Group action invariants
| Degree $n$ : | $19$ | |
| Transitive number $t$ : | $6$ | |
| Group : | $F_{19}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,3,5,9,17,14,8,15,10,19,18,16,12,4,7,13,6,11) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 9: $C_9$ 18: $C_{18}$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - 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 $ | $1$ | $1$ | $()$ |
| $ 18, 1 $ | $19$ | $18$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)$ |
| $ 18, 1 $ | $19$ | $18$ | $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)$ |
| $ 9, 9, 1 $ | $19$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)$ |
| $ 9, 9, 1 $ | $19$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)$ |
| $ 9, 9, 1 $ | $19$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)$ |
| $ 3, 3, 3, 3, 3, 3, 1 $ | $19$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$ |
| $ 6, 6, 6, 1 $ | $19$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)$ |
| $ 9, 9, 1 $ | $19$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)$ |
| $ 18, 1 $ | $19$ | $18$ | $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)$ |
| $ 3, 3, 3, 3, 3, 3, 1 $ | $19$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ |
| $ 6, 6, 6, 1 $ | $19$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)$ |
| $ 18, 1 $ | $19$ | $18$ | $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)$ |
| $ 18, 1 $ | $19$ | $18$ | $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)$ |
| $ 18, 1 $ | $19$ | $18$ | $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)$ |
| $ 9, 9, 1 $ | $19$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)$ |
| $ 9, 9, 1 $ | $19$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $19$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ |
| $ 19 $ | $18$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)$ |
Group invariants
| Order: | $342=2 \cdot 3^{2} \cdot 19$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [342, 7] |
| 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
19 1 . . . . . . . . . . . . . . . . .
1a 18a 18b 9a 9b 9c 3a 6a 9d 18c 3b 6b 18d 18e 18f 9e 9f 2a
2P 1a 9a 9d 9e 9c 9f 3b 3a 9b 9b 3a 3b 9f 9c 9e 9d 9a 1a
3P 1a 6a 6a 3a 3b 3a 1a 2a 3a 6b 1a 2a 6b 6a 6b 3b 3b 2a
5P 1a 18d 18f 9f 9d 9b 3b 6b 9e 18b 3a 6a 18e 18c 18a 9a 9c 2a
7P 1a 18e 18a 9c 9e 9d 3a 6a 9a 18f 3b 6b 18c 18b 18d 9f 9b 2a
11P 1a 18f 18c 9e 9c 9f 3b 6b 9b 18e 3a 6a 18a 18d 18b 9d 9a 2a
13P 1a 18b 18e 9d 9f 9a 3a 6a 9c 18d 3b 6b 18f 18a 18c 9b 9e 2a
17P 1a 18c 18d 9b 9a 9e 3b 6b 9f 18a 3a 6a 18b 18f 18e 9c 9d 2a
19P 1a 18a 18b 9a 9b 9c 3a 6a 9d 18c 3b 6b 18d 18e 18f 9e 9f 2a
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 -A -/A 1 -1 -/A /A 1 -1 /A A /A -A -A -1
X.4 1 /A /A -A -/A -A 1 -1 -A A 1 -1 A /A A -/A -/A -1
X.5 1 -/A -/A -A -/A -A 1 1 -A -A 1 1 -A -/A -A -/A -/A 1
X.6 1 -A -A -/A -A -/A 1 1 -/A -/A 1 1 -/A -A -/A -A -A 1
X.7 1 B D C /C /D -A -/A /B /B -/A -A /D /C C D B 1
X.8 1 /B /D /C C D -/A -A B B -A -/A D C /C /D /B 1
X.9 1 C /B D /D B -/A -A /C /C -A -/A B /D D /B C 1
X.10 1 D /C /B B C -A -/A /D /D -/A -A C B /B /C D 1
X.11 1 /D C B /B /C -/A -A D D -A -/A /C /B B C /D 1
X.12 1 /C B /D D /B -A -/A C C -/A -A /B D /D B /C 1
X.13 1 -/C -B /D D /B -A /A C -C -/A A -/B -D -/D B /C -1
X.14 1 -/D -C B /B /C -/A A D -D -A /A -/C -/B -B C /D -1
X.15 1 -D -/C /B B C -A /A /D -/D -/A A -C -B -/B /C D -1
X.16 1 -C -/B D /D B -/A A /C -/C -A /A -B -/D -D /B C -1
X.17 1 -/B -/D /C C D -/A A B -B -A /A -D -C -/C /D /B -1
X.18 1 -B -D C /C /D -A /A /B -/B -/A A -/D -/C -C D B -1
X.19 18 . . . . . . . . . . . . . . . . .
2 .
3 .
19 1
19a
2P 19a
3P 19a
5P 19a
7P 19a
11P 19a
13P 19a
17P 19a
19P 1a
X.1 1
X.2 1
X.3 1
X.4 1
X.5 1
X.6 1
X.7 1
X.8 1
X.9 1
X.10 1
X.11 1
X.12 1
X.13 1
X.14 1
X.15 1
X.16 1
X.17 1
X.18 1
X.19 -1
A = -E(3)
= (1-Sqrt(-3))/2 = -b3
B = -E(9)^2-E(9)^5
C = E(9)^7
D = E(9)^5
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