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Properties

Label 28T8
Order \(56\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times D_7$

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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)
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)
$|\Aut(F/K)|$:  $4$

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

28T8

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, 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