Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $22$ | |
| Group : | $D_4\times D_5$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,32,19,7,36,23,11,37,27,16,2,31,20,8,35,24,12,38,28,15)(3,30,18,5,34,22,9,40,26,13,4,29,17,6,33,21,10,39,25,14), (1,20,36,12,27)(2,19,35,11,28)(3,17,34,10,26)(4,18,33,9,25)(5,22,40,13,29,6,21,39,14,30)(7,23,37,16,31,8,24,38,15,32), (1,39,2,40)(3,38,4,37)(5,36,6,35)(7,34,8,33)(9,31,10,32)(11,29,12,30)(13,28,14,27)(15,26,16,25)(17,23,18,24)(19,21,20,22) | |
| $|\Aut(F/K)|$: | $20$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 8: $D_{4}$ x 2, $C_2^3$ 10: $D_{5}$ 16: $D_4\times C_2$ 20: $D_{10}$ x 3 Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 5: $D_{5}$
Degree 8: $D_4\times C_2$
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)(37,38)(39,40)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 3)( 2, 4)( 5,37)( 6,38)( 7,40)( 8,39)( 9,35)(10,36)(11,33)(12,34)(13,32) (14,31)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,35)(10,36)(11,33)(12,34)(13,31) (14,32)(15,30)(16,29)(17,27)(18,28)(19,25)(20,26)(21,23)(22,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 4)( 2, 3)( 5,38)( 6,37)( 7,39)( 8,40)( 9,36)(10,35)(11,34)(12,33)(13,31) (14,32)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,23)(22,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,38)(10,37)(11,39)(12,40)(13,35)(14,36)(15,34) (16,33)(17,31)(18,32)(19,30)(20,29)(21,27)(22,28)(23,25)(24,26)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,38,10,37)(11,39,12,40)(13,36,14,35)(15,33,16,34) (17,31,18,32)(19,30,20,29)(21,28,22,27)(23,26,24,25)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 7,11,16,20,24,28,32,36,37, 2, 8,12,15,19,23,27,31,35,38)( 3, 5, 9,13,17, 21,25,30,34,40, 4, 6,10,14,18,22,26,29,33,39)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 7,12,15,20,24,27,31,36,37)( 2, 8,11,16,19,23,28,32,35,38)( 3, 5,10,14,17, 21,26,29,34,40)( 4, 6, 9,13,18,22,25,30,33,39)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,11,20,28,36, 2,12,19,27,35)( 3, 9,17,25,34, 4,10,18,26,33)( 5,13,21,30,40, 6,14,22,29,39)( 7,16,24,32,37, 8,15,23,31,38)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $4$ | $10$ | $( 1,11,20,28,36, 2,12,19,27,35)( 3, 9,17,25,34, 4,10,18,26,33)( 5,14,21,29,40) ( 6,13,22,30,39)( 7,15,24,31,37)( 8,16,23,32,38)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,20,27,36)( 2,11,19,28,35)( 3,10,17,26,34)( 4, 9,18,25,33) ( 5,14,21,29,40)( 6,13,22,30,39)( 7,15,24,31,37)( 8,16,23,32,38)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,15,27,37,12,24,36, 7,20,31)( 2,16,28,38,11,23,35, 8,19,32)( 3,14,26,40,10, 21,34, 5,17,29)( 4,13,25,39, 9,22,33, 6,18,30)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,15,28,38,12,24,35, 8,20,31, 2,16,27,37,11,23,36, 7,19,32)( 3,14,25,39,10, 21,33, 6,17,29, 4,13,26,40, 9,22,34, 5,18,30)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $4$ | $10$ | $( 1,19,36,11,27, 2,20,35,12,28)( 3,18,34, 9,26, 4,17,33,10,25)( 5,21,40,14,29) ( 6,22,39,13,30)( 7,24,37,15,31)( 8,23,38,16,32)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,19,36,11,27, 2,20,35,12,28)( 3,18,34, 9,26, 4,17,33,10,25)( 5,22,40,13,29, 6,21,39,14,30)( 7,23,37,16,31, 8,24,38,15,32)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,20,36,12,27)( 2,19,35,11,28)( 3,17,34,10,26)( 4,18,33, 9,25) ( 5,21,40,14,29)( 6,22,39,13,30)( 7,24,37,15,31)( 8,23,38,16,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,23)( 2,24)( 3,22)( 4,21)( 5,25)( 6,26)( 7,28)( 8,27)( 9,29)(10,30)(11,31) (12,32)(13,34)(14,33)(15,35)(16,36)(17,39)(18,40)(19,37)(20,38)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,23, 2,24)( 3,22, 4,21)( 5,26, 6,25)( 7,27, 8,28)( 9,29,10,30)(11,31,12,32) (13,33,14,34)(15,36,16,35)(17,39,18,40)(19,37,20,38)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [80, 39] |
| Character table: |
2 4 3 4 4 3 4 3 3 2 2 3 2 3 2 2 2 3 3 3 3
5 1 1 1 . . . . . 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 2b 2c 2d 2e 2f 4a 20a 10a 10b 10c 5a 10d 20b 10e 10f 5b 2g 4b
2P 1a 1a 1a 1a 1a 1a 1a 2b 10b 5a 5b 5b 5b 5b 10f 5a 5a 5a 1a 2b
3P 1a 2a 2b 2c 2d 2e 2f 4a 20b 10d 10f 10e 5b 10a 20a 10c 10b 5a 2g 4b
5P 1a 2a 2b 2c 2d 2e 2f 4a 4b 2g 2b 2a 1a 2g 4b 2a 2b 1a 2g 4b
7P 1a 2a 2b 2c 2d 2e 2f 4a 20b 10d 10f 10e 5b 10a 20a 10c 10b 5a 2g 4b
11P 1a 2a 2b 2c 2d 2e 2f 4a 20a 10a 10b 10c 5a 10d 20b 10e 10f 5b 2g 4b
13P 1a 2a 2b 2c 2d 2e 2f 4a 20b 10d 10f 10e 5b 10a 20a 10c 10b 5a 2g 4b
17P 1a 2a 2b 2c 2d 2e 2f 4a 20b 10d 10f 10e 5b 10a 20a 10c 10b 5a 2g 4b
19P 1a 2a 2b 2c 2d 2e 2f 4a 20a 10a 10b 10c 5a 10d 20b 10e 10f 5b 2g 4b
X.1 1 1 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 1 1 -1
X.3 1 -1 1 -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 1 -1 1
X.5 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1
X.6 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1
X.7 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1
X.8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1
X.9 2 . -2 -2 . 2 . . . . -2 . 2 . . . -2 2 . .
X.10 2 . -2 2 . -2 . . . . -2 . 2 . . . -2 2 . .
X.11 2 -2 2 . . . . . A -A -*A *A -*A -*A *A A -A -A 2 -2
X.12 2 -2 2 . . . . . *A -*A -A A -A -A A *A -*A -*A 2 -2
X.13 2 -2 2 . . . . . -*A *A -A A -A A -A *A -*A -*A -2 2
X.14 2 -2 2 . . . . . -A A -*A *A -*A *A -*A A -A -A -2 2
X.15 2 2 2 . . . . . A A -*A -*A -*A *A *A -A -A -A -2 -2
X.16 2 2 2 . . . . . *A *A -A -A -A A A -*A -*A -*A -2 -2
X.17 2 2 2 . . . . . -*A -*A -A -A -A -A -A -*A -*A -*A 2 2
X.18 2 2 2 . . . . . -A -A -*A -*A -*A -*A -*A -A -A -A 2 2
X.19 4 . -4 . . . . . . . B . -B . . . *B -*B . .
X.20 4 . -4 . . . . . . . *B . -*B . . . B -B . .
A = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
B = -2*E(5)^2-2*E(5)^3
= 1+Sqrt(5) = 1+r5
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