Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $3$ | |
| Group : | $C_5:C_8$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,27,10,33,17,3,26,11,35,19,2,28,9,34,18,4,25,12,36,20)(5,31,14,40,23,8,30,15,37,22,6,32,13,39,24,7,29,16,38,21), (1,5,3,8,2,6,4,7)(9,37,12,39,10,38,11,40)(13,34,16,36,14,33,15,35)(17,29,19,31,18,30,20,32)(21,25,23,27,22,26,24,28) | |
| $|\Aut(F/K)|$: | $40$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 8: $C_8$ 10: $D_{5}$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $D_{5}$
Degree 8: $C_8$
Degree 10: $D_5$
Degree 20: 20T2
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, 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 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)(37,39,38,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 5, 3, 8, 2, 6, 4, 7)( 9,37,12,39,10,38,11,40)(13,34,16,36,14,33,15,35) (17,29,19,31,18,30,20,32)(21,25,23,27,22,26,24,28)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 6, 3, 7, 2, 5, 4, 8)( 9,38,12,40,10,37,11,39)(13,33,16,35,14,34,15,36) (17,30,19,32,18,29,20,31)(21,26,23,28,22,25,24,27)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 7, 4, 6, 2, 8, 3, 5)( 9,40,11,38,10,39,12,37)(13,35,15,33,14,36,16,34) (17,32,20,30,18,31,19,29)(21,28,24,26,22,27,23,25)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 8, 4, 5, 2, 7, 3, 6)( 9,39,11,37,10,40,12,38)(13,36,15,34,14,35,16,33) (17,31,20,29,18,32,19,30)(21,27,24,25,22,28,23,26)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 9,17,25,35)( 2,10,18,26,36)( 3,12,19,27,34)( 4,11,20,28,33) ( 5,13,23,29,37)( 6,14,24,30,38)( 7,15,21,32,40)( 8,16,22,31,39)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,10,17,26,35, 2, 9,18,25,36)( 3,11,19,28,34, 4,12,20,27,33)( 5,14,23,30,37, 6,13,24,29,38)( 7,16,21,31,40, 8,15,22,32,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,11,18,27,35, 4,10,19,25,33, 2,12,17,28,36, 3, 9,20,26,34)( 5,15,24,31,37, 7,14,22,29,40, 6,16,23,32,38, 8,13,21,30,39)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,12,18,28,35, 3,10,20,25,34, 2,11,17,27,36, 4, 9,19,26,33)( 5,16,24,32,37, 8,14,21,29,39, 6,15,23,31,38, 7,13,22,30,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,17,35, 9,25)( 2,18,36,10,26)( 3,19,34,12,27)( 4,20,33,11,28) ( 5,23,37,13,29)( 6,24,38,14,30)( 7,21,40,15,32)( 8,22,39,16,31)$ |
| $ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,18,35,10,25, 2,17,36, 9,26)( 3,20,34,11,27, 4,19,33,12,28)( 5,24,37,14,29, 6,23,38,13,30)( 7,22,40,16,32, 8,21,39,15,31)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,19,36,11,25, 3,18,33, 9,27, 2,20,35,12,26, 4,17,34,10,28)( 5,22,38,15,29, 8,24,40,13,31, 6,21,37,16,30, 7,23,39,14,32)$ |
| $ 20, 20 $ | $2$ | $20$ | $( 1,20,36,12,25, 4,18,34, 9,28, 2,19,35,11,26, 3,17,33,10,27)( 5,21,38,16,29, 7,24,39,13,32, 6,22,37,15,30, 8,23,40,14,31)$ |
Group invariants
| Order: | $40=2^{3} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [40, 1] |
| Character table: |
2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2
5 1 1 1 1 . . . . 1 1 1 1 1 1 1 1
1a 2a 4a 4b 8a 8b 8c 8d 5a 10a 20a 20b 5b 10b 20c 20d
2P 1a 1a 2a 2a 4a 4a 4b 4b 5b 5b 10b 10b 5a 5a 10a 10a
3P 1a 2a 4b 4a 8d 8c 8b 8a 5b 10b 20c 20d 5a 10a 20a 20b
5P 1a 2a 4a 4b 8b 8a 8d 8c 1a 2a 4b 4a 1a 2a 4a 4b
7P 1a 2a 4b 4a 8c 8d 8a 8b 5b 10b 20c 20d 5a 10a 20a 20b
11P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 20b 20a 5b 10b 20d 20c
13P 1a 2a 4a 4b 8b 8a 8d 8c 5b 10b 20d 20c 5a 10a 20b 20a
17P 1a 2a 4a 4b 8a 8b 8c 8d 5b 10b 20d 20c 5a 10a 20b 20a
19P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 20b 20a 5b 10b 20d 20c
X.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
X.3 1 -1 A -A C -C /C -/C 1 -1 -A A 1 -1 A -A
X.4 1 -1 A -A -C C -/C /C 1 -1 -A A 1 -1 A -A
X.5 1 -1 -A A -/C /C -C C 1 -1 A -A 1 -1 -A A
X.6 1 -1 -A A /C -/C C -C 1 -1 A -A 1 -1 -A A
X.7 1 1 -1 -1 A A -A -A 1 1 -1 -1 1 1 -1 -1
X.8 1 1 -1 -1 -A -A A A 1 1 -1 -1 1 1 -1 -1
X.9 2 -2 B -B . . . . D -D E -E *D -*D -F F
X.10 2 -2 B -B . . . . *D -*D F -F D -D -E E
X.11 2 -2 -B B . . . . D -D -E E *D -*D F -F
X.12 2 -2 -B B . . . . *D -*D -F F D -D E -E
X.13 2 2 -2 -2 . . . . D D -D -D *D *D -*D -*D
X.14 2 2 -2 -2 . . . . *D *D -*D -*D D D -D -D
X.15 2 2 2 2 . . . . D D D D *D *D *D *D
X.16 2 2 2 2 . . . . *D *D *D *D D D D D
A = -E(4)
= -Sqrt(-1) = -i
B = -2*E(4)
= -2*Sqrt(-1) = -2i
C = -E(8)^3
D = E(5)^2+E(5)^3
= (-1-Sqrt(5))/2 = -1-b5
E = E(20)^13+E(20)^17
F = E(20)+E(20)^9
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