Properties

Label 28T21
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 $t$ :  $21$
Group :  $C_2^2:F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,10)(2,11,9)(3,5,13)(4,6,14)(15,21,18,16,22,17)(19,23,26,20,24,25)(27,28), (1,19,8,24,5,18)(2,20,7,23,6,17)(3,26,12,22,13,27)(4,25,11,21,14,28)(9,16)(10,15)
$|\Aut(F/K)|$:  $2$

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: $F_7$

Low degree siblings

28T25

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 $ $2$ $2$ $(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
$ 6, 6, 3, 3, 3, 3, 2, 1, 1 $ $14$ $6$ $( 3, 5,10)( 4, 6, 9)( 7,14,11)( 8,13,12)(15,23,26,16,24,25)(17,27,20,18,28,19) (21,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3, 5,10)( 4, 6, 9)( 7,14,11)( 8,13,12)(15,24,26)(16,23,25)(17,28,20) (18,27,19)$
$ 6, 6, 3, 3, 3, 3, 2, 1, 1 $ $14$ $6$ $( 3,10, 5)( 4, 9, 6)( 7,11,14)( 8,12,13)(15,25,24,16,26,23)(17,19,28,18,20,27) (21,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,10, 5)( 4, 9, 6)( 7,11,14)( 8,12,13)(15,26,24)(16,25,23)(17,20,28) (18,19,27)$
$ 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 $ $7$ $6$ $( 1, 2)( 3, 6,10, 4, 5, 9)( 7,13,11, 8,14,12)(15,23,26,16,24,25) (17,27,20,18,28,19)(21,22)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3, 9, 5, 4,10, 6)( 7,12,14, 8,11,13)(15,25,24,16,26,23) (17,19,28,18,20,27)(21,22)$
$ 14, 7, 7 $ $6$ $14$ $( 1, 3, 5, 8,10,12,13)( 2, 4, 6, 7, 9,11,14)(15,17,19,21,24,25,27,16,18,20,22, 23,26,28)$
$ 7, 7, 7, 7 $ $6$ $7$ $( 1, 3, 5, 8,10,12,13)( 2, 4, 6, 7, 9,11,14)(15,18,19,22,24,26,27) (16,17,20,21,23,25,28)$
$ 14, 14 $ $6$ $14$ $( 1, 4, 5, 7,10,11,13, 2, 3, 6, 8, 9,12,14)(15,17,19,21,24,25,27,16,18,20,22, 23,26,28)$
$ 14, 7, 7 $ $6$ $14$ $( 1, 4, 5, 7,10,11,13, 2, 3, 6, 8, 9,12,14)(15,18,19,22,24,26,27) (16,17,20,21,23,25,28)$
$ 6, 6, 6, 6, 2, 2 $ $14$ $6$ $( 1,15, 5,27,13,24)( 2,16, 6,28,14,23)( 3,22,10,26, 8,19)( 4,21, 9,25, 7,20) (11,17)(12,18)$
$ 12, 12, 4 $ $14$ $12$ $( 1,15, 6,28,13,24, 2,16, 5,27,14,23)( 3,22, 9,25, 8,19, 4,21,10,26, 7,20) (11,17,12,18)$
$ 12, 12, 4 $ $14$ $12$ $( 1,15, 7,17, 3,26, 2,16, 8,18, 4,25)( 5,22, 9,28,12,24, 6,21,10,27,11,23) (13,19,14,20)$
$ 6, 6, 6, 6, 2, 2 $ $14$ $6$ $( 1,15, 8,18, 3,26)( 2,16, 7,17, 4,25)( 5,22,10,27,12,24)( 6,21, 9,28,11,23) (13,19)(14,20)$
$ 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:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 11]
Character table:   
      2  3  2   2   3   2   3  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 2a  6a  3a  6b  3b 2b  6c  6d 14a 7a 14b 14c  6e 12a 12b  6f 2c 4a
     2P 1a 1a  3b  3b  3a  3a 1a  3b  3a  7a 7a  7a  7a  3a  6c  6d  3b 1a 2b
     3P 1a 2a  2a  1a  2a  1a 2b  2b  2b 14c 7a 14b 14a  2c  4a  4a  2c 2c 4a
     5P 1a 2a  6b  3b  6a  3a 2b  6d  6c 14c 7a 14b 14a  6f 12b 12a  6e 2c 4a
     7P 1a 2a  6a  3a  6b  3b 2b  6c  6d  2a 1a  2b  2a  6e 12a 12b  6f 2c 4a
    11P 1a 2a  6b  3b  6a  3a 2b  6d  6c 14a 7a 14b 14c  6f 12b 12a  6e 2c 4a
    13P 1a 2a  6a  3a  6b  3b 2b  6c  6d 14c 7a 14b 14a  6e 12a 12b  6f 2c 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 -1   A  -A  /A -/A  1  -A -/A  -1  1   1  -1  /A -/A  -A   A -1  1
X.6      1 -1  /A -/A   A  -A  1 -/A  -A  -1  1   1  -1   A  -A -/A  /A -1  1
X.7      1 -1   A  -A  /A -/A  1  -A -/A  -1  1   1  -1 -/A  /A   A  -A  1 -1
X.8      1 -1  /A -/A   A  -A  1 -/A  -A  -1  1   1  -1  -A   A  /A -/A  1 -1
X.9      1  1 -/A -/A  -A  -A  1 -/A  -A   1  1   1   1   A   A  /A  /A -1 -1
X.10     1  1  -A  -A -/A -/A  1  -A -/A   1  1   1   1  /A  /A   A   A -1 -1
X.11     1  1 -/A -/A  -A  -A  1 -/A  -A   1  1   1   1  -A  -A -/A -/A  1  1
X.12     1  1  -A  -A -/A -/A  1  -A -/A   1  1   1   1 -/A -/A  -A  -A  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   .   .   .   .  6   .   .   1 -1  -1   1   .   .   .   .  .  .
X.17     6  6   .   .   .   .  6   .   .  -1 -1  -1  -1   .   .   .   .  .  .
X.18     6  .   .   .   .   . -6   .   .   C -1   1  -C   .   .   .   .  .  .
X.19     6  .   .   .   .   . -6   .   .  -C -1   1   C   .   .   .   .  .  .

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