Group action invariants
| Degree $n$ : | $41$ | |
| Transitive number $t$ : | $6$ | |
| Group : | $C_{41}:C_{10}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (1,25,10,4,18,40,16,31,37,23)(2,9,20,8,36,39,32,21,33,5)(3,34,30,12,13,38,7,11,29,28)(6,27,19,24,26,35,14,22,17,15) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 5: $C_5$ 10: $C_{10}$ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2, 5,17,24,11,41,38,26,19,32)( 3, 9,33, 6,21,40,34,10,37,22)( 4,13, 8,29,31, 39,30,35,14,12)( 7,25,15,16,20,36,18,28,27,23)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$ |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,24,38,32,17,41,19, 5,11,26)( 3, 6,34,22,33,40,37, 9,21,10)( 4,29,30,12, 8, 39,14,13,31,35)( 7,16,18,23,15,36,27,25,20,28)$ |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,26,11, 5,19,41,17,32,38,24)( 3,10,21, 9,37,40,33,22,34, 6)( 4,35,31,13,14, 39, 8,12,30,29)( 7,28,20,25,27,36,15,23,18,16)$ |
| $ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,32,19,26,38,41,11,24,17, 5)( 3,22,37,10,34,40,21, 6,33, 9)( 4,12,14,35,30, 39,31,29, 8,13)( 7,23,27,28,18,36,20,16,15,25)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $41$ | $2$ | $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31) (13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$ |
| $ 41 $ | $10$ | $41$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ |
| $ 41 $ | $10$ | $41$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ |
| $ 41 $ | $10$ | $41$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$ |
| $ 41 $ | $10$ | $41$ | $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$ |
Group invariants
| Order: | $410=2 \cdot 5 \cdot 41$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [410, 1] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1 . . . .
5 1 1 1 1 1 1 1 1 1 1 . . . .
41 1 . . . . . . . . . 1 1 1 1
1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 41a 41b 41c 41d
2P 1a 5b 5c 5a 5d 5d 5a 5c 5b 1a 41b 41a 41d 41c
3P 1a 10b 5b 5d 5a 10d 10a 10c 5c 2a 41c 41d 41b 41a
5P 1a 2a 1a 1a 1a 2a 2a 2a 1a 2a 41b 41a 41d 41c
7P 1a 10c 5c 5a 5d 10a 10d 10b 5b 2a 41c 41d 41b 41a
11P 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 41c 41d 41b 41a
13P 1a 10b 5b 5d 5a 10d 10a 10c 5c 2a 41c 41d 41b 41a
17P 1a 10c 5c 5a 5d 10a 10d 10b 5b 2a 41d 41c 41a 41b
19P 1a 10d 5d 5c 5b 10c 10b 10a 5a 2a 41d 41c 41a 41b
23P 1a 10b 5b 5d 5a 10d 10a 10c 5c 2a 41a 41b 41c 41d
29P 1a 10d 5d 5c 5b 10c 10b 10a 5a 2a 41c 41d 41b 41a
31P 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 41a 41b 41c 41d
37P 1a 10c 5c 5a 5d 10a 10d 10b 5b 2a 41a 41b 41c 41d
41P 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 1a 1a 1a 1a
X.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
X.3 1 A -/A -B -/B /B B /A -A -1 1 1 1 1
X.4 1 B -/B -/A -A A /A /B -B -1 1 1 1 1
X.5 1 /B -B -A -/A /A A B -/B -1 1 1 1 1
X.6 1 /A -A -/B -B B /B A -/A -1 1 1 1 1
X.7 1 -/A -A -/B -B -B -/B -A -/A 1 1 1 1 1
X.8 1 -/B -B -A -/A -/A -A -B -/B 1 1 1 1 1
X.9 1 -B -/B -/A -A -A -/A -/B -B 1 1 1 1 1
X.10 1 -A -/A -B -/B -/B -B -/A -A 1 1 1 1 1
X.11 10 . . . . . . . . . C D F E
X.12 10 . . . . . . . . . D C E F
X.13 10 . . . . . . . . . E F C D
X.14 10 . . . . . . . . . F E D C
A = -E(5)
B = -E(5)^2
C = E(41)^3+E(41)^7+E(41)^11+E(41)^12+E(41)^13+E(41)^28+E(41)^29+E(41)^30+E(41)^34+E(41)^38
D = E(41)^6+E(41)^14+E(41)^15+E(41)^17+E(41)^19+E(41)^22+E(41)^24+E(41)^26+E(41)^27+E(41)^35
E = E(41)+E(41)^4+E(41)^10+E(41)^16+E(41)^18+E(41)^23+E(41)^25+E(41)^31+E(41)^37+E(41)^40
F = E(41)^2+E(41)^5+E(41)^8+E(41)^9+E(41)^20+E(41)^21+E(41)^32+E(41)^33+E(41)^36+E(41)^39
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