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Properties

Label 8T23
8T23 1 2 1->2 3 1->3 2->3 4 3->4 8 3->8 5 4->5 4->5 6 5->6 7 5->7 6->7 7->4 7->8 8->1 8->1
Degree $8$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\textrm{GL(2,3)}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(8, 23);
 

Group invariants

Abstract group:  $\textrm{GL(2,3)}$
Copy content magma:IdentifyGroup(G);
 
Order:  $48=2^{4} \cdot 3$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $2S_{4}(8)=GL(2,3)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,3,4,5,6,7,8)$, $(1,3,8)(4,5,7)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Low degree siblings

8T23, 16T66, 24T22

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 $2^{3},1^{2}$ $12$ $2$ $3$ $(1,8)(2,6)(4,5)$
3A $3^{2},1^{2}$ $8$ $3$ $4$ $(1,2,4)(5,6,8)$
4A $4^{2}$ $6$ $4$ $6$ $(1,6,5,2)(3,4,7,8)$
6A $6,2$ $8$ $6$ $6$ $(1,8,2,5,4,6)(3,7)$
8A1 $8$ $6$ $8$ $7$ $(1,4,6,7,5,8,2,3)$
8A-1 $8$ $6$ $8$ $7$ $(1,2,3,4,5,6,7,8)$

Malle's constant $a(G)$:     $1/3$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 4A 6A 8A1 8A-1
Size 1 1 12 8 6 8 6 6
2 P 1A 1A 1A 3A 2A 3A 4A 4A
3 P 1A 2A 2B 1A 4A 2A 8A1 8A-1
Type
48.29.1a R 1 1 1 1 1 1 1 1
48.29.1b R 1 1 −1 1 1 1 −1 −1
48.29.2a R 2 2 0 −1 2 −1 0 0
48.29.2b1 C 2 −2 0 −1 0 1 −ζ8−ζ83 ζ8+ζ83
48.29.2b2 C 2 −2 0 −1 0 1 ζ8+ζ83 −ζ8−ζ83
48.29.3a R 3 3 1 0 −1 0 −1 −1
48.29.3b R 3 3 −1 0 −1 0 1 1
48.29.4a R 4 −4 0 1 0 −1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} + 4 t x^{6} + \left(132 t - 726\right) x^{4} + \left(129 t + 132\right) x^{2} + \left(32 t - 803\right)$ Copy content Toggle raw display