Group action invariants
| Degree $n$: | $40$ | |
| Transitive number $t$: | $35$ | |
| Group: | $D_4:D_5$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $4$ | |
| Generators: | (1,5,2,6)(3,7,4,8)(9,39,10,40)(11,37,12,38)(13,36,14,35)(15,34,16,33)(17,32,18,31)(19,30,20,29)(21,25,22,26)(23,28,24,27), (1,12)(2,11)(3,9)(4,10)(7,8)(13,38)(14,37)(15,40)(16,39)(17,33)(18,34)(19,36)(20,35)(21,30)(22,29)(23,31)(24,32)(27,28), (1,33,2,34)(3,35,4,36)(5,31,6,32)(7,29,8,30)(9,26,10,25)(11,27,12,28)(13,24,14,23)(15,21,16,22)(17,20,18,19)(37,40,38,39) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $C_2^3$ $10$: $D_{5}$ $16$: $Q_8:C_2$ $20$: $D_{10}$ x 3 Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 5: $D_{5}$
Degree 8: $Q_8:C_2$
Degree 10: $D_{10}$ x 3
Degree 20: 20T8
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, 1, 1, 1, 1 $ | $10$ | $2$ | $( 3, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,34)(10,33)(11,36)(12,35)(13,29)(14,30) (15,32)(16,31)(17,28)(18,27)(19,26)(20,25)(23,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 2, 4)( 5,40, 6,39)( 7,37, 8,38)( 9,35,10,36)(11,33,12,34)(13,32,14,31) (15,29,16,30)(17,26,18,25)(19,28,20,27)(21,24,22,23)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,39,10,40)(11,37,12,38)(13,36,14,35)(15,34,16,33) (17,32,18,31)(19,30,20,29)(21,25,22,26)(23,28,24,27)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 5,11,14,20,22,26,30,35,38, 2, 6,12,13,19,21,25,29,36,37)( 3, 8, 9,16,17, 24,28,32,34,39, 4, 7,10,15,18,23,27,31,33,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,40,10,39)(11,38,12,37)(13,35,14,36)(15,33,16,34) (17,31,18,32)(19,29,20,30)(21,26,22,25)(23,27,24,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,37,10,38)(11,39,12,40)(13,33,14,34)(15,35,16,36) (17,29,18,30)(19,31,20,32)(21,27,22,28)(23,26,24,25)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 7,12,16,20,23,25,32,35,40)( 2, 8,11,15,19,24,26,31,36,39)( 3, 6,10,14,17, 21,27,30,34,37)( 4, 5, 9,13,18,22,28,29,33,38)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 9,20,28,35, 4,12,18,25,33)( 2,10,19,27,36, 3,11,17,26,34)( 5,15,22,31,38, 8,13,24,29,39)( 6,16,21,32,37, 7,14,23,30,40)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,11,20,26,35, 2,12,19,25,36)( 3, 9,17,28,34, 4,10,18,27,33)( 5,14,22,30,38, 6,13,21,29,37)( 7,15,23,31,40, 8,16,24,32,39)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,12,20,25,35)( 2,11,19,26,36)( 3,10,17,27,34)( 4, 9,18,28,33) ( 5,13,22,29,38)( 6,14,21,30,37)( 7,16,23,32,40)( 8,15,24,31,39)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,13,26,37,12,22,36, 6,20,29, 2,14,25,38,11,21,35, 5,19,30)( 3,15,28,40,10, 24,33, 7,17,31, 4,16,27,39, 9,23,34, 8,18,32)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,15,25,39,12,24,35, 8,20,31)( 2,16,26,40,11,23,36, 7,19,32)( 3,13,27,38,10, 22,34, 5,17,29)( 4,14,28,37, 9,21,33, 6,18,30)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,17,35,10,25, 3,20,34,12,27)( 2,18,36, 9,26, 4,19,33,11,28)( 5,23,38,16,29, 7,22,40,13,32)( 6,24,37,15,30, 8,21,39,14,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,19,35,11,25, 2,20,36,12,26)( 3,18,34, 9,27, 4,17,33,10,28)( 5,21,38,14,29, 6,22,37,13,30)( 7,24,40,15,32, 8,23,39,16,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,20,35,12,25)( 2,19,36,11,26)( 3,17,34,10,27)( 4,18,33, 9,28) ( 5,22,38,13,29)( 6,21,37,14,30)( 7,23,40,16,32)( 8,24,39,15,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,21, 2,22)( 3,23, 4,24)( 5,25, 6,26)( 7,28, 8,27)( 9,31,10,32)(11,29,12,30) (13,35,14,36)(15,34,16,33)(17,40,18,39)(19,38,20,37)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,23)( 2,24)( 3,21)( 4,22)( 5,28)( 6,27)( 7,25)( 8,26)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,36)(16,35)(17,37)(18,38)(19,39)(20,40)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [80, 40] |
| Character table: |
2 4 3 4 3 3 4 2 4 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 4a 4b 20a 4c 4d 10a 10b 10c 5a 20b 10d 10e 10f 5b 4e 2d
2P 1a 1a 1a 1a 2b 2b 10c 2b 2b 5a 5b 5b 5b 10f 5b 5a 5a 5a 2b 1a
3P 1a 2a 2b 2c 4a 4c 20b 4b 4d 10d 10e 10f 5b 20a 10a 10b 10c 5a 4e 2d
5P 1a 2a 2b 2c 4a 4b 4e 4c 4d 2d 2c 2b 1a 4e 2d 2c 2b 1a 4e 2d
7P 1a 2a 2b 2c 4a 4c 20b 4b 4d 10d 10e 10f 5b 20a 10a 10b 10c 5a 4e 2d
11P 1a 2a 2b 2c 4a 4c 20a 4b 4d 10a 10b 10c 5a 20b 10d 10e 10f 5b 4e 2d
13P 1a 2a 2b 2c 4a 4b 20b 4c 4d 10d 10e 10f 5b 20a 10a 10b 10c 5a 4e 2d
17P 1a 2a 2b 2c 4a 4b 20b 4c 4d 10d 10e 10f 5b 20a 10a 10b 10c 5a 4e 2d
19P 1a 2a 2b 2c 4a 4c 20a 4b 4d 10a 10b 10c 5a 20b 10d 10e 10f 5b 4e 2d
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 . . A . -A . . . -2 2 . . . -2 2 . .
X.10 2 . -2 . . -A . A . . . -2 2 . . . -2 2 . .
X.11 2 . 2 -2 . . B . . -B *B -*B -*B *B -*B B -B -B -2 2
X.12 2 . 2 -2 . . *B . . -*B B -B -B B -B *B -*B -*B -2 2
X.13 2 . 2 -2 . . -*B . . *B B -B -B -B B *B -*B -*B 2 -2
X.14 2 . 2 -2 . . -B . . B *B -*B -*B -*B *B B -B -B 2 -2
X.15 2 . 2 2 . . B . . B -*B -*B -*B *B *B -B -B -B -2 -2
X.16 2 . 2 2 . . *B . . *B -B -B -B B B -*B -*B -*B -2 -2
X.17 2 . 2 2 . . -*B . . -*B -B -B -B -B -B -*B -*B -*B 2 2
X.18 2 . 2 2 . . -B . . -B -*B -*B -*B -*B -*B -B -B -B 2 2
X.19 4 . -4 . . . . . . . . C -C . . . *C -*C . .
X.20 4 . -4 . . . . . . . . *C -*C . . . C -C . .
A = -2*E(4)
= -2*Sqrt(-1) = -2i
B = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
C = -2*E(5)^2-2*E(5)^3
= 1+Sqrt(5) = 1+r5
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