Properties

Label 28T27
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_8:C_3$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $27$
Group :  $F_8:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,8,3,12,5)(2,10,7,4,11,6)(13,22,26,14,23,28)(15,21,27,16,24,25)(17,20,18), (1,27,10,5,20,22,13)(2,25,11,7,19,21,16)(3,28,12,6,18,23,15)(4,26,9,8,17,24,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7:C_3$

Degree 14: None

Low degree siblings

8T36, 14T11, 24T283, 42T26

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 43]
Character table:   
     2  3  3   1  1  1   1  .  .
     3  1  1   1  1  1   1  .  .
     7  1  .   .  .  .   .  1  1

       1a 2a  6a 3a 3b  6b 7a 7b
    2P 1a 1a  3b 3b 3a  3a 7a 7b
    3P 1a 2a  2a 1a 1a  2a 7b 7a
    5P 1a 2a  6b 3b 3a  6a 7b 7a
    7P 1a 2a  6a 3a 3b  6b 1a 1a

X.1     1  1   1  1  1   1  1  1
X.2     1  1   A  A /A  /A  1  1
X.3     1  1  /A /A  A   A  1  1
X.4     3  3   .  .  .   .  B /B
X.5     3  3   .  .  .   . /B  B
X.6     7 -1  -1  1  1  -1  .  .
X.7     7 -1  -A  A /A -/A  .  .
X.8     7 -1 -/A /A  A  -A  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7