Group action invariants
| Degree $n$ : | $33$ | |
| Transitive number $t$ : | $9$ | |
| Group : | $D_{33}:C_5$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,27,21,6,18,15,24,29,10,33)(2,26,19,5,16,14,22,28,11,32)(3,25,20,4,17,13,23,30,12,31)(8,9), (1,32)(2,31)(3,33)(4,28)(5,30)(6,29)(7,25)(8,27)(9,26)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(16,18) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 5: $C_5$ 6: $S_3$ 10: $C_{10}$ 30: $S_3 \times C_5$ 110: $F_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: $F_{11}$
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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,10,30,18,13)( 5,11,28,16,14)( 6,12,29,17,15)( 7,19,22,32,26) ( 8,20,23,33,27)( 9,21,24,31,25)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,13,18,30,10)( 5,14,16,28,11)( 6,15,17,29,12)( 7,26,32,22,19) ( 8,27,33,23,20)( 9,25,31,24,21)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,18,10,13,30)( 5,16,11,14,28)( 6,17,12,15,29)( 7,32,19,26,22) ( 8,33,20,27,23)( 9,31,21,25,24)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,30,13,10,18)( 5,28,14,11,16)( 6,29,15,12,17)( 7,22,26,19,32) ( 8,23,27,20,33)( 9,24,25,21,31)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4, 9,13,25,18,31,30,24,10,21)( 5, 8,14,27,16,33,28,23,11,20) ( 6, 7,15,26,17,32,29,22,12,19)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,21,10,24,30,31,18,25,13, 9)( 5,20,11,23,28,33,16,27,14, 8) ( 6,19,12,22,29,32,17,26,15, 7)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,24,18, 9,10,31,13,21,30,25)( 5,23,16, 8,11,33,14,20,28,27) ( 6,22,17, 7,12,32,15,19,29,26)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,25,30,21,13,31,10, 9,18,24)( 5,27,28,20,14,33,11, 8,16,23) ( 6,26,29,19,15,32,12, 7,17,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $33$ | $2$ | $( 2, 3)( 4,31)( 5,33)( 6,32)( 7,29)( 8,28)( 9,30)(10,25)(11,27)(12,26)(13,24) (14,23)(15,22)(16,20)(17,19)(18,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,11,29,18,14, 6,10,28,17,13, 5,12,30,16,15)( 7,20,24,32,27, 9,19, 23,31,26, 8,21,22,33,25)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,14,17,30,11, 6,13,16,29,10, 5,15,18,28,12)( 7,27,31,22,20, 9,26, 33,24,19, 8,25,32,23,21)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,16,12,13,28, 6,18,11,15,30, 5,17,10,14,29)( 7,33,21,26,23, 9,32, 20,25,22, 8,31,19,27,24)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,28,15,10,16, 6,30,14,12,18, 5,29,13,11,17)( 7,23,25,19,33, 9,22, 27,21,32, 8,24,26,20,31)$ |
| $ 11, 11, 11 $ | $10$ | $11$ | $( 1, 4, 9,10,13,18,21,24,25,30,31)( 2, 5, 7,11,14,16,19,22,26,28,32) ( 3, 6, 8,12,15,17,20,23,27,29,33)$ |
| $ 33 $ | $10$ | $33$ | $( 1, 5, 8,10,14,17,21,22,27,30,32, 3, 4, 7,12,13,16,20,24,26,29,31, 2, 6, 9, 11,15,18,19,23,25,28,33)$ |
| $ 33 $ | $10$ | $33$ | $( 1, 6, 7,10,15,16,21,23,26,30,33, 2, 4, 8,11,13,17,19,24,27,28,31, 3, 5, 9, 12,14,18,20,22,25,29,32)$ |
Group invariants
| Order: | $330=2 \cdot 3 \cdot 5 \cdot 11$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [330, 3] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1 . . . . . . . .
3 1 1 1 1 1 . . . . . 1 1 1 1 1 1 1 1
5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . .
11 1 . . . . . . . . . 1 . . . . 1 1 1
1a 5a 5b 5c 5d 10a 10b 10c 10d 2a 3a 15a 15b 15c 15d 11a 33a 33b
2P 1a 5d 5c 5a 5b 5b 5a 5c 5d 1a 3a 15d 15c 15a 15b 11a 33a 33b
3P 1a 5c 5d 5b 5a 10d 10c 10a 10b 2a 1a 5c 5d 5b 5a 11a 11a 11a
5P 1a 1a 1a 1a 1a 2a 2a 2a 2a 2a 3a 3a 3a 3a 3a 11a 33b 33a
7P 1a 5d 5c 5a 5b 10c 10d 10b 10a 2a 3a 15d 15c 15a 15b 11a 33b 33a
11P 1a 5a 5b 5c 5d 10a 10b 10c 10d 2a 3a 15a 15b 15c 15d 1a 3a 3a
13P 1a 5c 5d 5b 5a 10d 10c 10a 10b 2a 3a 15c 15d 15b 15a 11a 33b 33a
17P 1a 5d 5c 5a 5b 10c 10d 10b 10a 2a 3a 15d 15c 15a 15b 11a 33a 33b
19P 1a 5b 5a 5d 5c 10b 10a 10d 10c 2a 3a 15b 15a 15d 15c 11a 33b 33a
23P 1a 5c 5d 5b 5a 10d 10c 10a 10b 2a 3a 15c 15d 15b 15a 11a 33b 33a
29P 1a 5b 5a 5d 5c 10b 10a 10d 10c 2a 3a 15b 15a 15d 15c 11a 33a 33b
31P 1a 5a 5b 5c 5d 10a 10b 10c 10d 2a 3a 15a 15b 15c 15d 11a 33a 33b
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 /B B -B -/B -/A -A -1 1 A /A /B B 1 1 1
X.4 1 B /B A /A -/A -A -/B -B -1 1 B /B A /A 1 1 1
X.5 1 /B B /A A -A -/A -B -/B -1 1 /B B /A A 1 1 1
X.6 1 /A A B /B -/B -B -A -/A -1 1 /A A B /B 1 1 1
X.7 1 A /A /B B B /B /A A 1 1 A /A /B B 1 1 1
X.8 1 B /B A /A /A A /B B 1 1 B /B A /A 1 1 1
X.9 1 /B B /A A A /A B /B 1 1 /B B /A A 1 1 1
X.10 1 /A A B /B /B B A /A 1 1 /A A B /B 1 1 1
X.11 2 2 2 2 2 . . . . . -1 -1 -1 -1 -1 2 -1 -1
X.12 2 C /C /D D . . . . . -1 -B -/B -A -/A 2 -1 -1
X.13 2 /C C D /D . . . . . -1 -/B -B -/A -A 2 -1 -1
X.14 2 D /D C /C . . . . . -1 -/A -A -B -/B 2 -1 -1
X.15 2 /D D /C C . . . . . -1 -A -/A -/B -B 2 -1 -1
X.16 10 . . . . . . . . . 10 . . . . -1 -1 -1
X.17 10 . . . . . . . . . -5 . . . . -1 E *E
X.18 10 . . . . . . . . . -5 . . . . -1 *E E
A = E(5)^4
B = E(5)^3
C = 2*E(5)^3
D = 2*E(5)
E = E(33)^5+E(33)^7+E(33)^10+E(33)^13+E(33)^14+E(33)^19+E(33)^20+E(33)^23+E(33)^26+E(33)^28
= (1-Sqrt(33))/2 = -b33
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