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Properties

Label 8T49
Degree $8$
Order $20160$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_8$

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Show commands: Magma

magma: G := TransitiveGroup(8, 49);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_8$
CHM label:  $A8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(3,4,5,6,7,8), (1,2,3)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

15T72 x 2, 28T433, 35T36

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, 2 $ $105$ $2$ $(1,3)(2,6)(4,5)(7,8)$
$ 3, 3, 1, 1 $ $1120$ $3$ $(1,6,5)(2,4,3)$
$ 6, 2 $ $3360$ $6$ $(1,4,6,3,5,2)(7,8)$
$ 2, 2, 1, 1, 1, 1 $ $210$ $2$ $(4,6)(5,7)$
$ 4, 2, 1, 1 $ $2520$ $4$ $(1,2)(4,5,6,7)$
$ 4, 4 $ $1260$ $4$ $(1,3,2,8)(4,7,6,5)$
$ 7, 1 $ $2880$ $7$ $(1,7,4,5,6,8,2)$
$ 7, 1 $ $2880$ $7$ $(1,2,8,6,5,4,7)$
$ 3, 1, 1, 1, 1, 1 $ $112$ $3$ $(1,7,2)$
$ 3, 2, 2, 1 $ $1680$ $6$ $(1,2,7)(3,5)(4,6)$
$ 5, 1, 1, 1 $ $1344$ $5$ $(1,5,8,2,4)$
$ 5, 3 $ $1344$ $15$ $(1,8,4,5,2)(3,7,6)$
$ 5, 3 $ $1344$ $15$ $(1,8,4,5,2)(3,6,7)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  20160.a
magma: IdentifyGroup(G);
 
Character table:    
      2  6  2  .   .   .  .  .  6  1  1  5  2  3  4
      3  2  2  1   1   1  .  .  1  2  1  1  1  .  .
      5  1  1  1   1   1  .  .  .  .  .  .  .  .  .
      7  1  .  .   .   .  1  1  .  .  .  .  .  .  .

        1a 3a 5a 15a 15b 7a 7b 2a 3b 6a 2b 6b 4a 4b
     2P 1a 3a 5a 15a 15b 7a 7b 1a 3b 3b 1a 3a 2b 2a
     3P 1a 1a 5a  5a  5a 7b 7a 2a 1a 2a 2b 2b 4a 4b
     5P 1a 3a 1a  3a  3a 7b 7a 2a 3b 6a 2b 6b 4a 4b
     7P 1a 3a 5a 15b 15a 1a 1a 2a 3b 6a 2b 6b 4a 4b
    11P 1a 3a 5a 15b 15a 7a 7b 2a 3b 6a 2b 6b 4a 4b
    13P 1a 3a 5a 15b 15a 7b 7a 2a 3b 6a 2b 6b 4a 4b

X.1      1  1  1   1   1  1  1  1  1  1  1  1  1  1
X.2      7  4  2  -1  -1  .  . -1  1 -1  3  .  1 -1
X.3     14 -1 -1  -1  -1  .  .  6  2  .  2 -1  .  2
X.4     20  5  .   .   . -1 -1  4 -1  1  4  1  .  .
X.5     21  6  1   1   1  .  . -3  .  .  1 -2 -1  1
X.6     21 -3  1   A  /A  .  . -3  .  .  1  1 -1  1
X.7     21 -3  1  /A   A  .  . -3  .  .  1  1 -1  1
X.8     28  1 -2   1   1  .  . -4  1 -1  4  1  .  .
X.9     35  5  .   .   .  .  .  3  2  . -5  1 -1 -1
X.10    45  .  .   .   .  B /B -3  .  . -3  .  1  1
X.11    45  .  .   .   . /B  B -3  .  . -3  .  1  1
X.12    56 -4  1   1   1  .  .  8 -1 -1  .  .  .  .
X.13    64  4 -1  -1  -1  1  1  . -2  .  .  .  .  .
X.14    70 -5  .   .   .  .  . -2  1  1  2 -1  . -2

A = -E(15)-E(15)^2-E(15)^4-E(15)^8
  = (-1-Sqrt(-15))/2 = -1-b15
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7

magma: CharacterTable(G);