Properties

Label 28T10
Order \(56\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{28}$

Learn more about

Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $10$
Group :  $D_{28}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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), (1,28)(2,27)(3,25)(4,26)(5,23)(6,24)(7,22)(8,21)(9,20)(10,19)(11,17)(12,18)(13,16)(14,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
14:  $D_{7}$
28:  $D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Low degree siblings

28T10

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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, 1, 1 $ $14$ $2$ $( 3,27)( 4,28)( 5,25)( 6,26)( 7,24)( 8,23)( 9,22)(10,21)(11,19)(12,20)(13,18) (14,17)(15,16)$
$ 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 $ $14$ $2$ $( 1, 3)( 2, 4)( 5,28)( 6,27)( 7,25)( 8,26)( 9,23)(10,24)(11,22)(12,21)(13,19) (14,20)(15,17)(16,18)$
$ 28 $ $2$ $28$ $( 1, 3, 6, 8, 9,11,13,16,17,20,21,24,25,28, 2, 4, 5, 7,10,12,14,15,18,19,22, 23,26,27)$
$ 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 $ $2$ $4$ $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,27,14,28)$

Group invariants

Order:  $56=2^{3} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [56, 5]
Character table:   
      2  3  2  3  2   2   2  2   2   2   2  2   2   2   2   2  2  2
      7  1  .  1  .   1   1  1   1   1   1  1   1   1   1   1  1  1

        1a 2a 2b 2c 28a 28b 7a 14a 28c 28d 7b 14b 28e 28f 14c 7c 4a
     2P 1a 1a 1a 1a 14a 14a 7b  7b 14c 14c 7c  7c 14b 14b  7a 7a 2b
     3P 1a 2a 2b 2c 28d 28c 7c 14c 28e 28f 7a 14a 28b 28a 14b 7b 4a
     5P 1a 2a 2b 2c 28e 28f 7b 14b 28a 28b 7c 14c 28d 28c 14a 7a 4a
     7P 1a 2a 2b 2c  4a  4a 1a  2b  4a  4a 1a  2b  4a  4a  2b 1a 4a
    11P 1a 2a 2b 2c 28c 28d 7c 14c 28f 28e 7a 14a 28a 28b 14b 7b 4a
    13P 1a 2a 2b 2c 28b 28a 7a 14a 28d 28c 7b 14b 28f 28e 14c 7c 4a
    17P 1a 2a 2b 2c 28c 28d 7c 14c 28f 28e 7a 14a 28a 28b 14b 7b 4a
    19P 1a 2a 2b 2c 28f 28e 7b 14b 28b 28a 7c 14c 28c 28d 14a 7a 4a
    23P 1a 2a 2b 2c 28e 28f 7b 14b 28a 28b 7c 14c 28d 28c 14a 7a 4a

X.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
X.3      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
X.5      2  . -2  .   .   .  2  -2   .   .  2  -2   .   .  -2  2  .
X.6      2  .  2  .   A   A -B  -B   C   C -C  -C   B   B  -A -A -2
X.7      2  .  2  .   B   B -C  -C   A   A -A  -A   C   C  -B -B -2
X.8      2  .  2  .   C   C -A  -A   B   B -B  -B   A   A  -C -C -2
X.9      2  .  2  .  -A  -A -B  -B  -C  -C -C  -C  -B  -B  -A -A  2
X.10     2  .  2  .  -B  -B -C  -C  -A  -A -A  -A  -C  -C  -B -B  2
X.11     2  .  2  .  -C  -C -A  -A  -B  -B -B  -B  -A  -A  -C -C  2
X.12     2  . -2  .   D  -D -C   C  -F   F -A   A   E  -E   B -B  .
X.13     2  . -2  .   E  -E -A   A   D  -D -B   B   F  -F   C -C  .
X.14     2  . -2  .   F  -F -B   B   E  -E -C   C  -D   D   A -A  .
X.15     2  . -2  .  -F   F -B   B  -E   E -C   C   D  -D   A -A  .
X.16     2  . -2  .  -E   E -A   A  -D   D -B   B  -F   F   C -C  .
X.17     2  . -2  .  -D   D -C   C   F  -F -A   A  -E   E   B -B  .

A = -E(7)^3-E(7)^4
B = -E(7)-E(7)^6
C = -E(7)^2-E(7)^5
D = -E(28)^3+E(28)^11
E = -E(28)^15+E(28)^27
F = -E(28)^19+E(28)^23