Group action invariants
Degree $n$: | $21$ | |
Transitive number $t$: | $33$ | |
Group: | $A_7$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (4,6,5)(9,11,10)(13,15,14)(16,18,17)(19,20,21) |
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 7: None
Low degree siblings
7T6, 15T47 x 2, 35T28, 42T294, 42T299Siblings 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 $ | $1$ | $1$ | $()$ |
$ 7, 7, 7 $ | $360$ | $7$ | $( 1, 8,12,13,19,21, 6)( 2, 9,17,15, 4,10,18)( 3, 7,16,14,20, 5,11)$ |
$ 7, 7, 7 $ | $360$ | $7$ | $( 1, 6,21,19,13,12, 8)( 2,18,10, 4,15,17, 9)( 3,11, 5,20,14,16, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1, 7)( 3,15)( 4,13)( 5,14)( 6,12)( 8,11)(16,20)(17,21)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $280$ | $3$ | $( 1,19,15)( 2,10,20)( 3,17,18)( 4,14,11)( 5,21, 6)( 7, 9,13)( 8,16,12)$ |
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $( 1, 2, 4)( 7,13, 9)( 8,12,16)(10,14,19)(11,15,20)$ |
$ 6, 6, 3, 2, 2, 1, 1 $ | $210$ | $6$ | $( 1,19, 2,10, 4,14)( 3,21)( 6,17)( 7, 9,13)( 8,20,12,11,16,15)$ |
$ 5, 5, 5, 5, 1 $ | $504$ | $5$ | $( 2, 3, 5, 4, 6)( 7, 8,10, 9,11)(12,17,19,20,15)(13,18,14,16,21)$ |
$ 4, 4, 4, 4, 2, 2, 1 $ | $630$ | $4$ | $( 1, 3, 4, 6)( 2, 5)( 7,17,13,21)( 8,16,20,11)( 9,18)(10,12,19,15)$ |
Group invariants
Order: | $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
GAP id: | not available |
Character table: |
2 3 . . 3 2 2 2 . . 3 2 . . 1 . 2 1 2 . 5 1 . . . . . . . 1 7 1 1 1 . . . . . . 1a 7a 7b 2a 4a 3a 6a 3b 5a 2P 1a 7a 7b 1a 2a 3a 3a 3b 5a 3P 1a 7b 7a 2a 4a 1a 2a 1a 5a 5P 1a 7b 7a 2a 4a 3a 6a 3b 1a 7P 1a 1a 1a 2a 4a 3a 6a 3b 5a X.1 1 1 1 1 1 1 1 1 1 X.2 6 -1 -1 2 . 3 -1 . 1 X.3 10 A /A -2 . 1 1 1 . X.4 10 /A A -2 . 1 1 1 . X.5 14 . . 2 . 2 2 -1 -1 X.6 14 . . 2 . -1 -1 2 -1 X.7 15 1 1 -1 -1 3 -1 . . X.8 21 . . 1 -1 -3 1 . 1 X.9 35 . . -1 1 -1 -1 -1 . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |