Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $8$ | |
| Group : | $C_5:Q_8$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,38,35,31,28,24,19,16,12,7,2,37,36,32,27,23,20,15,11,8)(3,40,33,30,26,21,18,14,10,5,4,39,34,29,25,22,17,13,9,6), (1,4,2,3)(5,37,6,38)(7,40,8,39)(9,35,10,36)(11,34,12,33)(13,31,14,32)(15,29,16,30)(17,28,18,27)(19,26,20,25)(21,23,22,24) | |
| $|\Aut(F/K)|$: | $40$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 8: $Q_8$ 10: $D_{5}$ 20: $D_{10}$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: $D_{5}$
Degree 8: $Q_8$
Degree 10: $D_5$, $D_{10}$ x 2
Degree 20: 20T4
Low degree siblings
There are no siblings with degree $\leq 10$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 2, 4)( 5,38, 6,37)( 7,39, 8,40)( 9,36,10,35)(11,33,12,34)(13,32,14,31) (15,30,16,29)(17,27,18,28)(19,25,20,26)(21,24,22,23)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,38,10,37)(11,39,12,40)(13,35,14,36)(15,34,16,33) (17,31,18,32)(19,30,20,29)(21,27,22,28)(23,25,24,26)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1, 7,11,16,20,24,27,31,36,38, 2, 8,12,15,19,23,28,32,35,37)( 3, 5, 9,14,17, 21,25,30,34,40, 4, 6,10,13,18,22,26,29,33,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1, 8,11,15,20,23,27,32,36,37, 2, 7,12,16,19,24,28,31,35,38)( 3, 6, 9,13,17, 22,25,29,34,39, 4, 5,10,14,18,21,26,30,33,40)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,11,20,27,36, 2,12,19,28,35)( 3, 9,17,25,34, 4,10,18,26,33)( 5,14,21,30,40, 6,13,22,29,39)( 7,16,24,31,38, 8,15,23,32,37)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,20,28,36)( 2,11,19,27,35)( 3,10,17,26,34)( 4, 9,18,25,33) ( 5,13,21,29,40)( 6,14,22,30,39)( 7,15,24,32,38)( 8,16,23,31,37)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,15,27,37,12,24,35, 8,20,32, 2,16,28,38,11,23,36, 7,19,31)( 3,13,25,39,10, 21,33, 6,17,29, 4,14,26,40, 9,22,34, 5,18,30)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,16,27,38,12,23,35, 7,20,31, 2,15,28,37,11,24,36, 8,19,32)( 3,14,25,40,10, 22,33, 5,17,30, 4,13,26,39, 9,21,34, 6,18,29)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,19,36,11,28, 2,20,35,12,27)( 3,18,34, 9,26, 4,17,33,10,25)( 5,22,40,14,29, 6,21,39,13,30)( 7,23,38,16,32, 8,24,37,15,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,20,36,12,28)( 2,19,35,11,27)( 3,17,34,10,26)( 4,18,33, 9,25) ( 5,21,40,13,29)( 6,22,39,14,30)( 7,24,38,15,32)( 8,23,37,16,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,23, 2,24)( 3,22, 4,21)( 5,26, 6,25)( 7,28, 8,27)( 9,29,10,30)(11,32,12,31) (13,34,14,33)(15,36,16,35)(17,39,18,40)(19,38,20,37)$ |
Group invariants
| Order: | $40=2^{3} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [40, 4] |
| Character table: |
2 3 3 2 2 2 2 2 2 2 2 2 2 2
5 1 1 . . 1 1 1 1 1 1 1 1 1
1a 2a 4a 4b 20a 20b 10a 5a 20c 20d 10b 5b 4c
2P 1a 1a 2a 2a 10a 10a 5b 5b 10b 10b 5a 5a 2a
3P 1a 2a 4a 4b 20d 20c 10b 5b 20a 20b 10a 5a 4c
5P 1a 2a 4a 4b 4c 4c 2a 1a 4c 4c 2a 1a 4c
7P 1a 2a 4a 4b 20c 20d 10b 5b 20b 20a 10a 5a 4c
11P 1a 2a 4a 4b 20b 20a 10a 5a 20d 20c 10b 5b 4c
13P 1a 2a 4a 4b 20c 20d 10b 5b 20b 20a 10a 5a 4c
17P 1a 2a 4a 4b 20d 20c 10b 5b 20a 20b 10a 5a 4c
19P 1a 2a 4a 4b 20a 20b 10a 5a 20c 20d 10b 5b 4c
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 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1
X.4 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1
X.5 2 -2 . . . . -2 2 . . -2 2 .
X.6 2 -2 . . A -A *C -*C B -B C -C .
X.7 2 -2 . . B -B C -C -A A *C -*C .
X.8 2 -2 . . -B B C -C A -A *C -*C .
X.9 2 -2 . . -A A *C -*C -B B C -C .
X.10 2 2 . . C C -*C -*C *C *C -C -C -2
X.11 2 2 . . *C *C -C -C C C -*C -*C -2
X.12 2 2 . . -*C -*C -C -C -C -C -*C -*C 2
X.13 2 2 . . -C -C -*C -*C -*C -*C -C -C 2
A = -E(20)+E(20)^9
B = -E(20)^13+E(20)^17
C = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
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