Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $40$ | |
| Group : | $D_4\times D_5$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,32)(2,31)(3,30)(4,29)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,38)(34,37)(35,40)(36,39), (1,35)(2,36)(3,34)(4,33)(5,30)(6,29)(7,32)(8,31)(9,28)(10,27)(11,25)(12,26)(13,21)(14,22)(15,24)(16,23), (1,27,12,34,19,3,26,10,35,17)(2,28,11,33,20,4,25,9,36,18)(5,31,14,39,22,8,30,15,37,24)(6,32,13,40,21,7,29,16,38,23) | |
| $|\Aut(F/K)|$: | $8$ |
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: $D_4$
Degree 10: $D_{10}$ x 3
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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,37)( 6,38)( 7,40)( 8,39)( 9,33)(10,34)(11,36)(12,35)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,26)(20,25)$ |
| $ 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 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,34)(10,33)(11,35)(12,36)(13,30) (14,29)(15,32)(16,31)(17,28)(18,27)(19,25)(20,26)(21,22)(23,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2, 4)( 5,39)( 6,40)( 7,38)( 8,37)( 9,36)(10,35)(11,33)(12,34)(13,32) (14,31)(15,30)(16,29)(17,26)(18,25)(19,27)(20,28)(21,23)(22,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,39,10,40)(11,38,12,37)(13,35,14,36)(15,34,16,33) (17,32,18,31)(19,30,20,29)(21,26,22,25)(23,28,24,27)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 5,11,13,19,22,25,29,35,37, 2, 6,12,14,20,21,26,30,36,38)( 3, 7, 9,15,17, 23,28,31,34,40, 4, 8,10,16,18,24,27,32,33,39)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,38)(10,37)(11,39)(12,40)(13,33)(14,34)(15,36) (16,35)(17,30)(18,29)(19,32)(20,31)(21,28)(22,27)(23,26)(24,25)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 7,12,16,19,23,26,32,35,40)( 2, 8,11,15,20,24,25,31,36,39)( 3, 5,10,14,17, 22,27,30,34,37)( 4, 6, 9,13,18,21,28,29,33,38)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 9,19,28,35, 4,12,18,26,33)( 2,10,20,27,36, 3,11,17,25,34)( 5,16,22,32,37, 7,14,23,30,40)( 6,15,21,31,38, 8,13,24,29,39)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,11,19,25,35, 2,12,20,26,36)( 3, 9,17,28,34, 4,10,18,27,33)( 5,13,22,29,37, 6,14,21,30,38)( 7,15,23,31,40, 8,16,24,32,39)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,19,26,35)( 2,11,20,25,36)( 3,10,17,27,34)( 4, 9,18,28,33) ( 5,14,22,30,37)( 6,13,21,29,38)( 7,16,23,32,40)( 8,15,24,31,39)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,13,25,37,12,21,36, 5,19,29, 2,14,26,38,11,22,35, 6,20,30)( 3,15,28,40,10, 24,33, 7,17,31, 4,16,27,39, 9,23,34, 8,18,32)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,15,26,39,12,24,35, 8,19,31)( 2,16,25,40,11,23,36, 7,20,32)( 3,13,27,38,10, 21,34, 6,17,29)( 4,14,28,37, 9,22,33, 5,18,30)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,17,35,10,26, 3,19,34,12,27)( 2,18,36, 9,25, 4,20,33,11,28)( 5,24,37,15,30, 8,22,39,14,31)( 6,23,38,16,29, 7,21,40,13,32)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,35,12,26)( 2,20,36,11,25)( 3,17,34,10,27)( 4,18,33, 9,28) ( 5,22,37,14,30)( 6,21,38,13,29)( 7,23,40,16,32)( 8,24,39,15,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,20,35,11,26, 2,19,36,12,25)( 3,18,34, 9,27, 4,17,33,10,28)( 5,21,37,13,30, 6,22,38,14,29)( 7,24,40,15,32, 8,23,39,16,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,21, 2,22)( 3,24, 4,23)( 5,26, 6,25)( 7,27, 8,28)( 9,32,10,31)(11,30,12,29) (13,36,14,35)(15,33,16,34)(17,39,18,40)(19,38,20,37)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,23)( 2,24)( 3,22)( 4,21)( 5,27)( 6,28)( 7,26)( 8,25)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,36)(16,35)(17,37)(18,38)(19,40)(20,39)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [80, 39] |
| Character table: |
2 4 4 4 4 3 3 3 2 3 2 2 3 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 4a 20a 2f 10a 10b 10c 5a 20b 10d 10e 5b 10f 4b 2g
2P 1a 1a 1a 1a 1a 1a 2b 10c 1a 5a 5b 5b 5b 10f 5b 5a 5a 5a 2b 1a
3P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g
5P 1a 2a 2b 2c 2d 2e 4a 4b 2f 2g 2d 2b 1a 4b 2g 2d 1a 2b 4b 2g
7P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g
11P 1a 2a 2b 2c 2d 2e 4a 20a 2f 10a 10b 10c 5a 20b 10d 10e 5b 10f 4b 2g
13P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g
17P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g
19P 1a 2a 2b 2c 2d 2e 4a 20a 2f 10a 10b 10c 5a 20b 10d 10e 5b 10f 4b 2g
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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