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Properties

Label 24T284
Order \(168\)
n \(24\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\PSL(2,7)$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $284$
Group :  $\PSL(2,7)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,24,21,17,10,13)(2,9,22,19,18,11,14)(3,7,23,20,16,12,15), (1,7,12,13,22,5,18)(2,8,10,14,23,6,16)(3,9,11,15,24,4,17)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 8: $\PSL(2,7)$

Degree 12: None

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 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, 1 $ $1$ $1$ $()$
$ 7, 7, 7, 1, 1, 1 $ $24$ $7$ $( 4, 8,16,11,22,20,13)( 5, 9,17,12,23,21,14)( 6, 7,18,10,24,19,15)$
$ 7, 7, 7, 1, 1, 1 $ $24$ $7$ $( 4,11,13,16,20, 8,22)( 5,12,14,17,21, 9,23)( 6,10,15,18,19, 7,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $56$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7,17,22)( 8,18,23)( 9,16,24)(10,14,20)(11,15,21) (12,13,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,15)( 8,13)( 9,14)(10,16)(11,17)(12,18)(19,23)(20,24) (21,22)$
$ 4, 4, 4, 4, 4, 4 $ $42$ $4$ $( 1, 4,17,22)( 2, 5,18,23)( 3, 6,16,24)( 7,20,13,10)( 8,21,14,11)( 9,19,15,12)$

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

X.1     1  1  1  1  1  1
X.2     3  A /A  . -1  1
X.3     3 /A  A  . -1  1
X.4     6 -1 -1  .  2  .
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