Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $49$ | |
Group: | $C_2\times C_5:F_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,11,18,7)(2,12,17,8)(3,5,16,13)(4,6,15,14)(9,20,10,19), (1,11,6,7)(2,12,5,8)(3,17,16,9)(4,18,15,10)(13,20,14,19) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $20$: $F_5$ x 2 $40$: $F_{5}\times C_2$ x 2 $100$: $C_5^2 : C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: None
Degree 10: $C_5^2 : C_4$
Low degree siblings
20T49, 20T52 x 2, 40T157 x 2Siblings 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 $ | $25$ | $2$ | $( 5,17)( 6,18)( 7,20)( 8,19)( 9,13)(10,14)(11,16)(12,15)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $5$ | $( 3, 7,12,15,20)( 4, 8,11,16,19)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $5$ | $( 3,12,20, 7,15)( 4,11,19, 8,16)$ |
$ 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 $ | $25$ | $2$ | $( 1, 2)( 3, 4)( 5,18)( 6,17)( 7,19)( 8,20)( 9,14)(10,13)(11,15)(12,16)$ |
$ 10, 2, 2, 2, 2, 2 $ | $4$ | $10$ | $( 1, 2)( 3, 8,12,16,20, 4, 7,11,15,19)( 5, 6)( 9,10)(13,14)(17,18)$ |
$ 10, 2, 2, 2, 2, 2 $ | $4$ | $10$ | $( 1, 2)( 3,11,20, 8,15, 4,12,19, 7,16)( 5, 6)( 9,10)(13,14)(17,18)$ |
$ 4, 4, 4, 4, 4 $ | $25$ | $4$ | $( 1, 3, 2, 4)( 5, 7,18,19)( 6, 8,17,20)( 9,11,14,15)(10,12,13,16)$ |
$ 4, 4, 4, 4, 4 $ | $25$ | $4$ | $( 1, 3, 2, 4)( 5,20,18, 8)( 6,19,17, 7)( 9,16,14,12)(10,15,13,11)$ |
$ 4, 4, 4, 4, 4 $ | $25$ | $4$ | $( 1, 4, 2, 3)( 5, 8,18,20)( 6, 7,17,19)( 9,12,14,16)(10,11,13,15)$ |
$ 4, 4, 4, 4, 4 $ | $25$ | $4$ | $( 1, 4, 2, 3)( 5,19,18, 7)( 6,20,17, 8)( 9,15,14,11)(10,16,13,12)$ |
$ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,12,15,20)( 4, 8,11,16,19)$ |
$ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3,12,20, 7,15)( 4,11,19, 8,16)$ |
$ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3,15, 7,20,12)( 4,16, 8,19,11)$ |
$ 10, 10 $ | $4$ | $10$ | $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,12,16,20, 4, 7,11,15,19)$ |
$ 10, 10 $ | $4$ | $10$ | $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3,11,20, 8,15, 4,12,19, 7,16)$ |
$ 10, 10 $ | $4$ | $10$ | $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3,16, 7,19,12, 4,15, 8,20,11)$ |
$ 10, 10 $ | $4$ | $10$ | $( 1, 9,17, 6,14, 2,10,18, 5,13)( 3,11,20, 8,15, 4,12,19, 7,16)$ |
$ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,10,17, 5,14)( 2, 9,18, 6,13)( 3,12,20, 7,15)( 4,11,19, 8,16)$ |
Group invariants
Order: | $200=2^{3} \cdot 5^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [200, 48] |
Character table: |
2 3 3 1 1 3 3 1 1 3 3 3 3 1 1 1 1 1 1 1 1 5 2 . 2 2 2 . 2 2 . . . . 2 2 2 2 2 2 2 2 1a 2a 5a 5b 2b 2c 10a 10b 4a 4b 4c 4d 5c 5d 5e 10c 10d 10e 10f 5f 2P 1a 1a 5b 5a 1a 1a 5b 5a 2c 2c 2c 2c 5f 5d 5e 5f 5d 5e 5c 5c 3P 1a 2a 5b 5a 2b 2c 10b 10a 4d 4c 4b 4a 5f 5d 5e 10f 10d 10e 10c 5c 5P 1a 2a 1a 1a 2b 2c 2b 2b 4a 4b 4c 4d 1a 1a 1a 2b 2b 2b 2b 1a 7P 1a 2a 5b 5a 2b 2c 10b 10a 4d 4c 4b 4a 5f 5d 5e 10f 10d 10e 10c 5c X.1 1 1 1 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 -1 -1 -1 1 X.3 1 -1 1 1 -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 1 1 1 1 X.5 1 -1 1 1 1 -1 1 1 C -C C -C 1 1 1 1 1 1 1 1 X.6 1 -1 1 1 1 -1 1 1 -C C -C C 1 1 1 1 1 1 1 1 X.7 1 1 1 1 -1 -1 -1 -1 C C -C -C 1 1 1 -1 -1 -1 -1 1 X.8 1 1 1 1 -1 -1 -1 -1 -C -C C C 1 1 1 -1 -1 -1 -1 1 X.9 4 . -1 -1 4 . -1 -1 . . . . -1 4 -1 -1 4 -1 -1 -1 X.10 4 . -1 -1 -4 . 1 1 . . . . -1 4 -1 1 -4 1 1 -1 X.11 4 . -1 -1 4 . -1 -1 . . . . -1 -1 4 -1 -1 4 -1 -1 X.12 4 . -1 -1 -4 . 1 1 . . . . -1 -1 4 1 1 -4 1 -1 X.13 4 . A *A 4 . A *A . . . . B -1 -1 B -1 -1 *B *B X.14 4 . *A A 4 . *A A . . . . *B -1 -1 *B -1 -1 B B X.15 4 . B *B 4 . B *B . . . . *A -1 -1 *A -1 -1 A A X.16 4 . *B B 4 . *B B . . . . A -1 -1 A -1 -1 *A *A X.17 4 . A *A -4 . -A -*A . . . . B -1 -1 -B 1 1 -*B *B X.18 4 . *A A -4 . -*A -A . . . . *B -1 -1 -*B 1 1 -B B X.19 4 . B *B -4 . -B -*B . . . . *A -1 -1 -*A 1 1 -A A X.20 4 . *B B -4 . -*B -B . . . . A -1 -1 -A 1 1 -*A *A A = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 B = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 = (3+Sqrt(5))/2 = 2+b5 C = -E(4) = -Sqrt(-1) = -i |