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Properties

Label 28T32
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\PSL(2,7)$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $32$
Group :  $\PSL(2,7)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,28,18,22,6,10)(2,14,26,17,24,7,12)(3,16,25,19,23,8,11)(4,15,27,20,21,5,9), (1,14)(2,15)(3,16)(4,13)(5,28)(6,26)(7,25)(8,27)(11,12)(17,18)(19,20)(21,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

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

Degree 14: None

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 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, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $21$ $2$ $( 3, 4)( 5,26)( 6,28)( 7,25)( 8,27)( 9,22)(10,23)(11,24)(12,21)(13,16)(17,19) (18,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $56$ $3$ $( 2, 3, 4)( 5,13,25)( 6,15,28)( 7,16,26)( 8,14,27)( 9,24,20)(10,21,19) (11,22,18)(12,23,17)$
$ 4, 4, 4, 4, 4, 4, 2, 2 $ $42$ $4$ $( 1, 2, 3, 4)( 5,16,24,18)( 6,14,23,17)( 7,13,22,20)( 8,15,21,19)( 9,26) (10,28,12,27)(11,25)$
$ 7, 7, 7, 7 $ $24$ $7$ $( 1, 5,13,17,25,23,12)( 2, 6,14,18,26,21, 9)( 3, 7,15,19,27,24,11) ( 4, 8,16,20,28,22,10)$
$ 7, 7, 7, 7 $ $24$ $7$ $( 1, 5,15,22,11,28,19)( 2, 7,13,21,12,27,18)( 3, 6,14,24,10,25,20) ( 4, 8,16,23, 9,26,17)$

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