Group action invariants
Degree $n$: | $28$ | |
Transitive number $t$: | $8$ | |
Group: | $C_4\times D_7$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $4$ | |
Generators: | (1,27,2,28)(3,25,4,26)(5,24,6,23)(7,22,8,21)(9,19,10,20)(11,17,12,18)(13,15,14,16), (1,10)(2,9)(3,8)(4,7)(5,6)(11,27)(12,28)(13,25)(14,26)(15,24)(16,23)(17,21)(18,22)(19,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $14$: $D_{7}$ $28$: $D_{14}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 7: $D_{7}$
Degree 14: $D_{14}$
Low degree siblings
28T8Siblings 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, 1, 1, 1, 1 $ | $7$ | $2$ | $( 3,28)( 4,27)( 5,25)( 6,26)( 7,23)( 8,24)( 9,22)(10,21)(11,20)(12,19)(13,18) (14,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,21)(10,22)(11,19)(12,20) (13,17)(14,18)(15,16)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $7$ | $4$ | $( 1, 3, 2, 4)( 5,28, 6,27)( 7,26, 8,25)( 9,23,10,24)(11,21,12,22)(13,19,14,20) (15,18,16,17)$ |
$ 28 $ | $2$ | $28$ | $( 1, 3, 6, 8, 9,11,13,16,17,20,21,24,25,28, 2, 4, 5, 7,10,12,14,15,18,19,22, 23,26,27)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $7$ | $4$ | $( 1, 4, 2, 3)( 5,27, 6,28)( 7,25, 8,26)( 9,24,10,23)(11,22,12,21)(13,20,14,19) (15,17,16,18)$ |
$ 28 $ | $2$ | $28$ | $( 1, 4, 6, 7, 9,12,13,15,17,19,21,23,25,27, 2, 3, 5, 8,10,11,14,16,18,20,22, 24,26,28)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 5, 9,14,17,22,25)( 2, 6,10,13,18,21,26)( 3, 7,11,15,20,23,28) ( 4, 8,12,16,19,24,27)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1, 6, 9,13,17,21,25, 2, 5,10,14,18,22,26)( 3, 8,11,16,20,24,28, 4, 7,12,15, 19,23,27)$ |
$ 28 $ | $2$ | $28$ | $( 1, 7,13,19,25, 3,10,16,22,28, 6,12,17,23, 2, 8,14,20,26, 4, 9,15,21,27, 5, 11,18,24)$ |
$ 28 $ | $2$ | $28$ | $( 1, 8,13,20,25, 4,10,15,22,27, 6,11,17,24, 2, 7,14,19,26, 3, 9,16,21,28, 5, 12,18,23)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 9,17,25, 5,14,22)( 2,10,18,26, 6,13,21)( 3,11,20,28, 7,15,23) ( 4,12,19,27, 8,16,24)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1,10,17,26, 5,13,22, 2, 9,18,25, 6,14,21)( 3,12,20,27, 7,16,23, 4,11,19,28, 8,15,24)$ |
$ 28 $ | $2$ | $28$ | $( 1,11,21, 4,14,23, 6,16,25, 7,18,27, 9,20, 2,12,22, 3,13,24, 5,15,26, 8,17, 28,10,19)$ |
$ 28 $ | $2$ | $28$ | $( 1,12,21, 3,14,24, 6,15,25, 8,18,28, 9,19, 2,11,22, 4,13,23, 5,16,26, 7,17, 27,10,20)$ |
$ 14, 14 $ | $2$ | $14$ | $( 1,13,25,10,22, 6,17, 2,14,26, 9,21, 5,18)( 3,16,28,12,23, 8,20, 4,15,27,11, 24, 7,19)$ |
$ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,14,25, 9,22, 5,17)( 2,13,26,10,21, 6,18)( 3,15,28,11,23, 7,20) ( 4,16,27,12,24, 8,19)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,27,14,28)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,16, 2,15)( 3,17, 4,18)( 5,19, 6,20)( 7,22, 8,21)( 9,24,10,23)(11,25,12,26) (13,28,14,27)$ |
Group invariants
Order: | $56=2^{3} \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [56, 4] |
Character table: |
2 3 3 3 3 3 2 3 2 2 2 2 2 2 2 2 2 2 2 3 3 7 1 . 1 . . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 4a 28a 4b 28b 7a 14a 28c 28d 7b 14b 28e 28f 14c 7c 4c 4d 2P 1a 1a 1a 1a 2b 14a 2b 14a 7b 7b 14c 14c 7c 7c 14b 14b 7a 7a 2b 2b 3P 1a 2a 2b 2c 4b 28d 4a 28c 7c 14c 28f 28e 7a 14a 28b 28a 14b 7b 4d 4c 5P 1a 2a 2b 2c 4a 28e 4b 28f 7b 14b 28a 28b 7c 14c 28c 28d 14a 7a 4c 4d 7P 1a 2a 2b 2c 4b 4d 4a 4c 1a 2b 4d 4c 1a 2b 4d 4c 2b 1a 4d 4c 11P 1a 2a 2b 2c 4b 28d 4a 28c 7c 14c 28f 28e 7a 14a 28b 28a 14b 7b 4d 4c 13P 1a 2a 2b 2c 4a 28a 4b 28b 7a 14a 28c 28d 7b 14b 28e 28f 14c 7c 4c 4d 17P 1a 2a 2b 2c 4a 28c 4b 28d 7c 14c 28e 28f 7a 14a 28a 28b 14b 7b 4c 4d 19P 1a 2a 2b 2c 4b 28f 4a 28e 7b 14b 28b 28a 7c 14c 28d 28c 14a 7a 4d 4c 23P 1a 2a 2b 2c 4b 28f 4a 28e 7b 14b 28b 28a 7c 14c 28d 28c 14a 7a 4d 4c X.1 1 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 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.4 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.5 1 -1 -1 1 A -A -A A 1 -1 -A A 1 -1 -A A -1 1 -A A X.6 1 -1 -1 1 -A A A -A 1 -1 A -A 1 -1 A -A -1 1 A -A X.7 1 1 -1 -1 A A -A -A 1 -1 A -A 1 -1 A -A -1 1 A -A X.8 1 1 -1 -1 -A -A A A 1 -1 -A A 1 -1 -A A -1 1 -A A X.9 2 . -2 . . B . -B -F F D -D -G G C -C E -E H -H X.10 2 . -2 . . C . -C -G G B -B -E E D -D F -F H -H X.11 2 . -2 . . D . -D -E E C -C -F F B -B G -G H -H X.12 2 . -2 . . -D . D -E E -C C -F F -B B G -G -H H X.13 2 . -2 . . -C . C -G G -B B -E E -D D F -F -H H X.14 2 . -2 . . -B . B -F F -D D -G G -C C E -E -H H X.15 2 . 2 . . E . E -F -F G G -G -G F F -E -E -2 -2 X.16 2 . 2 . . F . F -G -G E E -E -E G G -F -F -2 -2 X.17 2 . 2 . . G . G -E -E F F -F -F E E -G -G -2 -2 X.18 2 . 2 . . -G . -G -E -E -F -F -F -F -E -E -G -G 2 2 X.19 2 . 2 . . -F . -F -G -G -E -E -E -E -G -G -F -F 2 2 X.20 2 . 2 . . -E . -E -F -F -G -G -G -G -F -F -E -E 2 2 A = -E(4) = -Sqrt(-1) = -i B = -E(28)^3-E(28)^11 C = -E(28)^15-E(28)^27 D = -E(28)^19-E(28)^23 E = -E(7)-E(7)^6 F = -E(7)^2-E(7)^5 G = -E(7)^3-E(7)^4 H = -2*E(4) = -2*Sqrt(-1) = -2i |