Properties

Label 28T25
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2:F_7$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $25$
Group :  $C_2^2:F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,13,15,5,28)(2,19,14,16,6,27)(3,10,7,17,24,22)(4,9,8,18,23,21)(11,25)(12,26), (1,16,2,15)(3,25,23,13,19,22,4,26,24,14,20,21)(5,7,17,12,9,27,6,8,18,11,10,28)
$|\Aut(F/K)|$:  $2$

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

28T21

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 11]
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 4a 14a 12a 12b  6e  6f 14b 7a 14c 2c
     2P 1a  3a  3b  3b  3a 1a 1a  3b  3a 2b  7a  6c  6d  3b  3a  7a 7a  7a 1a
     3P 1a  2a  2a  1a  1a 2a 2b  2b  2b 4a 14b  4a  4a  2c  2c 14a 7a 14c 2c
     5P 1a  6b  6a  3b  3a 2a 2b  6d  6c 4a 14b 12b 12a  6f  6e 14a 7a 14c 2c
     7P 1a  6a  6b  3a  3b 2a 2b  6c  6d 4a  2c 12a 12b  6e  6f  2c 1a  2b 2c
    11P 1a  6b  6a  3b  3a 2a 2b  6d  6c 4a 14a 12b 12a  6f  6e 14b 7a 14c 2c
    13P 1a  6a  6b  3a  3b 2a 2b  6c  6d 4a 14b 12a 12b  6e  6f 14a 7a 14c 2c

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(7)-E(7)^2+E(7)^3-E(7)^4+E(7)^5+E(7)^6
  = -Sqrt(-7) = -i7