Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $43$ | |
Group: | $C_2^4:D_5$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $4$ | |
Generators: | (1,15)(2,16)(3,14)(4,13)(5,11)(6,12)(7,9,17,19)(8,10,18,20), (1,13,11,3)(2,14,12,4)(5,9)(6,10)(7,18)(8,17)(15,19)(16,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: $D_{5}$
Degree 10: $D_5$, $(C_2^4 : C_5) : C_2$, $(C_2^4 : C_5) : C_2$
Low degree siblings
10T15 x 3, 10T16 x 3, 16T415, 20T38 x 6, 20T39, 20T43 x 2, 20T45 x 3, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 7,17)( 8,18)( 9,19)(10,20)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,15)( 6,16)( 9,19)(10,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12)(13,19)(14,20)(15,18)(16,17)$ |
$ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 9,13,19)( 4,10,14,20)( 5, 8,15,18)( 6, 7,16,17)(11,12)$ |
$ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 3)( 2, 4)( 5, 9,15,19)( 6,10,16,20)( 7,18)( 8,17)(11,13)(12,14)$ |
$ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 3,11,13)( 2, 4,12,14)( 5, 9)( 6,10)( 7,18)( 8,17)(15,19)(16,20)$ |
$ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10)(11,14,16,18,19)(12,13,15,17,20)$ |
$ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 6, 9, 4, 8)( 2, 5,10, 3, 7)(11,16,19,14,18)(12,15,20,13,17)$ |
Group invariants
Order: | $160=2^{5} \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [160, 234] |
Character table: |
2 5 5 5 5 3 3 3 3 . . 5 1 . . . . . . . 1 1 1a 2a 2b 2c 2d 4a 4b 4c 5a 5b 2P 1a 1a 1a 1a 1a 2c 2b 2a 5b 5a 3P 1a 2a 2b 2c 2d 4a 4b 4c 5b 5a 5P 1a 2a 2b 2c 2d 4a 4b 4c 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 -1 1 1 X.3 2 2 2 2 . . . . A *A X.4 2 2 2 2 . . . . *A A X.5 5 -3 1 1 -1 1 1 -1 . . X.6 5 -3 1 1 1 -1 -1 1 . . X.7 5 1 -3 1 -1 1 -1 1 . . X.8 5 1 -3 1 1 -1 1 -1 . . X.9 5 1 1 -3 -1 -1 1 1 . . X.10 5 1 1 -3 1 1 -1 -1 . . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 |