# Properties

 Label 10T7 Degree $10$ Order $60$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_{5}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 7);

## Group action invariants

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

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 5: None

## Low degree siblings

5T4, 6T12, 12T33, 15T5, 20T15, 30T9

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 1$ $20$ $3$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$ $2, 2, 2, 2, 1, 1$ $15$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)$ $5, 5$ $12$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$ $5, 5$ $12$ $5$ $( 1, 2, 9, 6, 7)( 3, 8, 4,10, 5)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $60=2^{2} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 60.5 magma: IdentifyGroup(G);
 Character table:  2 2 . 2 . . 3 1 1 . . . 5 1 . . 1 1 1a 3a 2a 5a 5b 2P 1a 3a 1a 5b 5a 3P 1a 1a 2a 5b 5a 5P 1a 3a 2a 1a 1a X.1 1 1 1 1 1 X.2 3 . -1 A *A X.3 3 . -1 *A A X.4 4 1 . -1 -1 X.5 5 -1 1 . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 

magma: CharacterTable(G);