Group action invariants
| Degree $n$ : | $41$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $C_{41}:C_{8}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,27,32,3,40,14,9,38)(2,13,23,6,39,28,18,35)(4,26,5,12,37,15,36,29)(7,25,19,21,34,16,22,20)(8,11,10,24,33,30,31,17), (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) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 8: $C_8$ 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 $ | $1$ | $1$ | $()$ |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2, 4,10,28,41,39,33,15)( 3, 7,19,14,40,36,24,29)( 5,13,37,27,38,30, 6,16) ( 8,22,23,26,35,21,20,17)( 9,25,32,12,34,18,11,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $41$ | $4$ | $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20) ( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$ |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2,15,33,39,41,28,10, 4)( 3,29,24,36,40,14,19, 7)( 5,16, 6,30,38,27,37,13) ( 8,17,20,21,35,26,23,22)( 9,31,11,18,34,12,32,25)$ |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2,28,33, 4,41,15,10,39)( 3,14,24, 7,40,29,19,36)( 5,27, 6,13,38,16,37,30) ( 8,26,20,22,35,17,23,21)( 9,12,11,25,34,31,32,18)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $41$ | $4$ | $( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23) ( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$ |
| $ 8, 8, 8, 8, 8, 1 $ | $41$ | $8$ | $( 2,39,10,15,41, 4,33,28)( 3,36,19,29,40, 7,24,14)( 5,30,37,16,38,13, 6,27) ( 8,21,23,17,35,22,20,26)( 9,18,32,31,34,25,11,12)$ |
| $ 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 $ | $8$ | $41$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ |
| $ 41 $ | $8$ | $41$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ |
| $ 41 $ | $8$ | $41$ | $( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15, 19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$ |
| $ 41 $ | $8$ | $41$ | $( 1, 8,15,22,29,36, 2, 9,16,23,30,37, 3,10,17,24,31,38, 4,11,18,25,32,39, 5, 12,19,26,33,40, 6,13,20,27,34,41, 7,14,21,28,35)$ |
| $ 41 $ | $8$ | $41$ | $( 1, 9,17,25,33,41, 8,16,24,32,40, 7,15,23,31,39, 6,14,22,30,38, 5,13,21,29, 37, 4,12,20,28,36, 3,11,19,27,35, 2,10,18,26,34)$ |
Group invariants
| Order: | $328=2^{3} \cdot 41$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [328, 12] |
| Character table: |
2 3 3 3 3 3 3 3 3 . . . . .
41 1 . . . . . . . 1 1 1 1 1
1a 8a 4a 8b 8c 4b 8d 2a 41a 41b 41c 41d 41e
2P 1a 4a 2a 4b 4b 2a 4a 1a 41b 41c 41e 41a 41d
3P 1a 8c 4b 8d 8a 4a 8b 2a 41a 41b 41c 41d 41e
5P 1a 8d 4a 8c 8b 4b 8a 2a 41c 41e 41d 41b 41a
7P 1a 8b 4b 8a 8d 4a 8c 2a 41d 41a 41b 41e 41c
11P 1a 8c 4b 8d 8a 4a 8b 2a 41e 41d 41a 41c 41b
13P 1a 8d 4a 8c 8b 4b 8a 2a 41b 41c 41e 41a 41d
17P 1a 8a 4a 8b 8c 4b 8d 2a 41e 41d 41a 41c 41b
19P 1a 8c 4b 8d 8a 4a 8b 2a 41d 41a 41b 41e 41c
23P 1a 8b 4b 8a 8d 4a 8c 2a 41b 41c 41e 41a 41d
29P 1a 8d 4a 8c 8b 4b 8a 2a 41c 41e 41d 41b 41a
31P 1a 8b 4b 8a 8d 4a 8c 2a 41e 41d 41a 41c 41b
37P 1a 8d 4a 8c 8b 4b 8a 2a 41c 41e 41d 41b 41a
41P 1a 8a 4a 8b 8c 4b 8d 2a 1a 1a 1a 1a 1a
X.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
X.3 1 A -1 -A -A -1 A 1 1 1 1 1 1
X.4 1 -A -1 A A -1 -A 1 1 1 1 1 1
X.5 1 B -A /B -/B A -B -1 1 1 1 1 1
X.6 1 -/B A -B B -A /B -1 1 1 1 1 1
X.7 1 /B A B -B -A -/B -1 1 1 1 1 1
X.8 1 -B -A -/B /B A B -1 1 1 1 1 1
X.9 8 . . . . . . . C F G D E
X.10 8 . . . . . . . D C F E G
X.11 8 . . . . . . . E D C G F
X.12 8 . . . . . . . F G E C D
X.13 8 . . . . . . . G E D F C
A = -E(4)
= -Sqrt(-1) = -i
B = -E(8)
C = E(41)^7+E(41)^16+E(41)^19+E(41)^20+E(41)^21+E(41)^22+E(41)^25+E(41)^34
D = E(41)^8+E(41)^10+E(41)^11+E(41)^17+E(41)^24+E(41)^30+E(41)^31+E(41)^33
E = E(41)^4+E(41)^5+E(41)^12+E(41)^15+E(41)^26+E(41)^29+E(41)^36+E(41)^37
F = E(41)+E(41)^3+E(41)^9+E(41)^14+E(41)^27+E(41)^32+E(41)^38+E(41)^40
G = E(41)^2+E(41)^6+E(41)^13+E(41)^18+E(41)^23+E(41)^28+E(41)^35+E(41)^39
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