Group action invariants
| Degree $n$: | $41$ | |
| Transitive number $t$: | $3$ | |
| Group: | $C_{41}:C_{4}$ | |
| Parity: | $1$ | |
| Primitive: | yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,32,40,9)(2,23,39,18)(3,14,38,27)(4,5,37,36)(6,28,35,13)(7,19,34,22)(8,10,33,31)(11,24,30,17)(12,15,29,26)(16,20,25,21), (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) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ 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 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $41$ | $4$ | $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20) ( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $41$ | $4$ | $( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23) ( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$ |
| $ 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 $ | $4$ | $41$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ |
| $ 41 $ | $4$ | $41$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ |
| $ 41 $ | $4$ | $41$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$ |
| $ 41 $ | $4$ | $41$ | $( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15, 19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$ |
| $ 41 $ | $4$ | $41$ | $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$ |
| $ 41 $ | $4$ | $41$ | $( 1, 8,15,22,29,36, 2, 9,16,23,30,37, 3,10,17,24,31,38, 4,11,18,25,32,39, 5, 12,19,26,33,40, 6,13,20,27,34,41, 7,14,21,28,35)$ |
| $ 41 $ | $4$ | $41$ | $( 1, 9,17,25,33,41, 8,16,24,32,40, 7,15,23,31,39, 6,14,22,30,38, 5,13,21,29, 37, 4,12,20,28,36, 3,11,19,27,35, 2,10,18,26,34)$ |
| $ 41 $ | $4$ | $41$ | $( 1,12,23,34, 4,15,26,37, 7,18,29,40,10,21,32, 2,13,24,35, 5,16,27,38, 8,19, 30,41,11,22,33, 3,14,25,36, 6,17,28,39, 9,20,31)$ |
| $ 41 $ | $4$ | $41$ | $( 1,13,25,37, 8,20,32, 3,15,27,39,10,22,34, 5,17,29,41,12,24,36, 7,19,31, 2, 14,26,38, 9,21,33, 4,16,28,40,11,23,35, 6,18,30)$ |
| $ 41 $ | $4$ | $41$ | $( 1,17,33, 8,24,40,15,31, 6,22,38,13,29, 4,20,36,11,27, 2,18,34, 9,25,41,16, 32, 7,23,39,14,30, 5,21,37,12,28, 3,19,35,10,26)$ |
Group invariants
| Order: | $164=2^{2} \cdot 41$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [164, 3] |
| Character table: |
2 2 2 2 2 . . . . . . . . . .
41 1 . . . 1 1 1 1 1 1 1 1 1 1
1a 4a 4b 2a 41a 41b 41c 41d 41e 41f 41g 41h 41i 41j
2P 1a 2a 2a 1a 41b 41d 41e 41g 41i 41c 41j 41f 41h 41a
3P 1a 4b 4a 2a 41c 41e 41a 41i 41b 41j 41h 41g 41d 41f
5P 1a 4a 4b 2a 41d 41g 41i 41j 41h 41e 41a 41c 41f 41b
7P 1a 4b 4a 2a 41f 41c 41j 41e 41a 41g 41i 41d 41b 41h
11P 1a 4b 4a 2a 41h 41f 41g 41c 41j 41d 41e 41b 41a 41i
13P 1a 4a 4b 2a 41e 41i 41b 41h 41d 41a 41f 41j 41g 41c
17P 1a 4a 4b 2a 41h 41f 41g 41c 41j 41d 41e 41b 41a 41i
19P 1a 4b 4a 2a 41f 41c 41j 41e 41a 41g 41i 41d 41b 41h
23P 1a 4b 4a 2a 41b 41d 41e 41g 41i 41c 41j 41f 41h 41a
29P 1a 4a 4b 2a 41i 41h 41d 41f 41g 41b 41c 41a 41j 41e
31P 1a 4b 4a 2a 41g 41j 41h 41a 41f 41i 41b 41e 41c 41d
37P 1a 4a 4b 2a 41d 41g 41i 41j 41h 41e 41a 41c 41f 41b
41P 1a 4a 4b 2a 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 1
X.2 1 -1 -1 1 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 1
X.4 1 -A A -1 1 1 1 1 1 1 1 1 1 1
X.5 4 . . . B E G D C J H F K I
X.6 4 . . . C K E F D B J I H G
X.7 4 . . . D H K I F C B G J E
X.8 4 . . . E D C H K G I J F B
X.9 4 . . . F J H G I D C E B K
X.10 4 . . . G C B K E I F H D J
X.11 4 . . . H I F B J K E C G D
X.12 4 . . . I B J E G F D K C H
X.13 4 . . . J G I C B H K D E F
X.14 4 . . . K F D J H E G B I C
A = -E(4)
= -Sqrt(-1) = -i
B = E(41)^2+E(41)^18+E(41)^23+E(41)^39
C = E(41)^12+E(41)^15+E(41)^26+E(41)^29
D = E(41)^8+E(41)^10+E(41)^31+E(41)^33
E = E(41)^4+E(41)^5+E(41)^36+E(41)^37
F = E(41)^7+E(41)^19+E(41)^22+E(41)^34
G = E(41)^6+E(41)^13+E(41)^28+E(41)^35
H = E(41)^16+E(41)^20+E(41)^21+E(41)^25
I = E(41)+E(41)^9+E(41)^32+E(41)^40
J = E(41)^3+E(41)^14+E(41)^27+E(41)^38
K = E(41)^11+E(41)^17+E(41)^24+E(41)^30
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