Properties

Label 28T23
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{28}:C_3$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $23$
Group :  $D_{28}:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,22,16,18,8,2,3,21,15,17,7)(5,12,10,19,26,23,6,11,9,20,25,24)(13,28,14,27), (1,13,21,18,5,26)(2,14,22,17,6,25)(3,20,11,15,27,8)(4,19,12,16,28,7)(23,24)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $D_{4}$
12:  $C_6\times C_2$
24:  $D_4 \times C_3$
42:  $F_7$
84:  $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

Low degree siblings

28T23

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 9]
Character table:   
      2  3   2   2   3   3  2  3   3   3  2   2   2   2   2   2   2  2   2  2
      3  1   1   1   1   1  1  1   1   1  1   .   1   1   1   1   .  .   .  1
      7  1   .   .   .   .  .  1   .   .  .   1   .   .   .   .   1  1   1  1

        1a  6a  6b  3a  3b 2a 2b  6c  6d 2c 28a  6e  6f 12a 12b 28b 7a 14a 4a
     2P 1a  3a  3b  3b  3a 1a 1a  3b  3a 1a 14a  3a  3b  6d  6c 14a 7a  7a 2b
     3P 1a  2a  2a  1a  1a 2a 2b  2b  2b 2c 28a  2c  2c  4a  4a 28b 7a 14a 4a
     5P 1a  6b  6a  3b  3a 2a 2b  6d  6c 2c 28b  6f  6e 12b 12a 28a 7a 14a 4a
     7P 1a  6a  6b  3a  3b 2a 2b  6c  6d 2c  4a  6e  6f 12a 12b  4a 1a  2b 4a
    11P 1a  6b  6a  3b  3a 2a 2b  6d  6c 2c 28b  6f  6e 12b 12a 28a 7a 14a 4a
    13P 1a  6a  6b  3a  3b 2a 2b  6c  6d 2c 28b  6e  6f 12a 12b 28a 7a 14a 4a
    17P 1a  6b  6a  3b  3a 2a 2b  6d  6c 2c 28b  6f  6e 12b 12a 28a 7a 14a 4a
    19P 1a  6a  6b  3a  3b 2a 2b  6c  6d 2c 28a  6e  6f 12a 12b 28b 7a 14a 4a
    23P 1a  6b  6a  3b  3a 2a 2b  6d  6c 2c 28b  6f  6e 12b 12a 28a 7a 14a 4a

X.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
X.3      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
X.5      1   A  /A -/A  -A -1  1 -/A  -A -1   1   A  /A -/A  -A   1  1   1  1
X.6      1  /A   A  -A -/A -1  1  -A -/A -1   1  /A   A  -A -/A   1  1   1  1
X.7      1   A  /A -/A  -A -1  1 -/A  -A  1  -1  -A -/A  /A   A  -1  1   1 -1
X.8      1  /A   A  -A -/A -1  1  -A -/A  1  -1 -/A  -A   A  /A  -1  1   1 -1
X.9      1 -/A  -A  -A -/A  1  1  -A -/A -1  -1  /A   A   A  /A  -1  1   1 -1
X.10     1  -A -/A -/A  -A  1  1 -/A  -A -1  -1   A  /A  /A   A  -1  1   1 -1
X.11     1 -/A  -A  -A -/A  1  1  -A -/A  1   1 -/A  -A  -A -/A   1  1   1  1
X.12     1  -A -/A -/A  -A  1  1 -/A  -A  1   1  -A -/A -/A  -A   1  1   1  1
X.13     2   .   .   2   2  . -2  -2  -2  .   .   .   .   .   .   .  2  -2  .
X.14     2   .   .   B  /B  . -2  -B -/B  .   .   .   .   .   .   .  2  -2  .
X.15     2   .   .  /B   B  . -2 -/B  -B  .   .   .   .   .   .   .  2  -2  .
X.16     6   .   .   .   .  .  6   .   .  .  -1   .   .   .   .  -1 -1  -1  6
X.17     6   .   .   .   .  .  6   .   .  .   1   .   .   .   .   1 -1  -1 -6
X.18     6   .   .   .   .  . -6   .   .  .   C   .   .   .   .  -C -1   1  .
X.19     6   .   .   .   .  . -6   .   .  .  -C   .   .   .   .   C -1   1  .

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
C = -E(28)^3+E(28)^11+E(28)^15-E(28)^19+E(28)^23-E(28)^27
  = -Sqrt(7) = -r7