Group action invariants
| Degree $n$ : | $31$ | |
| Transitive number $t$ : | $3$ | |
| Group : | $C_{31}:C_{3}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31), (1,25,5)(2,19,10)(3,13,15)(4,7,20)(6,26,30)(8,14,9)(11,27,24)(12,21,29)(16,28,18)(17,22,23) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $31$ | $3$ | $( 2, 6,26)( 3,11,20)( 4,16,14)( 5,21, 8)( 7,31,27)( 9,10,15)(12,25,28) (13,30,22)(17,19,29)(18,24,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $31$ | $3$ | $( 2,26, 6)( 3,20,11)( 4,14,16)( 5, 8,21)( 7,27,31)( 9,15,10)(12,28,25) (13,22,30)(17,29,19)(18,23,24)$ |
| $ 31 $ | $3$ | $31$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31)$ |
| $ 31 $ | $3$ | $31$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31, 2, 4, 6, 8,10,12,14,16,18, 20,22,24,26,28,30)$ |
| $ 31 $ | $3$ | $31$ | $( 1, 4, 7,10,13,16,19,22,25,28,31, 3, 6, 9,12,15,18,21,24,27,30, 2, 5, 8,11, 14,17,20,23,26,29)$ |
| $ 31 $ | $3$ | $31$ | $( 1, 5, 9,13,17,21,25,29, 2, 6,10,14,18,22,26,30, 3, 7,11,15,19,23,27,31, 4, 8,12,16,20,24,28)$ |
| $ 31 $ | $3$ | $31$ | $( 1, 7,13,19,25,31, 6,12,18,24,30, 5,11,17,23,29, 4,10,16,22,28, 3, 9,15,21, 27, 2, 8,14,20,26)$ |
| $ 31 $ | $3$ | $31$ | $( 1, 9,17,25, 2,10,18,26, 3,11,19,27, 4,12,20,28, 5,13,21,29, 6,14,22,30, 7, 15,23,31, 8,16,24)$ |
| $ 31 $ | $3$ | $31$ | $( 1,12,23, 3,14,25, 5,16,27, 7,18,29, 9,20,31,11,22, 2,13,24, 4,15,26, 6,17, 28, 8,19,30,10,21)$ |
| $ 31 $ | $3$ | $31$ | $( 1,13,25, 6,18,30,11,23, 4,16,28, 9,21, 2,14,26, 7,19,31,12,24, 5,17,29,10, 22, 3,15,27, 8,20)$ |
| $ 31 $ | $3$ | $31$ | $( 1,17, 2,18, 3,19, 4,20, 5,21, 6,22, 7,23, 8,24, 9,25,10,26,11,27,12,28,13, 29,14,30,15,31,16)$ |
| $ 31 $ | $3$ | $31$ | $( 1,18, 4,21, 7,24,10,27,13,30,16, 2,19, 5,22, 8,25,11,28,14,31,17, 3,20, 6, 23, 9,26,12,29,15)$ |
Group invariants
| Order: | $93=3 \cdot 31$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [93, 1] |
| Character table: |
3 1 1 1 . . . . . . . . . .
31 1 . . 1 1 1 1 1 1 1 1 1 1
1a 3a 3b 31a 31b 31c 31d 31e 31f 31g 31h 31i 31j
2P 1a 3b 3a 31b 31d 31e 31f 31h 31i 31j 31g 31a 31c
3P 1a 1a 1a 31c 31e 31f 31h 31i 31g 31b 31a 31j 31d
5P 1a 3b 3a 31a 31b 31c 31d 31e 31f 31g 31h 31i 31j
7P 1a 3a 3b 31d 31f 31h 31i 31g 31a 31c 31j 31b 31e
11P 1a 3b 3a 31g 31j 31b 31c 31d 31e 31i 31f 31h 31a
13P 1a 3a 3b 31c 31e 31f 31h 31i 31g 31b 31a 31j 31d
17P 1a 3b 3a 31j 31c 31d 31e 31f 31h 31a 31i 31g 31b
19P 1a 3a 3b 31b 31d 31e 31f 31h 31i 31j 31g 31a 31c
23P 1a 3b 3a 31j 31c 31d 31e 31f 31h 31a 31i 31g 31b
29P 1a 3b 3a 31h 31g 31a 31j 31b 31c 31f 31d 31e 31i
31P 1a 3a 3b 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 A /A 1 1 1 1 1 1 1 1 1 1
X.3 1 /A A 1 1 1 1 1 1 1 1 1 1
X.4 3 . . B F E /C /B /D C /F /E D
X.5 3 . . C D F E /C /B /E /D /F B
X.6 3 . . D E /C /B /D /F B /E C F
X.7 3 . . E /B /D /F /E C F B D /C
X.8 3 . . /C /D /F /E C B E D F /B
X.9 3 . . F /C /B /D /F /E D C B E
X.10 3 . . /E B D F E /C /F /B /D C
X.11 3 . . /D /E C B D F /B E /C /F
X.12 3 . . /B /F /E C B D /C F E /D
X.13 3 . . /F C B D F E /D /C /B /E
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(31)^2+E(31)^10+E(31)^19
C = E(31)^17+E(31)^22+E(31)^23
D = E(31)^3+E(31)^13+E(31)^15
E = E(31)^6+E(31)^26+E(31)^30
F = E(31)^4+E(31)^7+E(31)^20
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