Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $6$ | |
Group: | $C_4\times D_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $4$ | |
Generators: | (1,15,2,16)(3,14,4,13)(5,12,6,11)(7,9,8,10)(17,20,18,19), (1,3,5,8,10,11,14,16,18,19,2,4,6,7,9,12,13,15,17,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $10$: $D_{5}$ $20$: $D_{10}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $D_{5}$
Degree 10: $D_{10}$
Low degree siblings
20T6, 40T9Siblings 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, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,19)( 4,20)( 5,17)( 6,18)( 7,15)( 8,16)( 9,14)(10,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,20)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,13)(10,14)(11,12)$ |
$ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5,20, 6,19)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$ |
$ 20 $ | $2$ | $20$ | $( 1, 3, 5, 8,10,11,14,16,18,19, 2, 4, 6, 7, 9,12,13,15,17,20)$ |
$ 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 4, 2, 3)( 5,19, 6,20)( 7,18, 8,17)( 9,15,10,16)(11,13,12,14)$ |
$ 20 $ | $2$ | $20$ | $( 1, 4, 5, 7,10,12,14,15,18,20, 2, 3, 6, 8, 9,11,13,16,17,19)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 5,10,14,18, 2, 6, 9,13,17)( 3, 8,11,16,19, 4, 7,12,15,20)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 6,10,13,18)( 2, 5, 9,14,17)( 3, 7,11,15,19)( 4, 8,12,16,20)$ |
$ 20 $ | $2$ | $20$ | $( 1, 7,14,20, 6,11,17, 4,10,15, 2, 8,13,19, 5,12,18, 3, 9,16)$ |
$ 20 $ | $2$ | $20$ | $( 1, 8,14,19, 6,12,17, 3,10,16, 2, 7,13,20, 5,11,18, 4, 9,15)$ |
$ 10, 10 $ | $2$ | $10$ | $( 1, 9,18, 5,13, 2,10,17, 6,14)( 3,12,19, 8,15, 4,11,20, 7,16)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,18, 6,13)( 2, 9,17, 5,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$ |
$ 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,17, 8,18)( 9,20,10,19)$ |
$ 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,12, 2,11)( 3,13, 4,14)( 5,15, 6,16)( 7,18, 8,17)( 9,19,10,20)$ |
Group invariants
Order: | $40=2^{3} \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [40, 5] |
Character table: |
2 3 3 3 3 3 2 3 2 2 2 2 2 2 2 3 3 5 1 . 1 . . 1 . 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 4a 20a 4b 20b 10a 5a 20c 20d 10b 5b 4c 4d 2P 1a 1a 1a 1a 2b 10a 2b 10a 5b 5b 10b 10b 5a 5a 2b 2b 3P 1a 2a 2b 2c 4b 20d 4a 20c 10b 5b 20b 20a 10a 5a 4d 4c 5P 1a 2a 2b 2c 4a 4c 4b 4d 2b 1a 4c 4d 2b 1a 4c 4d 7P 1a 2a 2b 2c 4b 20d 4a 20c 10b 5b 20b 20a 10a 5a 4d 4c 11P 1a 2a 2b 2c 4b 20b 4a 20a 10a 5a 20d 20c 10b 5b 4d 4c 13P 1a 2a 2b 2c 4a 20c 4b 20d 10b 5b 20a 20b 10a 5a 4c 4d 17P 1a 2a 2b 2c 4a 20c 4b 20d 10b 5b 20a 20b 10a 5a 4c 4d 19P 1a 2a 2b 2c 4b 20b 4a 20a 10a 5a 20d 20c 10b 5b 4d 4c X.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 X.3 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 X.5 1 -1 -1 1 A -A -A A -1 1 -A A -1 1 -A A X.6 1 -1 -1 1 -A A A -A -1 1 A -A -1 1 A -A X.7 1 1 -1 -1 A A -A -A -1 1 A -A -1 1 A -A X.8 1 1 -1 -1 -A -A A A -1 1 -A A -1 1 -A A X.9 2 . -2 . . B . -B *D -*D C -C D -D E -E X.10 2 . -2 . . C . -C D -D B -B *D -*D E -E X.11 2 . -2 . . -C . C D -D -B B *D -*D -E E X.12 2 . -2 . . -B . B *D -*D -C C D -D -E E X.13 2 . 2 . . D . D -*D -*D *D *D -D -D -2 -2 X.14 2 . 2 . . *D . *D -D -D D D -*D -*D -2 -2 X.15 2 . 2 . . -*D . -*D -D -D -D -D -*D -*D 2 2 X.16 2 . 2 . . -D . -D -*D -*D -*D -*D -D -D 2 2 A = -E(4) = -Sqrt(-1) = -i B = -E(20)-E(20)^9 C = -E(20)^13-E(20)^17 D = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 E = -2*E(4) = -2*Sqrt(-1) = -2i |