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Properties

Label 8T37
Degree $8$
Order $168$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(2,7)$

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Show commands: Magma

magma: G := TransitiveGroup(8, 37);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,7)$
CHM label:   $L(8)=PSL(2,7)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,4)(3,6,5), (1,2,3,4,5,6,8), (1,6)(2,3)(4,5)(7,8)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

7T5 x 2, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

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 $2^{4}$ $21$ $2$ $4$ $(1,8)(2,3)(4,6)(5,7)$
3A $3^{2},1^{2}$ $56$ $3$ $4$ $(2,4,8)(5,7,6)$
4A $4^{2}$ $42$ $4$ $6$ $(1,2,8,3)(4,7,6,5)$
7A1 $7,1$ $24$ $7$ $6$ $(1,7,8,2,5,3,6)$
7A-1 $7,1$ $24$ $7$ $6$ $(1,3,2,7,6,5,8)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  168.42
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 4A 7A1 7A-1
Size 1 21 56 42 24 24
2 P 1A 1A 3A 2A 7A1 7A-1
3 P 1A 2A 1A 4A 7A-1 7A1
7 P 1A 2A 3A 4A 1A 1A
Type
168.42.1a R 1 1 1 1 1 1
168.42.3a1 C 3 −1 0 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
168.42.3a2 C 3 −1 0 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
168.42.6a R 6 2 0 0 −1 −1
168.42.7a R 7 −1 1 −1 0 0
168.42.8a R 8 0 −1 0 1 1

magma: CharacterTable(G);