Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $5$ | |
Group: | $F_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $20$ | |
Generators: | (1,4,17,15)(2,3,18,16)(5,12,14,7)(6,11,13,8)(9,19,10,20), (1,6,10,14,18)(2,5,9,13,17)(3,8,12,15,19)(4,7,11,16,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $F_5$
Degree 10: $F_5$
Low degree siblings
5T3, 10T4Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,20)( 4,19)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$ |
$ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 9, 7)( 2, 4,10, 8)( 5,16, 6,15)(11,14,19,17)(12,13,20,18)$ |
$ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 4,17,15)( 2, 3,18,16)( 5,12,14, 7)( 6,11,13, 8)( 9,19,10,20)$ |
$ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 6,10,14,18)( 2, 5, 9,13,17)( 3, 8,12,15,19)( 4, 7,11,16,20)$ |
Group invariants
Order: | $20=2^{2} \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [20, 3] |
Character table: |
2 2 2 2 2 . 5 1 . . . 1 1a 2a 4a 4b 5a 2P 1a 1a 2a 2a 5a 3P 1a 2a 4b 4a 5a 5P 1a 2a 4a 4b 1a X.1 1 1 1 1 1 X.2 1 1 -1 -1 1 X.3 1 -1 A -A 1 X.4 1 -1 -A A 1 X.5 4 . . . -1 A = -E(4) = -Sqrt(-1) = -i |