Properties

Label 8T37
Order \(168\)
n \(8\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,7)$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $37$
Group :  $\PSL(2,7)$
CHM label :  $L(8)=PSL(2,7)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4)(3,6,5), (1,2,3,4,5,6,8), (1,6)(2,3)(4,5)(7,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

7T5 x 2, 14T10 x 2, 21T14, 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 $ $1$ $1$ $()$
$ 3, 3, 1, 1 $ $56$ $3$ $(3,7,8)(4,5,6)$
$ 7, 1 $ $24$ $7$ $(2,3,7,6,8,5,4)$
$ 7, 1 $ $24$ $7$ $(2,4,5,8,6,7,3)$
$ 2, 2, 2, 2 $ $21$ $2$ $(1,2)(3,4)(5,8)(6,7)$
$ 4, 4 $ $42$ $4$ $(1,2,3,7)(4,6,5,8)$

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