Group action invariants
Degree $n$ : | $40$ | |
Transitive number $t$ : | $44$ | |
Group : | $C_2^2\times F_5$ | |
Parity: | $1$ | |
Primitive: | No | |
Nilpotency class: | $-1$ (not nilpotent) | |
Generators: | (1,17,35,10,25,4,20,34,12,28)(2,18,36,9,26,3,19,33,11,27)(5,22,37,13,31,8,24,40,16,30)(6,21,38,14,32,7,23,39,15,29), (1,21,4,23)(2,22,3,24)(5,36,40,9)(6,35,39,10)(7,34,38,12)(8,33,37,11)(13,18,31,26)(14,17,32,25)(15,20,29,28)(16,19,30,27), (1,5,17,13)(2,6,18,14)(3,7,19,15)(4,8,20,16)(9,29,11,32)(10,30,12,31)(21,26,38,33)(22,25,37,34)(23,27,39,36)(24,28,40,35) | |
$|\Aut(F/K)|$: | $8$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_4$ x 4, $C_2^2$ x 7 8: $C_4\times C_2$ x 6, $C_2^3$ 20: $F_5$ 40: $F_{5}\times C_2$ x 3 Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 5: $F_5$
Degree 8: $C_4\times C_2$
Degree 10: $F_5$, $F_{5}\times C_2$ x 2
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 $ | $5$ | $2$ | $( 5,37)( 6,38)( 7,39)( 8,40)( 9,33)(10,34)(11,36)(12,35)(13,30)(14,29)(15,32) (16,31)(17,28)(18,27)(19,26)(20,25)$ |
$ 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 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5,38)( 6,37)( 7,40)( 8,39)( 9,34)(10,33)(11,35)(12,36)(13,29) (14,30)(15,31)(16,32)(17,27)(18,28)(19,25)(20,26)(21,22)(23,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 3)( 2, 4)( 5,39)( 6,40)( 7,37)( 8,38)( 9,35)(10,36)(11,34)(12,33)(13,32) (14,31)(15,30)(16,29)(17,26)(18,25)(19,28)(20,27)(21,24)(22,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 4)( 2, 3)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,33)(12,34)(13,31) (14,32)(15,29)(16,30)(17,25)(18,26)(19,27)(20,28)(21,23)(22,24)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 5,17,13)( 2, 6,18,14)( 3, 7,19,15)( 4, 8,20,16)( 9,29,11,32)(10,30,12,31) (21,26,38,33)(22,25,37,34)(23,27,39,36)(24,28,40,35)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 5,34,30)( 2, 6,33,29)( 3, 7,36,32)( 4, 8,35,31)( 9,21,26,15)(10,22,25,16) (11,23,27,14)(12,24,28,13)(17,40,20,37)(18,39,19,38)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 6,17,14)( 2, 5,18,13)( 3, 8,19,16)( 4, 7,20,15)( 9,30,11,31)(10,29,12,32) (21,25,38,34)(22,26,37,33)(23,28,39,35)(24,27,40,36)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 6,34,29)( 2, 5,33,30)( 3, 8,36,31)( 4, 7,35,32)( 9,22,26,16)(10,21,25,15) (11,24,27,13)(12,23,28,14)(17,39,20,38)(18,40,19,37)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 7,17,15)( 2, 8,18,16)( 3, 5,19,13)( 4, 6,20,14)( 9,31,11,30)(10,32,12,29) (21,28,38,35)(22,27,37,36)(23,25,39,34)(24,26,40,33)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 7,34,32)( 2, 8,33,31)( 3, 5,36,30)( 4, 6,35,29)( 9,24,26,13)(10,23,25,14) (11,22,27,16)(12,21,28,15)(17,38,20,39)(18,37,19,40)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 8,17,16)( 2, 7,18,15)( 3, 6,19,14)( 4, 5,20,13)( 9,32,11,29)(10,31,12,30) (21,27,38,36)(22,28,37,35)(23,26,39,33)(24,25,40,34)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 8,34,31)( 2, 7,33,32)( 3, 6,36,29)( 4, 5,35,30)( 9,23,26,14)(10,24,25,13) (11,21,27,15)(12,22,28,16)(17,37,20,40)(18,38,19,39)$ |
$ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1, 9,20,27,35, 3,12,18,25,33)( 2,10,19,28,36, 4,11,17,26,34)( 5,14,24,29,37, 7,16,21,31,39)( 6,13,23,30,38, 8,15,22,32,40)$ |
$ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,10,20,28,35, 4,12,17,25,34)( 2, 9,19,27,36, 3,11,18,26,33)( 5,13,24,30,37, 8,16,22,31,40)( 6,14,23,29,38, 7,15,21,32,39)$ |
$ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,11,20,26,35, 2,12,19,25,36)( 3,10,18,28,33, 4, 9,17,27,34)( 5,15,24,32,37, 6,16,23,31,38)( 7,13,21,30,39, 8,14,22,29,40)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,12,20,25,35)( 2,11,19,26,36)( 3, 9,18,27,33)( 4,10,17,28,34) ( 5,16,24,31,37)( 6,15,23,32,38)( 7,14,21,29,39)( 8,13,22,30,40)$ |
Group invariants
Order: | $80=2^{4} \cdot 5$ | |
Cyclic: | No | |
Abelian: | No | |
Solvable: | Yes | |
GAP id: | [80, 50] |
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 2d 2e 2f 2g 4a 4b 4c 4d 4e 4f 4g 4h 10a 10b 10c 5a 2P 1a 1a 1a 1a 1a 1a 1a 1a 2g 2g 2g 2g 2g 2g 2g 2g 5a 5a 5a 5a 3P 1a 2a 2b 2c 2d 2e 2f 2g 4h 4g 4f 4e 4d 4c 4b 4a 10a 10b 10c 5a 5P 1a 2a 2b 2c 2d 2e 2f 2g 4a 4b 4c 4d 4e 4f 4g 4h 2d 2f 2b 1a 7P 1a 2a 2b 2c 2d 2e 2f 2g 4h 4g 4f 4e 4d 4c 4b 4a 10a 10b 10c 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 -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 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 1 -1 1 -1 A -A A -A A -A A -A 1 1 1 1 X.12 1 -1 1 -1 1 -1 1 -1 -A A -A A -A A -A A 1 1 1 1 X.13 1 1 -1 -1 1 1 -1 -1 A A -A -A A A -A -A 1 -1 -1 1 X.14 1 1 -1 -1 1 1 -1 -1 -A -A A A -A -A A A 1 -1 -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 . -4 . -4 . . . . . . . . . 1 1 -1 -1 X.20 4 . 4 . 4 . 4 . . . . . . . . . -1 -1 -1 -1 A = -E(4) = -Sqrt(-1) = -i |