Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $35$ | |
Group: | $S_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,3)(2,4)(5,13)(6,14)(7,8)(9,10)(11,12)(15,17)(16,18), (1,6,10,14,18)(2,5,9,13,17)(3,8,11,16,19)(4,7,12,15,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: None
Degree 10: $S_5$
Low degree siblings
5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 24T202, 30T22, 30T25, 30T27, 40T62Siblings are shown with degree $\leq 47$
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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 1, 1 $ | $20$ | $3$ | $( 3, 7,10)( 4, 8, 9)( 5,11,17)( 6,12,18)(13,16,20)(14,15,19)$ |
$ 6, 6, 6, 1, 1 $ | $20$ | $6$ | $( 3,14,10,19, 7,15)( 4,13, 9,20, 8,16)( 5,12,17, 6,11,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $10$ | $2$ | $( 3,19)( 4,20)( 5, 6)( 7,14)( 8,13)( 9,16)(10,15)(11,12)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,20)( 8,19)( 9,15)(10,16)(11,18)(12,17)$ |
$ 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 3, 6,12,15)( 2, 4, 5,11,16)( 7,17,14, 9,20)( 8,18,13,10,19)$ |
$ 4, 4, 4, 4, 4 $ | $30$ | $4$ | $( 1, 3, 8,10)( 2, 4, 7, 9)( 5,12,16,20)( 6,11,15,19)(13,18,14,17)$ |
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
GAP id: | [120, 34] |
Character table: |
2 3 1 1 2 3 . 2 3 1 1 1 1 . . . 5 1 . . . . 1 . 1a 3a 6a 2a 2b 5a 4a 2P 1a 3a 3a 1a 1a 5a 2b 3P 1a 1a 2a 2a 2b 5a 4a 5P 1a 3a 6a 2a 2b 1a 4a X.1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 -1 X.3 4 1 1 -2 . -1 . X.4 4 1 -1 2 . -1 . X.5 5 -1 1 1 1 . -1 X.6 5 -1 -1 -1 1 . 1 X.7 6 . . . -2 1 . |