Group action invariants
Degree $n$: | $10$ | |
Transitive number $t$: | $34$ | |
Group: | $C_2^4 : A_5$ | |
CHM label: | $[2^{4}]A(5)$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $2$ | |
Generators: | (2,4,10)(5,7,9), (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: $A_5$
Low degree siblings
16T1081, 20T172, 20T177, 30T214, 30T217, 40T888, 40T889, 40T932, 40T942, 40T944, 40T945Siblings 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, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 4, 9)( 5,10)$ |
$ 2, 2, 2, 2, 1, 1 $ | $5$ | $2$ | $( 1, 6)( 2, 7)( 4, 9)( 5,10)$ |
$ 2, 2, 2, 2, 1, 1 $ | $60$ | $2$ | $(1,2)(3,4)(6,7)(8,9)$ |
$ 4, 2, 2, 2 $ | $120$ | $4$ | $( 1, 2)( 3, 9, 8, 4)( 5,10)( 6, 7)$ |
$ 4, 4, 1, 1 $ | $60$ | $4$ | $(1,7,6,2)(3,9,8,4)$ |
$ 3, 3, 1, 1, 1, 1 $ | $80$ | $3$ | $(1,2,3)(6,7,8)$ |
$ 3, 3, 2, 2 $ | $80$ | $6$ | $( 1, 2, 3)( 4, 9)( 5,10)( 6, 7, 8)$ |
$ 6, 2, 1, 1 $ | $80$ | $6$ | $( 1, 7, 8, 6, 2, 3)( 5,10)$ |
$ 6, 2, 1, 1 $ | $80$ | $6$ | $(1,7,8,6,2,3)(4,9)$ |
$ 5, 5 $ | $192$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$ |
$ 5, 5 $ | $192$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)$ |
Group invariants
Order: | $960=2^{6} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
Label: | 960.11358 |
Character table: |
2 6 5 6 4 3 4 2 2 2 2 . . 3 1 1 1 . . . 1 1 1 1 . . 5 1 . . . . . . . . . 1 1 1a 2a 2b 2c 4a 4b 3a 6a 6b 6c 5a 5b 2P 1a 1a 1a 1a 2a 2b 3a 3a 3a 3a 5b 5a 3P 1a 2a 2b 2c 4a 4b 1a 2a 2b 2b 5b 5a 5P 1a 2a 2b 2c 4a 4b 3a 6a 6c 6b 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 3 3 3 -1 -1 -1 . . . . B *B X.3 3 3 3 -1 -1 -1 . . . . *B B X.4 4 4 4 . . . 1 1 1 1 -1 -1 X.5 5 5 5 1 1 1 -1 -1 -1 -1 . . X.6 5 1 -3 1 -1 1 2 -2 . . . . X.7 5 1 -3 1 -1 1 -1 1 A -A . . X.8 5 1 -3 1 -1 1 -1 1 -A A . . X.9 10 -2 2 -2 . 2 1 1 -1 -1 . . X.10 10 -2 2 2 . -2 1 1 -1 -1 . . X.11 15 3 -9 -1 1 -1 . . . . . . X.12 20 -4 4 . . . -1 -1 1 1 . . A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 B = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |