Group action invariants
Degree $n$: | $21$ | |
Transitive number $t$: | $2$ | |
Group: | $C_7:C_3$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $21$ | |
Generators: | (1,5,10)(2,6,11)(3,4,12)(7,16,13)(8,17,14)(9,18,15)(19,20,21), (1,4,7,11,14,18,19)(2,5,8,12,15,16,20)(3,6,9,10,13,17,21) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: $C_7:C_3$
Low degree siblings
7T3Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 2, 3)( 4, 8,13)( 5, 9,14)( 6, 7,15)(10,19,16)(11,20,17)(12,21,18)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 2)( 4,13, 8)( 5,14, 9)( 6,15, 7)(10,16,19)(11,17,20)(12,18,21)$ |
$ 7, 7, 7 $ | $3$ | $7$ | $( 1, 4, 7,11,14,18,19)( 2, 5, 8,12,15,16,20)( 3, 6, 9,10,13,17,21)$ |
$ 7, 7, 7 $ | $3$ | $7$ | $( 1,11,19, 7,18, 4,14)( 2,12,20, 8,16, 5,15)( 3,10,21, 9,17, 6,13)$ |
Group invariants
Order: | $21=3 \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [21, 1] |
Character table: |
3 1 1 1 . . 7 1 . . 1 1 1a 3a 3b 7a 7b 2P 1a 3b 3a 7a 7b 3P 1a 1a 1a 7b 7a 5P 1a 3b 3a 7b 7a 7P 1a 3a 3b 1a 1a X.1 1 1 1 1 1 X.2 1 A /A 1 1 X.3 1 /A A 1 1 X.4 3 . . B /B X.5 3 . . /B B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)+E(7)^2+E(7)^4 = (-1+Sqrt(-7))/2 = b7 |