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Properties

Label 10T8
Degree $10$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4 : C_5$

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Show commands: Magma

magma: G := TransitiveGroup(10, 8);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4 : C_5$
CHM label:   $[2^{4}]5$
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:  (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $C_5$

Low degree siblings

10T8 x 2, 16T178, 20T17 x 6, 20T23, 40T57 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{6}$ $5$ $2$ $2$ $(2,7)(4,9)$
2B $2^{2},1^{6}$ $5$ $2$ $2$ $(1,6)(2,7)$
2C $2^{4},1^{2}$ $5$ $2$ $4$ $( 1, 6)( 2, 7)( 4, 9)( 5,10)$
5A1 $5^{2}$ $16$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
5A-1 $5^{2}$ $16$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
5A2 $5^{2}$ $16$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A-2 $5^{2}$ $16$ $5$ $8$ $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$

Malle's constant $a(G)$:     $1/2$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $80=2^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  80.49
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 5A1 5A-1 5A2 5A-2
Size 1 5 5 5 16 16 16 16
2 P 1A 1A 1A 1A 5A2 5A-2 5A-1 5A1
5 P 1A 2A 2B 2C 1A 1A 1A 1A
Type
80.49.1a R 1 1 1 1 1 1 1 1
80.49.1b1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1
80.49.1b2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5
80.49.1b3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52
80.49.1b4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2
80.49.5a R 5 −3 1 1 0 0 0 0
80.49.5b R 5 1 −3 1 0 0 0 0
80.49.5c R 5 1 1 −3 0 0 0 0

magma: CharacterTable(G);