Group action invariants
Degree $n$: | $21$ | |
Transitive number $t$: | $27$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,16,19)(2,18,20,3,17,21)(4,7,13)(5,9,14,6,8,15)(11,12), (1,18,8)(2,16,9)(3,17,7)(4,6,5)(10,21,14)(11,19,15)(12,20,13) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $168$: $\GL(3,2)$ $336$: 14T17 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: $\GL(3,2)$
Low degree siblings
21T27, 24T2671, 42T169 x 2, 42T170 x 2, 42T171 x 2, 42T175 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $2$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$ |
$ 6, 6, 3, 3, 3 $ | $42$ | $6$ | $( 1, 2, 3)( 4,11, 6,10, 5,12)( 7,20, 9,19, 8,21)(13,14,15)(16,17,18)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $63$ | $2$ | $( 1, 3)( 4,12)( 5,11)( 6,10)( 7,21)( 8,20)( 9,19)(13,15)(16,18)$ |
$ 12, 6, 3 $ | $84$ | $12$ | $( 1, 3, 2)( 4,12, 5,10, 6,11)( 7,18,20,13, 9,17,19,15, 8,16,21,14)$ |
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $42$ | $4$ | $( 4,10)( 5,11)( 6,12)( 7,16,19,13)( 8,17,20,14)( 9,18,21,15)$ |
$ 4, 4, 4, 2, 2, 2, 2, 1 $ | $126$ | $4$ | $( 1, 2)( 4,11)( 5,10)( 6,12)( 7,17,19,14)( 8,16,20,13)( 9,18,21,15)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $112$ | $3$ | $( 1, 2, 3)( 4,17,21)( 5,18,19)( 6,16,20)( 7,11,15)( 8,12,13)( 9,10,14)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $56$ | $3$ | $( 4,16,19)( 5,17,20)( 6,18,21)( 7,10,13)( 8,11,14)( 9,12,15)$ |
$ 6, 6, 3, 3, 2, 1 $ | $168$ | $6$ | $( 1, 3)( 4,18,19, 6,16,21)( 5,17,20)( 7,12,13, 9,10,15)( 8,11,14)$ |
$ 7, 7, 7 $ | $24$ | $7$ | $( 1,16,19, 7,10,13, 4)( 2,17,20, 8,11,14, 5)( 3,18,21, 9,12,15, 6)$ |
$ 21 $ | $48$ | $21$ | $( 1,18,20, 7,12,14, 4, 3,17,19, 9,11,13, 6, 2,16,21, 8,10,15, 5)$ |
$ 14, 7 $ | $72$ | $14$ | $( 1,16,19, 7,10,13, 4)( 2,18,20, 9,11,15, 5, 3,17,21, 8,12,14, 6)$ |
$ 21 $ | $48$ | $21$ | $( 1,17,21,13, 5,12, 7, 2,18,19,14, 6,10, 8, 3,16,20,15, 4,11, 9)$ |
$ 7, 7, 7 $ | $24$ | $7$ | $( 1,16,19,13, 4,10, 7)( 2,17,20,14, 5,11, 8)( 3,18,21,15, 6,12, 9)$ |
$ 14, 7 $ | $72$ | $14$ | $( 1,18,19,15, 4,12, 7, 3,16,21,13, 6,10, 9)( 2,17,20,14, 5,11, 8)$ |
Group invariants
Order: | $1008=2^{4} \cdot 3^{2} \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
GAP id: | [1008, 883] |
Character table: |
2 4 3 4 3 4 4 2 3 3 . 1 1 1 . 1 . 1 1 3 2 2 1 1 1 . 1 1 . 2 2 1 1 1 . 1 1 . 7 1 1 1 . . . . . . . . . 1 1 1 1 1 1 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7a 21a 14a 21b 7b 14b 2P 1a 3a 1a 3a 1a 1a 6a 2b 2b 3b 3c 3c 7a 21a 7a 21b 7b 7b 3P 1a 1a 2a 2b 2b 2c 4a 4a 4b 1a 1a 2a 7b 7b 14b 7a 7a 14a 5P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a 7P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 1a 3a 2a 3a 1a 2a 11P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7a 21a 14a 21b 7b 14b 13P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a 17P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a 19P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 X.3 2 -1 . -1 2 . -1 2 . -1 2 . 2 -1 . -1 2 . X.4 3 3 -3 -1 -1 1 1 1 -1 . . . A A -A /A /A -/A X.5 3 3 -3 -1 -1 1 1 1 -1 . . . /A /A -/A A A -A X.6 3 3 3 -1 -1 -1 1 1 1 . . . A A A /A /A /A X.7 3 3 3 -1 -1 -1 1 1 1 . . . /A /A /A A A A X.8 6 6 6 2 2 2 . . . . . . -1 -1 -1 -1 -1 -1 X.9 6 6 -6 2 2 -2 . . . . . . -1 -1 1 -1 -1 1 X.10 6 -3 . 1 -2 . -1 2 . . . . B -A . -/A /B . X.11 6 -3 . 1 -2 . -1 2 . . . . /B -/A . -A B . X.12 7 7 7 -1 -1 -1 -1 -1 -1 1 1 1 . . . . . . X.13 7 7 -7 -1 -1 1 -1 -1 1 1 1 -1 . . . . . . X.14 8 8 8 . . . . . . -1 -1 -1 1 1 1 1 1 1 X.15 8 8 -8 . . . . . . -1 -1 1 1 1 -1 1 1 -1 X.16 12 -6 . -2 4 . . . . . . . -2 1 . 1 -2 . X.17 14 -7 . 1 -2 . 1 -2 . -1 2 . . . . . . . X.18 16 -8 . . . . . . . 1 -2 . 2 -1 . -1 2 . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 B = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7 |