Properties

Label 8T5
Degree $8$
Order $8$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_8$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(8, 5);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_8$
CHM label:  $Q_{8}(8)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,8)(4,5,6,7), (1,7,3,5)(2,6,8,4)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 4, 4 $ $2$ $4$ $(1,2,3,8)(4,5,6,7)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$ 4, 4 $ $2$ $4$ $(1,4,3,6)(2,7,8,5)$
$ 4, 4 $ $2$ $4$ $(1,5,3,7)(2,4,8,6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $8=2^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  8.4
magma: IdentifyGroup(G);
 
Character table:   
     2  3  2  3  2  2

       1a 4a 2a 4b 4c
    2P 1a 2a 1a 2a 2a
    3P 1a 4a 2a 4b 4c

X.1     1  1  1  1  1
X.2     1 -1  1 -1  1
X.3     1 -1  1  1 -1
X.4     1  1  1 -1 -1
X.5     2  . -2  .  .

magma: CharacterTable(G);