Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $31$ | |
| Group : | $Q_8:D_5$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,14,25,37,12,22,36,5,19,29,2,13,26,38,11,21,35,6,20,30)(3,15,28,40,9,24,33,7,18,31,4,16,27,39,10,23,34,8,17,32), (1,32,3,30,2,31,4,29)(5,25,8,28,6,26,7,27)(9,21,11,24,10,22,12,23)(13,20,15,17,14,19,16,18)(33,38,35,40,34,37,36,39) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 8: $D_{4}$ 10: $D_{5}$ 16: $QD_{16}$ 20: $D_{10}$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: $D_{5}$
Degree 8: $QD_{16}$
Degree 10: $D_{10}$
Degree 20: 20T7
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, 1, 1 $ | $20$ | $2$ | $( 3, 4)( 5,40)( 6,39)( 7,37)( 8,38)( 9,33)(10,34)(11,36)(12,35)(13,32)(14,31) (15,29)(16,30)(17,27)(18,28)(19,26)(20,25)(21,23)(22,24)$ |
| $ 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,35,34,36)(37,39,38,40)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 5,11,14,19,21,25,29,35,37, 2, 6,12,13,20,22,26,30,36,38)( 3, 7,10,15,18, 23,28,31,34,40, 4, 8, 9,16,17,24,27,32,33,39)$ |
| $ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,39,12,37,10,40,11,38)(13,33,16,36,14,34,15,35) (17,32,20,29,18,31,19,30)(21,28,23,25,22,27,24,26)$ |
| $ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 6, 4, 8, 2, 5, 3, 7)( 9,40,12,38,10,39,11,37)(13,34,16,35,14,33,15,36) (17,31,20,30,18,32,19,29)(21,27,23,26,22,28,24,25)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 7,11,15,19,23,25,31,35,40, 2, 8,12,16,20,24,26,32,36,39)( 3, 6,10,13,18, 22,28,30,34,38, 4, 5, 9,14,17,21,27,29,33,37)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 9,20,28,35, 3,11,17,26,34, 2,10,19,27,36, 4,12,18,25,33)( 5,15,22,32,37, 8,14,23,30,39, 6,16,21,31,38, 7,13,24,29,40)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,11,19,25,35, 2,12,20,26,36)( 3,10,18,28,34, 4, 9,17,27,33)( 5,14,21,29,37, 6,13,22,30,38)( 7,15,23,31,40, 8,16,24,32,39)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,19,26,35)( 2,11,20,25,36)( 3, 9,18,27,34)( 4,10,17,28,33) ( 5,13,21,30,37)( 6,14,22,29,38)( 7,16,23,32,40)( 8,15,24,31,39)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,13,25,38,12,21,36, 6,19,30, 2,14,26,37,11,22,35, 5,20,29)( 3,16,28,39, 9, 23,33, 8,18,32, 4,15,27,40,10,24,34, 7,17,31)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,15,25,40,12,24,36, 7,19,31, 2,16,26,39,11,23,35, 8,20,32)( 3,13,28,38, 9, 21,33, 6,18,30, 4,14,27,37,10,22,34, 5,17,29)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,17,36, 9,26, 4,20,34,12,28, 2,18,35,10,25, 3,19,33,11,27)( 5,23,38,15,30, 7,22,39,13,32, 6,24,37,16,29, 8,21,40,14,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,35,12,26)( 2,20,36,11,25)( 3,18,34, 9,27)( 4,17,33,10,28) ( 5,21,37,13,30)( 6,22,38,14,29)( 7,23,40,16,32)( 8,24,39,15,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,20,35,11,26, 2,19,36,12,25)( 3,17,34,10,27, 4,18,33, 9,28)( 5,22,37,14,30, 6,21,38,13,29)( 7,24,40,15,32, 8,23,39,16,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,21, 2,22)( 3,23, 4,24)( 5,25, 6,26)( 7,28, 8,27)( 9,32,10,31)(11,29,12,30) (13,36,14,35)(15,34,16,33)(17,39,18,40)(19,37,20,38)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [80, 17] |
| Character table: |
2 4 2 4 3 2 3 3 2 2 3 3 2 2 2 3 3 2
5 1 . 1 1 1 . . 1 1 1 1 1 1 1 1 1 1
1a 2a 2b 4a 20a 8a 8b 20b 20c 10a 5a 20d 20e 20f 5b 10b 4b
2P 1a 1a 1a 2b 10a 4a 4a 10a 10b 5b 5b 10b 10b 10a 5a 5a 2b
3P 1a 2a 2b 4a 20d 8a 8b 20e 20f 10b 5b 20b 20a 20c 5a 10a 4b
5P 1a 2a 2b 4a 4b 8b 8a 4b 4a 2b 1a 4b 4b 4a 1a 2b 4b
7P 1a 2a 2b 4a 20e 8b 8a 20d 20f 10b 5b 20a 20b 20c 5a 10a 4b
11P 1a 2a 2b 4a 20a 8a 8b 20b 20c 10a 5a 20d 20e 20f 5b 10b 4b
13P 1a 2a 2b 4a 20d 8b 8a 20e 20f 10b 5b 20b 20a 20c 5a 10a 4b
17P 1a 2a 2b 4a 20e 8a 8b 20d 20f 10b 5b 20a 20b 20c 5a 10a 4b
19P 1a 2a 2b 4a 20b 8a 8b 20a 20c 10a 5a 20e 20d 20f 5b 10b 4b
X.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
X.3 1 -1 1 1 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 1 1 1 -1
X.5 2 . 2 -2 . . . . -2 2 2 . . -2 2 2 .
X.6 2 . -2 . . D -D . . -2 2 . . . 2 -2 .
X.7 2 . -2 . . -D D . . -2 2 . . . 2 -2 .
X.8 2 . 2 -2 A . . -A *C -*C -*C -B B C -C -C .
X.9 2 . 2 -2 B . . -B C -C -C A -A *C -*C -*C .
X.10 2 . 2 -2 -B . . B C -C -C -A A *C -*C -*C .
X.11 2 . 2 -2 -A . . A *C -*C -*C B -B C -C -C .
X.12 2 . 2 2 C . . C -*C -*C -*C *C *C -C -C -C -2
X.13 2 . 2 2 *C . . *C -C -C -C C C -*C -*C -*C -2
X.14 2 . 2 2 -*C . . -*C -C -C -C -C -C -*C -*C -*C 2
X.15 2 . 2 2 -C . . -C -*C -*C -*C -*C -*C -C -C -C 2
X.16 4 . -4 . . . . . . E -E . . . -*E *E .
X.17 4 . -4 . . . . . . *E -*E . . . -E E .
A = -E(5)+E(5)^4
B = -E(5)^2+E(5)^3
C = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
D = -E(8)-E(8)^3
= -Sqrt(-2) = -i2
E = -2*E(5)^2-2*E(5)^3
= 1+Sqrt(5) = 1+r5
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