Group action invariants
Degree $n$: | $28$ | |
Transitive number $t$: | $4$ | |
Group: | $D_{14}$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $28$ | |
Generators: | (1,25)(2,26)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,28), (1,4,6,8,10,12,14,16,18,20,22,24,26,27)(2,3,5,7,9,11,13,15,17,19,21,23,25,28) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $14$: $D_{7}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 7: $D_{7}$
Low degree siblings
14T3 x 2Siblings 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, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,20)(12,19) (13,18)(14,17)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 3)( 2, 4)( 5,27)( 6,28)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21)(13,20) (14,19)(15,18)(16,17)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1, 4, 6, 8,10,12,14,16,18,20,22,24,26,27)( 2, 3, 5, 7, 9,11,13,15,17,19,21, 23,25,28)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 6,10,14,18,22,26)( 2, 5, 9,13,17,21,25)( 3, 7,11,15,19,23,28) ( 4, 8,12,16,20,24,27)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1, 8,14,20,26, 4,10,16,22,27, 6,12,18,24)( 2, 7,13,19,25, 3, 9,15,21,28, 5, 11,17,23)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,10,18,26, 6,14,22)( 2, 9,17,25, 5,13,21)( 3,11,19,28, 7,15,23) ( 4,12,20,27, 8,16,24)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1,12,22, 4,14,24, 6,16,26, 8,18,27,10,20)( 2,11,21, 3,13,23, 5,15,25, 7,17, 28, 9,19)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,14,26,10,22, 6,18)( 2,13,25, 9,21, 5,17)( 3,15,28,11,23, 7,19) ( 4,16,27,12,24, 8,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,16)( 2,15)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$ |
Group invariants
Order: | $28=2^{2} \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [28, 3] |
Character table: |
2 2 2 2 1 1 1 1 1 1 2 7 1 . . 1 1 1 1 1 1 1 1a 2a 2b 14a 7a 14b 7b 14c 7c 2c 2P 1a 1a 1a 7a 7b 7c 7c 7b 7a 1a 3P 1a 2a 2b 14b 7c 14c 7a 14a 7b 2c 5P 1a 2a 2b 14c 7b 14a 7c 14b 7a 2c 7P 1a 2a 2b 2c 1a 2c 1a 2c 1a 2c 11P 1a 2a 2b 14b 7c 14c 7a 14a 7b 2c 13P 1a 2a 2b 14a 7a 14b 7b 14c 7c 2c X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 -1 1 -1 1 -1 X.5 2 . . A -B C -C B -A -2 X.6 2 . . B -C A -A C -B -2 X.7 2 . . C -A B -B A -C -2 X.8 2 . . -C -A -B -B -A -C 2 X.9 2 . . -B -C -A -A -C -B 2 X.10 2 . . -A -B -C -C -B -A 2 A = -E(7)-E(7)^6 B = -E(7)^2-E(7)^5 C = -E(7)^3-E(7)^4 |