# Properties

 Label 10T1 Degree $10$ Order $10$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group no Group: $C_{10}$

# Learn more

Show commands: Magma

magma: G := TransitiveGroup(10, 1);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{10}$ CHM label: $C(10)=5[x]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $1$ magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $10$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,4,5,6,7,8,9,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: $C_5$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $10$ $1$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ $5, 5$ $1$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $10$ $1$ $10$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$ $5, 5$ $1$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$ $2, 2, 2, 2, 2$ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $5, 5$ $1$ $5$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$ $10$ $1$ $10$ $( 1, 8, 5, 2, 9, 6, 3,10, 7, 4)$ $5, 5$ $1$ $5$ $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$ $10$ $1$ $10$ $( 1,10, 9, 8, 7, 6, 5, 4, 3, 2)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $10=2 \cdot 5$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 10.2 magma: IdentifyGroup(G);
 Character table:  2 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1a 10a 5a 10b 5b 2a 5c 10c 5d 10d X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 -1 X.3 1 A B /B /A 1 A B /B /A X.4 1 -A B -/B /A -1 A -B /B -/A X.5 1 B /A A /B 1 B /A A /B X.6 1 -B /A -A /B -1 B -/A A -/B X.7 1 /B A /A B 1 /B A /A B X.8 1 -/B A -/A B -1 /B -A /A -B X.9 1 /A /B B A 1 /A /B B A X.10 1 -/A /B -B A -1 /A -/B B -A A = E(5)^3 B = E(5) 

magma: CharacterTable(G);