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Properties

Label 8T15
Degree $8$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Z_8 : Z_8^\times$

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Show commands: Magma

magma: G := TransitiveGroup(8, 15);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Z_8 : Z_8^\times$
CHM label:   $[1/4.cD(4)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8), (1,5)(3,7), (1,6)(2,5)(3,4)(7,8)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T15, 16T35, 16T38 x 2, 16T45, 32T21

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{4}$ $1$ $2$ $4$ $(1,5)(2,6)(3,7)(4,8)$
2B $2^{2},1^{4}$ $2$ $2$ $2$ $(2,6)(4,8)$
2C $2^{3},1^{2}$ $4$ $2$ $3$ $(1,7)(2,6)(3,5)$
2D $2^{4}$ $4$ $2$ $4$ $(1,4)(2,3)(5,8)(6,7)$
2E $2^{3},1^{2}$ $4$ $2$ $3$ $(1,7)(3,5)(4,8)$
4A $4^{2}$ $2$ $4$ $6$ $(1,3,5,7)(2,4,6,8)$
4B $4^{2}$ $2$ $4$ $6$ $(1,3,5,7)(2,8,6,4)$
4C $4^{2}$ $4$ $4$ $6$ $(1,4,5,8)(2,7,6,3)$
8A $8$ $4$ $8$ $7$ $(1,6,3,8,5,2,7,4)$
8B $8$ $4$ $8$ $7$ $(1,6,7,4,5,2,3,8)$

Malle's constant $a(G)$:     $1/2$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  32.43
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B
Size 1 1 2 4 4 4 2 2 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 4A 4A
Type
32.43.1a R 1 1 1 1 1 1 1 1 1 1 1
32.43.1b R 1 1 −1 −1 1 −1 1 −1 1 1 −1
32.43.1c R 1 1 −1 −1 1 1 1 −1 −1 −1 1
32.43.1d R 1 1 −1 1 −1 −1 1 −1 1 −1 1
32.43.1e R 1 1 −1 1 −1 1 1 −1 −1 1 −1
32.43.1f R 1 1 1 −1 −1 −1 1 1 −1 1 1
32.43.1g R 1 1 1 −1 −1 1 1 1 1 −1 −1
32.43.1h R 1 1 1 1 1 −1 1 1 −1 −1 −1
32.43.2a R 2 2 −2 0 0 0 −2 2 0 0 0
32.43.2b R 2 2 2 0 0 0 −2 −2 0 0 0
32.43.4a R 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);