Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $17$ | |
Group: | $C_2^4:C_5$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $4$ | |
Generators: | (1,6,10,14,18)(2,5,9,13,17)(3,7,12,15,19)(4,8,11,16,20), (1,2)(3,14)(4,13)(5,6)(9,20)(10,19)(11,12)(15,16) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $5$: $C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $C_5$
Degree 10: $C_2^4 : C_5$ x 2
Low degree siblings
10T8 x 3, 16T178, 20T17 x 5, 20T23, 40T57 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,16)( 6,15)( 7,17)( 8,18)( 9,10)(19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,14)( 4,13)( 5, 6)( 7,18)( 8,17)( 9,10)(15,16)(19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,14)( 4,13)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)(11,12)$ |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$ |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$ |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)(11,17,13,19,15)(12,18,14,20,16)$ |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)(11,19,17,15,13)(12,20,18,16,14)$ |
Group invariants
Order: | $80=2^{4} \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [80, 49] |
Character table: |
2 4 4 4 4 . . . . 5 1 . . . 1 1 1 1 1a 2a 2b 2c 5a 5b 5c 5d 2P 1a 1a 1a 1a 5b 5d 5a 5c 3P 1a 2a 2b 2c 5c 5a 5d 5b 5P 1a 2a 2b 2c 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 A B /B /A X.3 1 1 1 1 B /A A /B X.4 1 1 1 1 /B A /A B X.5 1 1 1 1 /A /B B A X.6 5 -3 1 1 . . . . X.7 5 1 -3 1 . . . . X.8 5 1 1 -3 . . . . A = E(5)^4 B = E(5)^3 |