Group action invariants
Degree $n$: | $21$ | |
Transitive number $t$: | $4$ | |
Group: | $F_7$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $3$ | |
Generators: | (1,9,5,18,10,14)(2,8,6,16,12,13)(3,7,4,17,11,15)(19,21,20), (1,4,8,12,15,18,19)(2,5,7,11,14,16,20)(3,6,9,10,13,17,21) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: $F_7$
Low degree siblings
7T4, 14T4, 42T4Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $7$ | $2$ | $( 4,19)( 5,20)( 6,21)( 7,16)( 8,18)( 9,17)(10,13)(11,14)(12,15)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 2, 3)( 4, 7,13)( 5, 9,15)( 6, 8,14)(10,19,16)(11,21,18)(12,20,17)$ |
$ 6, 6, 6, 3 $ | $7$ | $6$ | $( 1, 2, 3)( 4,16,13,19, 7,10)( 5,17,15,20, 9,12)( 6,18,14,21, 8,11)$ |
$ 6, 6, 6, 3 $ | $7$ | $6$ | $( 1, 3, 2)( 4,10, 7,19,13,16)( 5,12, 9,20,15,17)( 6,11, 8,21,14,18)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 2)( 4,13, 7)( 5,15, 9)( 6,14, 8)(10,16,19)(11,18,21)(12,17,20)$ |
$ 7, 7, 7 $ | $6$ | $7$ | $( 1, 4, 8,12,15,18,19)( 2, 5, 7,11,14,16,20)( 3, 6, 9,10,13,17,21)$ |
Group invariants
Order: | $42=2 \cdot 3 \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [42, 1] |
Character table: |
2 1 1 1 1 1 1 . 3 1 1 1 1 1 1 . 7 1 . . . . . 1 1a 2a 3a 6a 6b 3b 7a 2P 1a 1a 3b 3b 3a 3a 7a 3P 1a 2a 1a 2a 2a 1a 7a 5P 1a 2a 3b 6b 6a 3a 7a 7P 1a 2a 3a 6a 6b 3b 1a X.1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 1 X.3 1 -1 A -A -/A /A 1 X.4 1 -1 /A -/A -A A 1 X.5 1 1 A A /A /A 1 X.6 1 1 /A /A A A 1 X.7 6 . . . . . -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 |