1 A -A A -A -/A /A -1 1 -1 1 A -A /A -/A /A -/A
X.8 1 -1 /A -/A /A -/A -A A -1 1 -1 1 /A -/A A -A A -A
X.9 1 -1 A -A /A -/A -1 1 -A A -/A /A 1 -1 A -A /A -/A
X.10 1 -1 /A -/A A -A -1 1 -/A /A -A A 1 -1 /A -/A A -A
X.11 1 1 1 1 A A A A A A /A /A /A /A /A /A 1 1
X.12 1 1 1 1 /A /A /A /A /A /A A A A A A A 1 1
X.13 1 1 A A 1 1 A A /A /A A A /A /A 1 1 /A /A
X.14 1 1 /A /A 1 1 /A /A A A /A /A A A 1 1 A A
X.15 1 1 A A A A /A /A 1 1 1 1 A A /A /A /A /A
X.16 1 1 /A /A /A /A A A 1 1 1 1 /A /A A A A A
X.17 1 1 A A /A /A 1 1 A A /A /A 1 1 A A /A /A
X.18 1 1 /A /A A A 1 1 /A /A A A 1 1 /A /A A A
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
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