Properties

Label 28T6
Order \(56\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{14}:C_2$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $6$
Group :  $D_{14}:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24)(2,23)(3,22)(4,21)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,28)(12,27)(13,26)(14,25), (1,10,3,12,5,13,8)(2,9,4,11,6,14,7)(15,23,18,25,19,28,22,16,24,17,26,20,27,21)
$|\Aut(F/K)|$:  $14$

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: $D_{7}$

Low degree siblings

28T7

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

Group invariants

Order:  $56=2^{3} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [56, 7]
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 14a 7a 14b 14c 7b 14d 14e 14f 14g 14h 14i 7c 2c 4a
     2P 1a 1a 1a  7b 7b  7b  7b 7c  7c  7c  7c  7a  7a  7a 7a 1a 2b
     3P 1a 2a 2b 14i 7c 14g 14h 7a 14c 14a 14b 14f 14e 14d 7b 2c 4a
     5P 1a 2a 2b 14e 7b 14f 14d 7c 14i 14h 14g 14b 14c 14a 7a 2c 4a
     7P 1a 2a 2b  2a 1a  2b  2a 1a  2a  2a  2b  2b  2a  2a 1a 2c 4a
    11P 1a 2a 2b 14h 7c 14g 14i 7a 14a 14c 14b 14f 14d 14e 7b 2c 4a
    13P 1a 2a 2b 14c 7a 14b 14a 7b 14e 14d 14f 14g 14i 14h 7c 2c 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  2   A -A  -A   A -B   B   B  -B  -C   C   C -C  .  .
X.7      2 -2  2   B -B  -B   B -C   C   C  -C  -A   A   A -A  .  .
X.8      2 -2  2   C -C  -C   C -A   A   A  -A  -B   B   B -B  .  .
X.9      2  2  2  -A -A  -A  -A -B  -B  -B  -B  -C  -C  -C -C  .  .
X.10     2  2  2  -B -B  -B  -B -C  -C  -C  -C  -A  -A  -A -A  .  .
X.11     2  2  2  -C -C  -C  -C -A  -A  -A  -A  -B  -B  -B -B  .  .
X.12     2  . -2   D -A   A  -D -B   E  -E   B   C  -F   F -C  .  .
X.13     2  . -2   E -B   B  -E -C  -F   F   C   A   D  -D -A  .  .
X.14     2  . -2   F -C   C  -F -A  -D   D   A   B  -E   E -B  .  .
X.15     2  . -2  -F -C   C   F -A   D  -D   A   B   E  -E -B  .  .
X.16     2  . -2  -E -B   B   E -C   F  -F   C   A  -D   D -A  .  .
X.17     2  . -2  -D -A   A   D -B  -E   E   B   C   F  -F -C  .  .

A = -E(7)-E(7)^6
B = -E(7)^2-E(7)^5
C = -E(7)^3-E(7)^4
D = -E(7)+E(7)^6
E = -E(7)^2+E(7)^5
F = -E(7)^3+E(7)^4