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Properties

Label 8T19
Degree $8$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3 : C_4 $

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Show commands: Magma

magma: G := TransitiveGroup(8, 19);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3 : C_4 $
CHM label:   $E(8):4=[1/4.eD(4)^{2}]2$
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:  (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8), (1,3)(4,5,6,7)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T19, 8T20, 8T21, 16T33 x 2, 16T52, 16T53, 32T19

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{4}$ $1$ $2$ $4$ $(1,3)(2,8)(4,6)(5,7)$
2B $2^{4}$ $2$ $2$ $4$ $(1,8)(2,3)(4,5)(6,7)$
2C $2^{2},1^{4}$ $2$ $2$ $2$ $(4,6)(5,7)$
2D $2^{4}$ $2$ $2$ $4$ $(1,8)(2,3)(4,7)(5,6)$
2E $2^{4}$ $4$ $2$ $4$ $(1,5)(2,6)(3,7)(4,8)$
4A $4^{2}$ $4$ $4$ $6$ $(1,7,2,6)(3,5,8,4)$
4B1 $4^{2}$ $4$ $4$ $6$ $(1,5,8,6)(2,4,3,7)$
4B-1 $4,2,1^{2}$ $4$ $4$ $4$ $(2,8)(4,5,6,7)$
4C1 $4^{2}$ $4$ $4$ $6$ $(1,7,3,5)(2,4,8,6)$
4C-1 $4,2,1^{2}$ $4$ $4$ $4$ $(2,8)(4,7,6,5)$

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:  32.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B1 4B-1 4C1 4C-1
Size 1 1 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2D 2D 2C 2A 2C
Type
32.6.1a R 1 1 1 1 1 1 1 1 1 1 1
32.6.1b R 1 1 1 1 1 −1 −1 −1 −1 1 1
32.6.1c R 1 1 1 1 1 −1 −1 1 1 −1 −1
32.6.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1
32.6.1e1 C 1 1 1 −1 −1 −1 1 −i i i −i
32.6.1e2 C 1 1 1 −1 −1 −1 1 i −i −i i
32.6.1f1 C 1 1 1 −1 −1 1 −1 −i i −i i
32.6.1f2 C 1 1 1 −1 −1 1 −1 i −i i −i
32.6.2a R 2 2 −2 −2 2 0 0 0 0 0 0
32.6.2b R 2 2 −2 2 −2 0 0 0 0 0 0
32.6.4a R 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);