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Properties

Label 8T43
Order \(336\)
n \(8\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PGL(2,7)$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $43$
Group :  $\PGL(2,7)$
CHM label :  $L(8):2=PGL(2,7)$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,2,6,4,5), (1,2,3,4,5,6,8), (1,6)(2,3)(4,5)(7,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

14T16, 16T713, 21T20, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83

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 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1 $ $28$ $2$ $(3,4)(5,7)(6,8)$
$ 6, 1, 1 $ $56$ $6$ $(3,5,8,4,7,6)$
$ 3, 3, 1, 1 $ $56$ $3$ $(3,7,8)(4,5,6)$
$ 7, 1 $ $48$ $7$ $(2,3,7,6,8,5,4)$
$ 2, 2, 2, 2 $ $21$ $2$ $(1,2)(3,4)(5,8)(6,7)$
$ 8 $ $42$ $8$ $(1,2,3,6,4,7,8,5)$
$ 4, 4 $ $42$ $4$ $(1,2,3,7)(4,6,5,8)$
$ 8 $ $42$ $8$ $(1,2,3,8,6,7,5,4)$

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [336, 208]
Character table:    
     2  4  2  1  1  .  4  3  3  3
     3  1  1  1  1  .  .  .  .  .
     7  1  .  .  .  1  .  .  .  .

       1a 2a 6a 3a 7a 2b 8a 4a 8b
    2P 1a 1a 3a 3a 7a 1a 4a 2b 4a
    3P 1a 2a 2a 1a 7a 2b 8b 4a 8a
    5P 1a 2a 6a 3a 7a 2b 8b 4a 8a
    7P 1a 2a 6a 3a 1a 2b 8a 4a 8b

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1 -1  1  1  1 -1  1 -1
X.3     6  .  .  . -1 -2  .  2  .
X.4     6  .  .  . -1  2  A  . -A
X.5     6  .  .  . -1  2 -A  .  A
X.6     7 -1 -1  1  . -1  1 -1  1
X.7     7  1  1  1  . -1 -1 -1 -1
X.8     8 -2  1 -1  1  .  .  .  .
X.9     8  2 -1 -1  1  .  .  .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2