Group action invariants
| Degree $n$: | $40$ | |
| Transitive number $t$: | $21$ | |
| Group: | $D_4:D_5$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $20$ | |
| Generators: | (1,31,20,7,36,24,12,37,27,16,2,32,19,8,35,23,11,38,28,15)(3,29,17,5,34,22,9,39,25,13,4,30,18,6,33,21,10,40,26,14), (1,15,27,37,11,23,36,7,19,32)(2,16,28,38,12,24,35,8,20,31)(3,14,25,39,10,21,34,5,18,30)(4,13,26,40,9,22,33,6,17,29), (1,30,2,29)(3,31,4,32)(5,28,6,27)(7,25,8,26)(9,23,10,24)(11,21,12,22)(13,19,14,20)(15,18,16,17)(33,37,34,38)(35,40,36,39) |
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: $D_5$, $D_{10}$ x 2
Degree 20: 20T4
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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 2, 4)( 5,37, 6,38)( 7,40, 8,39)( 9,36,10,35)(11,34,12,33)(13,31,14,32) (15,29,16,30)(17,27,18,28)(19,25,20,26)(21,23,22,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5,38, 6,37)( 7,39, 8,40)( 9,36,10,35)(11,34,12,33)(13,32,14,31) (15,30,16,29)(17,27,18,28)(19,25,20,26)(21,24,22,23)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 4, 2, 3)( 5,37, 6,38)( 7,40, 8,39)( 9,35,10,36)(11,33,12,34)(13,31,14,32) (15,29,16,30)(17,28,18,27)(19,26,20,25)(21,23,22,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,37)(10,38)(11,39)(12,40)(13,35)(14,36)(15,33) (16,34)(17,32)(18,31)(19,30)(20,29)(21,27)(22,28)(23,26)(24,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,37,10,38)(11,39,12,40)(13,36,14,35)(15,34,16,33) (17,32,18,31)(19,30,20,29)(21,28,22,27)(23,25,24,26)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 7,12,16,19,23,28,31,36,37, 2, 8,11,15,20,24,27,32,35,38)( 3, 5, 9,13,18, 21,26,29,34,39, 4, 6,10,14,17,22,25,30,33,40)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 7,11,15,19,23,27,32,36,37)( 2, 8,12,16,20,24,28,31,35,38)( 3, 5,10,14,18, 21,25,30,34,39)( 4, 6, 9,13,17,22,26,29,33,40)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $4$ | $10$ | $( 1,11,19,27,36)( 2,12,20,28,35)( 3,10,18,25,34)( 4, 9,17,26,33) ( 5,13,21,29,39, 6,14,22,30,40)( 7,16,23,31,37, 8,15,24,32,38)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,19,27,36)( 2,12,20,28,35)( 3,10,18,25,34)( 4, 9,17,26,33) ( 5,14,21,30,39)( 6,13,22,29,40)( 7,15,23,32,37)( 8,16,24,31,38)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,12,19,28,36, 2,11,20,27,35)( 3, 9,18,26,34, 4,10,17,25,33)( 5,13,21,29,39, 6,14,22,30,40)( 7,16,23,31,37, 8,15,24,32,38)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,15,28,38,11,23,35, 8,19,32, 2,16,27,37,12,24,36, 7,20,31)( 3,14,26,40,10, 21,33, 6,18,30, 4,13,25,39, 9,22,34, 5,17,29)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,15,27,37,11,23,36, 7,19,32)( 2,16,28,38,12,24,35, 8,20,31)( 3,14,25,39,10, 21,34, 5,18,30)( 4,13,26,40, 9,22,33, 6,17,29)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,19,36,11,27)( 2,20,35,12,28)( 3,18,34,10,25)( 4,17,33, 9,26) ( 5,21,39,14,30)( 6,22,40,13,29)( 7,23,37,15,32)( 8,24,38,16,31)$ |
| $ 10, 10, 5, 5, 5, 5 $ | $4$ | $10$ | $( 1,19,36,11,27)( 2,20,35,12,28)( 3,18,34,10,25)( 4,17,33, 9,26) ( 5,22,39,13,30, 6,21,40,14,29)( 7,24,37,16,32, 8,23,38,15,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,20,36,12,27, 2,19,35,11,28)( 3,17,34, 9,25, 4,18,33,10,26)( 5,22,39,13,30, 6,21,40,14,29)( 7,24,37,16,32, 8,23,38,15,31)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,23)( 2,24)( 3,21)( 4,22)( 5,25)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,32) (12,31)(13,33)(14,34)(15,36)(16,35)(17,40)(18,39)(19,37)(20,38)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,23, 2,24)( 3,21, 4,22)( 5,26, 6,25)( 7,28, 8,27)( 9,29,10,30)(11,32,12,31) (13,34,14,33)(15,35,16,36)(17,40,18,39)(19,37,20,38)$ |
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 4 4 3 3 2 2 2 3 3 2 2 3 2 3 3 3
5 1 1 1 . . . . . 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 2b 4a 4b 4c 2c 4d 20a 10a 10b 5a 10c 20b 10d 5b 10e 10f 2d 4e
2P 1a 1a 1a 2b 2b 2b 1a 2b 10c 5a 5b 5b 5b 10f 5b 5a 5a 5a 1a 2b
3P 1a 2a 2b 4a 4c 4b 2c 4d 20b 10d 10e 5b 10f 20a 10a 5a 10b 10c 2d 4e
5P 1a 2a 2b 4a 4b 4c 2c 4d 4e 2d 2a 1a 2b 4e 2d 1a 2a 2b 2d 4e
7P 1a 2a 2b 4a 4c 4b 2c 4d 20b 10d 10e 5b 10f 20a 10a 5a 10b 10c 2d 4e
11P 1a 2a 2b 4a 4c 4b 2c 4d 20a 10a 10b 5a 10c 20b 10d 5b 10e 10f 2d 4e
13P 1a 2a 2b 4a 4b 4c 2c 4d 20b 10d 10e 5b 10f 20a 10a 5a 10b 10c 2d 4e
17P 1a 2a 2b 4a 4b 4c 2c 4d 20b 10d 10e 5b 10f 20a 10a 5a 10b 10c 2d 4e
19P 1a 2a 2b 4a 4c 4b 2c 4d 20a 10a 10b 5a 10c 20b 10d 5b 10e 10f 2d 4e
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 . . . . . B -B *B -*B -*B *B -*B -B B -B 2 -2
X.10 2 -2 2 . . . . . *B -*B B -B -B B -B -*B *B -*B 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 . A -A . . . . . 2 -2 . . 2 . -2 . .
X.18 2 . -2 . -A A . . . . . 2 -2 . . 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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