Properties

Label 21T14
Order \(168\)
n \(21\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\PSL(2,7)$

Related objects

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Group action invariants

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

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: $\GL(3,2)$ x 2

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 24T284, 28T32, 42T37, 42T38 x 2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $21$ $2$ $( 5, 6)( 7,19)( 8,20)( 9,21)(11,12)(13,16)(14,17)(15,18)$
$ 4, 4, 4, 4, 2, 2, 1 $ $42$ $4$ $( 2, 3)( 4,10)( 5,11, 6,12)( 7,16,19,13)( 8,18,20,15)( 9,17,21,14)$
$ 3, 3, 3, 3, 3, 3, 3 $ $56$ $3$ $( 1, 2, 3)( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,16)(11,21,17)(12,20,18)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 5, 9,12,13,17,19)( 2, 4, 8,11,15,16,20)( 3, 6, 7,10,14,18,21)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 5,13,17,21,11, 7)( 2, 4,14,16,20,12, 9)( 3, 6,15,18,19,10, 8)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [168, 42]
Character table:   
     2  3  3  2  .  .  .
     3  1  .  .  1  .  .
     7  1  .  .  .  1  1

       1a 2a 4a 3a 7a 7b
    2P 1a 1a 2a 3a 7a 7b
    3P 1a 2a 4a 1a 7b 7a
    5P 1a 2a 4a 3a 7b 7a
    7P 1a 2a 4a 3a 1a 1a

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

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