Group action invariants
| Degree $n$ : | $40$ | |
| Transitive number $t$ : | $42$ | |
| Group : | $D_5:C_8$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,24,3,21,2,23,4,22)(5,33,40,11,6,34,39,12)(7,36,37,9,8,35,38,10)(13,20,32,28,14,19,31,27)(15,17,29,25,16,18,30,26), (1,18,35,9,26,4,19,33,12,27,2,17,36,10,25,3,20,34,11,28)(5,22,38,14,32,7,23,40,15,30,6,21,37,13,31,8,24,39,16,29) | |
| $|\Aut(F/K)|$: | $8$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $C_8$ x 2, $C_4\times C_2$ 20: $F_5$ 40: $F_{5}\times C_2$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $F_5$
Degree 8: $C_8$
Degree 10: $F_5$
Degree 20: 20T9
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 $ | $5$ | $2$ | $( 5,37)( 6,38)( 7,39)( 8,40)( 9,33)(10,34)(11,35)(12,36)(13,30)(14,29)(15,32) (16,31)(17,28)(18,27)(19,25)(20,26)$ |
| $ 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 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5,38)( 6,37)( 7,40)( 8,39)( 9,34)(10,33)(11,36)(12,35)(13,29) (14,30)(15,31)(16,32)(17,27)(18,28)(19,26)(20,25)(21,22)(23,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)(37,40,38,39)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5,40, 6,39)( 7,37, 8,38)( 9,35,10,36)(11,34,12,33)(13,32,14,31) (15,29,16,30)(17,25,18,26)(19,27,20,28)(21,23,22,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,36,34,35)(37,39,38,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 4, 2, 3)( 5,39, 6,40)( 7,38, 8,37)( 9,36,10,35)(11,33,12,34)(13,31,14,32) (15,30,16,29)(17,26,18,25)(19,28,20,27)(21,24,22,23)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 5,17,14, 2, 6,18,13)( 3, 8,19,16, 4, 7,20,15)( 9,29,11,31,10,30,12,32) (21,25,38,34,22,26,37,33)(23,27,39,36,24,28,40,35)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 5,33,29, 2, 6,34,30)( 3, 8,35,31, 4, 7,36,32)( 9,21,25,16,10,22,26,15) (11,23,27,13,12,24,28,14)(17,40,19,38,18,39,20,37)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 6,17,13, 2, 5,18,14)( 3, 7,19,15, 4, 8,20,16)( 9,30,11,32,10,29,12,31) (21,26,38,33,22,25,37,34)(23,28,39,35,24,27,40,36)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 6,33,30, 2, 5,34,29)( 3, 7,35,32, 4, 8,36,31)( 9,22,25,15,10,21,26,16) (11,24,27,14,12,23,28,13)(17,39,19,37,18,40,20,38)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 7,18,16, 2, 8,17,15)( 3, 5,20,13, 4, 6,19,14)( 9,32,12,30,10,31,11,29) (21,28,37,36,22,27,38,35)(23,25,40,33,24,26,39,34)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 7,34,31, 2, 8,33,32)( 3, 5,36,30, 4, 6,35,29)( 9,24,26,13,10,23,25,14) (11,21,28,15,12,22,27,16)(17,37,20,39,18,38,19,40)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 8,18,15, 2, 7,17,16)( 3, 6,20,14, 4, 5,19,13)( 9,31,12,29,10,32,11,30) (21,27,37,35,22,28,38,36)(23,26,40,34,24,25,39,33)$ |
| $ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 8,34,32, 2, 7,33,31)( 3, 6,36,29, 4, 5,35,30)( 9,23,26,14,10,24,25,13) (11,22,28,16,12,21,27,15)(17,38,20,40,18,37,19,39)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1, 9,19,27,36, 3,11,18,26,33, 2,10,20,28,35, 4,12,17,25,34)( 5,14,23,30,37, 8,16,22,32,40, 6,13,24,29,38, 7,15,21,31,39)$ |
| $ 20, 20 $ | $4$ | $20$ | $( 1,10,19,28,36, 4,11,17,26,34, 2, 9,20,27,35, 3,12,18,25,33)( 5,13,23,29,37, 7,16,21,32,39, 6,14,24,30,38, 8,15,22,31,40)$ |
| $ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,11,20,25,36, 2,12,19,26,35)( 3,10,17,27,33, 4, 9,18,28,34)( 5,16,24,31,37, 6,15,23,32,38)( 7,14,22,29,39, 8,13,21,30,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,12,20,26,36)( 2,11,19,25,35)( 3, 9,17,28,33)( 4,10,18,27,34) ( 5,15,24,32,37)( 6,16,23,31,38)( 7,13,22,30,39)( 8,14,21,29,40)$ |
Group invariants
| Order: | $80=2^{4} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [80, 28] |
| Character table: |
2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2
5 1 . 1 . 1 . 1 . . . . . . . . . 1 1 1 1
1a 2a 2b 2c 4a 4b 4c 4d 8a 8b 8c 8d 8e 8f 8g 8h 20a 20b 10a 5a
2P 1a 1a 1a 1a 2b 2b 2b 2b 4b 4b 4b 4b 4d 4d 4d 4d 10a 10a 5a 5a
3P 1a 2a 2b 2c 4c 4d 4a 4b 8h 8g 8f 8e 8d 8c 8b 8a 20b 20a 10a 5a
5P 1a 2a 2b 2c 4a 4b 4c 4d 8c 8d 8a 8b 8g 8h 8e 8f 4a 4c 2b 1a
7P 1a 2a 2b 2c 4c 4d 4a 4b 8f 8e 8h 8g 8b 8a 8d 8c 20b 20a 10a 5a
11P 1a 2a 2b 2c 4c 4d 4a 4b 8h 8g 8f 8e 8d 8c 8b 8a 20b 20a 10a 5a
13P 1a 2a 2b 2c 4a 4b 4c 4d 8c 8d 8a 8b 8g 8h 8e 8f 20a 20b 10a 5a
17P 1a 2a 2b 2c 4a 4b 4c 4d 8a 8b 8c 8d 8e 8f 8g 8h 20a 20b 10a 5a
19P 1a 2a 2b 2c 4c 4d 4a 4b 8h 8g 8f 8e 8d 8c 8b 8a 20b 20a 10a 5a
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 A -A -A A C -C -C C -/C /C /C -/C A -A -1 1
X.6 1 -1 -1 1 A -A -A A -C C C -C /C -/C -/C /C A -A -1 1
X.7 1 -1 -1 1 -A A A -A -/C /C /C -/C C -C -C C -A A -1 1
X.8 1 -1 -1 1 -A A A -A /C -/C -/C /C -C C C -C -A A -1 1
X.9 1 -1 1 -1 1 -1 1 -1 A -A A -A A -A A -A 1 1 1 1
X.10 1 -1 1 -1 1 -1 1 -1 -A A -A A -A A -A A 1 1 1 1
X.11 1 1 -1 -1 A A -A -A -/C -/C /C /C -C -C C C A -A -1 1
X.12 1 1 -1 -1 A A -A -A /C /C -/C -/C C C -C -C A -A -1 1
X.13 1 1 -1 -1 -A -A A A C C -C -C /C /C -/C -/C -A A -1 1
X.14 1 1 -1 -1 -A -A A A -C -C C C -/C -/C /C /C -A A -1 1
X.15 1 1 1 1 -1 -1 -1 -1 A A A A -A -A -A -A -1 -1 1 1
X.16 1 1 1 1 -1 -1 -1 -1 -A -A -A -A A A A A -1 -1 1 1
X.17 4 . 4 . -4 . -4 . . . . . . . . . 1 1 -1 -1
X.18 4 . 4 . 4 . 4 . . . . . . . . . -1 -1 -1 -1
X.19 4 . -4 . B . -B . . . . . . . . . -A A 1 -1
X.20 4 . -4 . -B . B . . . . . . . . . A -A 1 -1
A = -E(4)
= -Sqrt(-1) = -i
B = -4*E(4)
= -4*Sqrt(-1) = -4i
C = -E(8)
|