# Properties

 Label 10T13 Degree $10$ Order $120$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $S_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 13);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $13$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ CHM label: $S_{5}(10d)$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,5,7,9)(2,4,6,8,10), (1,2)(3,7)(8,9) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 5: None

## Low degree siblings

5T5, 6T14, 10T12, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1, 1, 1$ $10$ $2$ $(3,9)(4,5)(7,8)$ $3, 3, 3, 1$ $20$ $3$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$ $2, 2, 2, 2, 1, 1$ $15$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)$ $6, 3, 1$ $20$ $6$ $( 2, 7, 5,10, 4, 8)( 3, 6, 9)$ $5, 5$ $24$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$ $4, 4, 2$ $30$ $4$ $( 1, 2, 3, 7)( 4,10)( 5, 9, 6, 8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 120.34 magma: IdentifyGroup(G);
 Character table:  2 3 2 1 3 1 . 2 3 1 1 1 . 1 . . 5 1 . . . . 1 . 1a 2a 3a 2b 6a 5a 4a 2P 1a 1a 3a 1a 3a 5a 2b 3P 1a 2a 1a 2b 2a 5a 4a 5P 1a 2a 3a 2b 6a 1a 4a X.1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 1 -1 X.3 4 -2 1 . 1 -1 . X.4 4 2 1 . -1 -1 . X.5 5 1 -1 1 1 . -1 X.6 5 -1 -1 1 -1 . 1 X.7 6 . . -2 . 1 . 

magma: CharacterTable(G);