Group action invariants
Degree $n$: | $10$ | |
Transitive number $t$: | $11$ | |
Group: | $A_5\times C_2$ | |
CHM label: | $A(5)[x]2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $2$ | |
Generators: | (2,4,10)(5,7,9), (1,6)(2,7)(3,8)(4,9)(5,10), (1,3,5,7,9)(2,4,6,8,10) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: $A_5$
Low degree siblings
12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 30T30, 40T61Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1, 1, 1 $ | $20$ | $3$ | $( 3, 5, 9)( 4, 8,10)$ |
$ 2, 2, 2, 2, 1, 1 $ | $15$ | $2$ | $( 2, 4)( 3, 5)( 7, 9)( 8,10)$ |
$ 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9)$ |
$ 10 $ | $12$ | $10$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ |
$ 6, 2, 2 $ | $20$ | $6$ | $( 1, 2, 3, 6, 7, 8)( 4, 9)( 5,10)$ |
$ 10 $ | $12$ | $10$ | $( 1, 2, 3,10, 9, 6, 7, 8, 5, 4)$ |
$ 5, 5 $ | $12$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ |
$ 5, 5 $ | $12$ | $5$ | $( 1, 3, 5, 9, 7)( 2, 6, 8,10, 4)$ |
$ 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
Label: | 120.35 |
Character table: |
2 3 1 3 3 1 1 1 1 1 3 3 1 1 . . . 1 . . . 1 5 1 . . . 1 . 1 1 1 1 1a 3a 2a 2b 10a 6a 10b 5a 5b 2c 2P 1a 3a 1a 1a 5a 3a 5b 5b 5a 1a 3P 1a 1a 2a 2b 10b 2c 10a 5b 5a 2c 5P 1a 3a 2a 2b 2c 6a 2c 1a 1a 2c 7P 1a 3a 2a 2b 10b 6a 10a 5b 5a 2c 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 3 . -1 -1 A . *A *A A 3 X.4 3 . -1 -1 *A . A A *A 3 X.5 3 . -1 1 -*A . -A A *A -3 X.6 3 . -1 1 -A . -*A *A A -3 X.7 4 1 . . -1 1 -1 -1 -1 4 X.8 4 1 . . 1 -1 1 -1 -1 -4 X.9 5 -1 1 1 . -1 . . . 5 X.10 5 -1 1 -1 . 1 . . . -5 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |