Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $32$ | |
| Group : | $C_5:D_8$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (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)(21,24)(22,23), (1,38,4,39,2,37,3,40)(5,34,7,35,6,33,8,36)(9,32,12,29,10,31,11,30)(13,27,16,26,14,28,15,25)(17,23,20,22,18,24,19,21) | |
| $|\Aut(F/K)|$: | $2$ |
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: $D_{8}$ 20: $D_{10}$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: $D_{5}$
Degree 8: $D_{8}$
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 $ | $20$ | $2$ | $( 3, 4)( 5,40)( 6,39)( 7,37)( 8,38)( 9,33)(10,34)(11,36)(12,35)(13,32)(14,31) (15,29)(16,30)(17,27)(18,28)(19,26)(20,25)(21,24)(22,23)$ |
| $ 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,35,34,36)(37,39,38,40)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 5,11,14,19,21,26,30,36,38)( 2, 6,12,13,20,22,25,29,35,37)( 3, 7,10,15,18, 23,27,32,33,39)( 4, 8, 9,16,17,24,28,31,34,40)$ |
| $ 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 $ | $10$ | $8$ | $( 1, 6, 4, 8, 2, 5, 3, 7)( 9,40,12,38,10,39,11,37)(13,34,16,35,14,33,15,36) (17,31,20,30,18,32,19,29)(21,27,23,26,22,28,24,25)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 7,11,15,19,23,26,32,36,39)( 2, 8,12,16,20,24,25,31,35,40)( 3, 6,10,13,18, 22,27,29,33,37)( 4, 5, 9,14,17,21,28,30,34,38)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 9,20,27,36, 4,12,18,26,34, 2,10,19,28,35, 3,11,17,25,33)( 5,15,22,31,38, 7,13,24,30,39, 6,16,21,32,37, 8,14,23,29,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,19,26,36)( 2,12,20,25,35)( 3,10,18,27,33)( 4, 9,17,28,34) ( 5,14,21,30,38)( 6,13,22,29,37)( 7,15,23,32,39)( 8,16,24,31,40)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,12,19,25,36, 2,11,20,26,35)( 3, 9,18,28,33, 4,10,17,27,34)( 5,13,21,29,38, 6,14,22,30,37)( 7,16,23,31,39, 8,15,24,32,40)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,13,26,37,11,22,36, 6,19,29)( 2,14,25,38,12,21,35, 5,20,30)( 3,16,27,40,10, 24,33, 8,18,31)( 4,15,28,39, 9,23,34, 7,17,32)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,15,26,39,11,23,36, 7,19,32)( 2,16,25,40,12,24,35, 8,20,31)( 3,13,27,37,10, 22,33, 6,18,29)( 4,14,28,38, 9,21,34, 5,17,30)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,17,35,10,26, 4,20,33,11,28, 2,18,36, 9,25, 3,19,34,12,27)( 5,23,37,16,30, 7,22,40,14,32, 6,24,38,15,29, 8,21,39,13,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,36,11,26)( 2,20,35,12,25)( 3,18,33,10,27)( 4,17,34, 9,28) ( 5,21,38,14,30)( 6,22,37,13,29)( 7,23,39,15,32)( 8,24,40,16,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,20,36,12,26, 2,19,35,11,25)( 3,17,33, 9,27, 4,18,34,10,28)( 5,22,38,13,30, 6,21,37,14,29)( 7,24,39,16,32, 8,23,40,15,31)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1,21)( 2,22)( 3,23)( 4,24)( 5,26)( 6,25)( 7,27)( 8,28)( 9,31)(10,32)(11,30) (12,29)(13,35)(14,36)(15,33)(16,34)(17,40)(18,39)(19,38)(20,37)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [80, 15] |
| 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 10a 8a 8b 10b 20a 5a 10c 10d 10e 20b 5b 10f 2c
2P 1a 1a 1a 2b 5a 4a 4a 5a 10f 5b 5b 5b 5b 10c 5a 5a 1a
3P 1a 2a 2b 4a 10d 8b 8a 10e 20b 5b 10f 10b 10a 20a 5a 10c 2c
5P 1a 2a 2b 4a 2c 8b 8a 2c 4a 1a 2b 2c 2c 4a 1a 2b 2c
7P 1a 2a 2b 4a 10e 8a 8b 10d 20b 5b 10f 10a 10b 20a 5a 10c 2c
11P 1a 2a 2b 4a 10a 8b 8a 10b 20a 5a 10c 10d 10e 20b 5b 10f 2c
13P 1a 2a 2b 4a 10d 8b 8a 10e 20b 5b 10f 10b 10a 20a 5a 10c 2c
17P 1a 2a 2b 4a 10e 8a 8b 10d 20b 5b 10f 10a 10b 20a 5a 10c 2c
19P 1a 2a 2b 4a 10b 8b 8a 10a 20a 5a 10c 10e 10d 20b 5b 10f 2c
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) = -r2
E = 2*E(5)^2+2*E(5)^3
= -1-Sqrt(5) = -1-r5
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