Properties

Label 28T4
Order \(28\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{14}$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $4$
Group :  $D_{14}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $28$

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: $D_{7}$

Degree 14: $D_{7}$, $D_{14}$ x 2

Low degree siblings

14T3 x 2

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, 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