Group action invariants
| Degree $n$ : | $38$ | |
| Transitive number $t$ : | $4$ | |
| Group : | $C_2\times C_{19}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,29,36,2,30,35)(3,5,19,4,6,20)(7,33,26,8,34,25)(9,10)(11,23,31,12,24,32)(13,38,15,14,37,16)(17,28,22,18,27,21), (1,24,25,2,23,26)(3,37,10,4,38,9)(5,14,31,6,13,32)(7,28,15,8,27,16)(11,18,22,12,17,21)(19,35,33,20,36,34)(29,30) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 57: $C_{19}:C_{3}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: $C_{19}:C_{3}$
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $19$ | $3$ | $( 3,16,23)( 4,15,24)( 5,29, 7)( 6,30, 8)( 9,20,13)(10,19,14)(11,34,35) (12,33,36)(17,37,26)(18,38,25)(21,28,32)(22,27,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $19$ | $3$ | $( 3,23,16)( 4,24,15)( 5, 7,29)( 6, 8,30)( 9,13,20)(10,14,19)(11,35,34) (12,36,33)(17,26,37)(18,25,38)(21,32,28)(22,31,27)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1, 2)( 3,15,23, 4,16,24)( 5,30, 7, 6,29, 8)( 9,19,13,10,20,14) (11,33,35,12,34,36)(17,38,26,18,37,25)(21,27,32,22,28,31)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1, 2)( 3,24,16, 4,23,15)( 5, 8,29, 6, 7,30)( 9,14,20,10,13,19) (11,36,34,12,35,33)(17,25,37,18,26,38)(21,31,28,22,32,27)$ |
| $ 19, 19 $ | $3$ | $19$ | $( 1, 3, 6, 8,10,12,14,16,18,19,21,23,25,28,30,32,33,36,38)( 2, 4, 5, 7, 9,11, 13,15,17,20,22,24,26,27,29,31,34,35,37)$ |
| $ 38 $ | $3$ | $38$ | $( 1, 4, 6, 7,10,11,14,15,18,20,21,24,25,27,30,31,33,35,38, 2, 3, 5, 8, 9,12, 13,16,17,19,22,23,26,28,29,32,34,36,37)$ |
| $ 38 $ | $3$ | $38$ | $( 1, 5,10,13,18,22,25,29,33,37, 3, 7,12,15,19,24,28,31,36, 2, 6, 9,14,17,21, 26,30,34,38, 4, 8,11,16,20,23,27,32,35)$ |
| $ 19, 19 $ | $3$ | $19$ | $( 1, 6,10,14,18,21,25,30,33,38, 3, 8,12,16,19,23,28,32,36)( 2, 5, 9,13,17,22, 26,29,34,37, 4, 7,11,15,20,24,27,31,35)$ |
| $ 38 $ | $3$ | $38$ | $( 1, 9,18,26,33, 4,12,20,28,35, 6,13,21,29,38, 7,16,24,32, 2,10,17,25,34, 3, 11,19,27,36, 5,14,22,30,37, 8,15,23,31)$ |
| $ 19, 19 $ | $3$ | $19$ | $( 1,10,18,25,33, 3,12,19,28,36, 6,14,21,30,38, 8,16,23,32)( 2, 9,17,26,34, 4, 11,20,27,35, 5,13,22,29,37, 7,15,24,31)$ |
| $ 38 $ | $3$ | $38$ | $( 1,11,21,31, 3,13,23,34, 6,15,25,35, 8,17,28,37,10,20,30, 2,12,22,32, 4,14, 24,33, 5,16,26,36, 7,18,27,38, 9,19,29)$ |
| $ 19, 19 $ | $3$ | $19$ | $( 1,12,21,32, 3,14,23,33, 6,16,25,36, 8,18,28,38,10,19,30)( 2,11,22,31, 4,13, 24,34, 5,15,26,35, 7,17,27,37, 9,20,29)$ |
| $ 38 $ | $3$ | $38$ | $( 1,17,33,11,28, 5,21,37,16,31,10,26, 3,20,36,13,30, 7,23, 2,18,34,12,27, 6, 22,38,15,32, 9,25, 4,19,35,14,29, 8,24)$ |
| $ 19, 19 $ | $3$ | $19$ | $( 1,18,33,12,28, 6,21,38,16,32,10,25, 3,19,36,14,30, 8,23)( 2,17,34,11,27, 5, 22,37,15,31, 9,26, 4,20,35,13,29, 7,24)$ |
| $ 19, 19 $ | $3$ | $19$ | $( 1,21, 3,23, 6,25, 8,28,10,30,12,32,14,33,16,36,18,38,19)( 2,22, 4,24, 5,26, 7,27, 9,29,11,31,13,34,15,35,17,37,20)$ |
| $ 38 $ | $3$ | $38$ | $( 1,22, 3,24, 6,26, 8,27,10,29,12,31,14,34,16,35,18,37,19, 2,21, 4,23, 5,25, 7,28, 9,30,11,32,13,33,15,36,17,38,20)$ |
Group invariants
| Order: | $114=2 \cdot 3 \cdot 19$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [114, 2] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 1 1 1 1 1 . . . . . . . . . . . .
19 1 . . 1 . . 1 1 1 1 1 1 1 1 1 1 1 1
1a 3a 3b 2a 6a 6b 19a 38a 38b 19b 38c 19c 38d 19d 38e 19e 19f 38f
2P 1a 3b 3a 1a 3b 3a 19b 19b 19c 19c 19e 19e 19f 19f 19d 19d 19a 19a
3P 1a 1a 1a 2a 2a 2a 19b 38b 38c 19c 38e 19e 38f 19f 38d 19d 19a 38a
5P 1a 3b 3a 2a 6b 6a 19d 38d 38f 19f 38a 19a 38c 19c 38b 19b 19e 38e
7P 1a 3a 3b 2a 6a 6b 19a 38a 38b 19b 38c 19c 38d 19d 38e 19e 19f 38f
11P 1a 3b 3a 2a 6b 6a 19a 38a 38b 19b 38c 19c 38d 19d 38e 19e 19f 38f
13P 1a 3a 3b 2a 6a 6b 19f 38f 38a 19a 38b 19b 38e 19e 38c 19c 19d 38d
17P 1a 3b 3a 2a 6b 6a 19d 38d 38f 19f 38a 19a 38c 19c 38b 19b 19e 38e
19P 1a 3a 3b 2a 6a 6b 1a 2a 2a 1a 2a 1a 2a 1a 2a 1a 1a 2a
23P 1a 3b 3a 2a 6b 6a 19c 38c 38e 19e 38d 19d 38a 19a 38f 19f 19b 38b
29P 1a 3b 3a 2a 6b 6a 19f 38f 38a 19a 38b 19b 38e 19e 38c 19c 19d 38d
31P 1a 3a 3b 2a 6a 6b 19e 38e 38d 19d 38f 19f 38b 19b 38a 19a 19c 38c
37P 1a 3a 3b 2a 6a 6b 19e 38e 38d 19d 38f 19f 38b 19b 38a 19a 19c 38c
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 -1 -A -/A 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1
X.4 1 /A A -1 -/A -A 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1
X.5 1 A /A 1 A /A 1 1 1 1 1 1 1 1 1 1 1 1
X.6 1 /A A 1 /A A 1 1 1 1 1 1 1 1 1 1 1 1
X.7 3 . . 3 . . B B C C /D /D /C /C /B /B D D
X.8 3 . . 3 . . /B /B /C /C D D C C B B /D /D
X.9 3 . . 3 . . C C /D /D /B /B D D /C /C B B
X.10 3 . . 3 . . D D B B C C /B /B /D /D /C /C
X.11 3 . . 3 . . /C /C D D B B /D /D C C /B /B
X.12 3 . . 3 . . /D /D /B /B /C /C B B D D C C
X.13 3 . . -3 . . B -B -C C -/D /D -/C /C -/B /B D -D
X.14 3 . . -3 . . /B -/B -/C /C -D D -C C -B B /D -/D
X.15 3 . . -3 . . C -C -/D /D -/B /B -D D -/C /C B -B
X.16 3 . . -3 . . D -D -B B -C C -/B /B -/D /D /C -/C
X.17 3 . . -3 . . /C -/C -D D -B B -/D /D -C C /B -/B
X.18 3 . . -3 . . /D -/D -/B /B -/C /C -B B -D D C -C
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(19)^2+E(19)^3+E(19)^14
C = E(19)^4+E(19)^6+E(19)^9
D = E(19)+E(19)^7+E(19)^11
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