Group action invariants
Degree $n$: | $10$ | |
Transitive number $t$: | $40$ | |
Group: | $A_5 \wr C_2$ | |
CHM label: | $[A(5)^{2}]2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $1$ | |
Generators: | (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,4,10) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: None
Low degree siblings
12T269, 20T363, 24T9631, 25T88, 30T652, 36T7075, 40T5410Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1 $ | $225$ | $2$ | $( 1, 3)( 2,10)( 5, 7)( 6, 8)$ |
$ 3, 3, 1, 1, 1, 1 $ | $400$ | $3$ | $( 1, 3, 5)( 6, 8,10)$ |
$ 5, 5 $ | $144$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ |
$ 5, 5 $ | $144$ | $5$ | $( 1, 3, 5, 9, 7)( 2, 6, 8,10, 4)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1 $ | $30$ | $2$ | $( 2,10)( 6, 8)$ |
$ 3, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $( 6, 8,10)$ |
$ 5, 1, 1, 1, 1, 1 $ | $24$ | $5$ | $( 2, 4, 6, 8,10)$ |
$ 5, 1, 1, 1, 1, 1 $ | $24$ | $5$ | $( 2, 6, 8,10, 4)$ |
$ 3, 2, 2, 1, 1, 1 $ | $600$ | $6$ | $( 1, 3)( 5, 7)( 6, 8,10)$ |
$ 5, 2, 2, 1 $ | $360$ | $10$ | $( 1, 3)( 2, 4, 6, 8,10)( 5, 7)$ |
$ 5, 2, 2, 1 $ | $360$ | $10$ | $( 1, 3)( 2, 6, 8,10, 4)( 5, 7)$ |
$ 5, 3, 1, 1 $ | $480$ | $15$ | $( 1, 3, 5)( 2, 4, 6, 8,10)$ |
$ 5, 3, 1, 1 $ | $480$ | $15$ | $( 1, 3, 5)( 2, 6, 8,10, 4)$ |
$ 5, 5 $ | $288$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 6, 8,10, 4)$ |
$ 2, 2, 2, 2, 2 $ | $60$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
$ 10 $ | $720$ | $10$ | $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$ |
$ 10 $ | $720$ | $10$ | $( 1, 8, 3,10, 5, 4, 9, 2, 7, 6)$ |
$ 4, 4, 2 $ | $900$ | $4$ | $( 1, 8, 3, 6)( 2, 7,10, 5)( 4, 9)$ |
$ 6, 2, 2 $ | $1200$ | $6$ | $( 1, 8, 3,10, 5, 6)( 2, 7)( 4, 9)$ |
Group invariants
Order: | $7200=2^{5} \cdot 3^{2} \cdot 5^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
Label: | not available |
Character table: |
2 5 5 1 1 1 4 2 2 2 2 2 2 . . . 3 1 1 3 1 3 2 . 2 . . 1 2 1 1 1 . . 1 1 . 1 . . . 1 5 2 . . 2 2 1 1 2 2 . 1 1 1 1 2 1 1 1 . . 1a 2a 3a 5a 5b 2b 3b 5c 5d 6a 10a 10b 15a 15b 5e 2c 10c 10d 4a 6b 2P 1a 1a 3a 5b 5a 1a 3b 5d 5c 3b 5d 5c 15b 15a 5e 1a 5a 5b 2a 3a 3P 1a 2a 1a 5b 5a 2b 1a 5d 5c 2b 10b 10a 5d 5c 5e 2c 10d 10c 4a 2c 5P 1a 2a 3a 1a 1a 2b 3b 1a 1a 6a 2b 2b 3b 3b 1a 2c 2c 2c 4a 6b 7P 1a 2a 3a 5b 5a 2b 3b 5d 5c 6a 10b 10a 15b 15a 5e 2c 10d 10c 4a 6b 11P 1a 2a 3a 5a 5b 2b 3b 5c 5d 6a 10a 10b 15a 15b 5e 2c 10c 10d 4a 6b 13P 1a 2a 3a 5b 5a 2b 3b 5d 5c 6a 10b 10a 15b 15a 5e 2c 10d 10c 4a 6b 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 6 -2 . A *A 2 3 C *C -1 G *G -*G -G 1 . . . . . X.4 6 -2 . *A A 2 3 *C C -1 *G G -G -*G 1 . . . . . X.5 8 . 2 -2 -2 4 5 3 3 1 -1 -1 . . -2 . . . . . X.6 9 1 . B *B -3 . D *D . *G G . . -1 -3 *G G 1 . X.7 9 1 . *B B -3 . *D D . G *G . . -1 -3 G *G 1 . X.8 9 1 . B *B -3 . D *D . *G G . . -1 3 -*G -G -1 . X.9 9 1 . *B B -3 . *D D . G *G . . -1 3 -G -*G -1 . X.10 10 2 -2 . . 6 4 5 5 . 1 1 -1 -1 . . . . . . X.11 16 . 1 1 1 . 4 -4 -4 . . . -1 -1 1 -4 1 1 . -1 X.12 16 . 1 1 1 . 4 -4 -4 . . . -1 -1 1 4 -1 -1 . 1 X.13 18 2 . -2 -2 -6 . 3 3 . -1 -1 . . 3 . . . . . X.14 24 . . -*A -A -4 3 E *E -1 1 1 -G -*G -1 . . . . . X.15 24 . . -A -*A -4 3 *E E -1 1 1 -*G -G -1 . . . . . X.16 25 1 1 . . 5 -5 . . -1 . . . . . -5 . . -1 1 X.17 25 1 1 . . 5 -5 . . -1 . . . . . 5 . . 1 -1 X.18 30 -2 . . . -2 -3 F *F 1 -*G -G *G G . . . . . . X.19 30 -2 . . . -2 -3 *F F 1 -G -*G G *G . . . . . . X.20 40 . -2 . . 4 1 -5 -5 1 -1 -1 1 1 . . . . . . A = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 B = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 = (3-Sqrt(5))/2 = 1-b5 C = -4*E(5)-3*E(5)^2-3*E(5)^3-4*E(5)^4 = (7-Sqrt(5))/2 = 3-b5 D = -3*E(5)-3*E(5)^4 = (3-3*Sqrt(5))/2 = -3b5 E = 3*E(5)-E(5)^2-E(5)^3+3*E(5)^4 = -1+2*Sqrt(5) = 1+4b5 F = -5*E(5)-5*E(5)^4 = (5-5*Sqrt(5))/2 = -5b5 G = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 |