# Properties

 Label 10T12 Degree $10$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 12);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $12$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ CHM label: $1/2[S(5)]2=S_{5}(10a)$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) 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), (1,4)(2,7)(3,8)(5,10)(6,9) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: $S_5$

## Low degree siblings

5T5, 6T14, 10T13, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1, 1, 1$ $20$ $3$ $( 3, 5, 9)( 4, 8,10)$ $2, 2, 2, 2, 1, 1$ $15$ $2$ $( 2, 4)( 3, 5)( 7, 9)( 8,10)$ $6, 2, 2$ $20$ $6$ $( 1, 2)( 3, 4, 5, 8, 9,10)( 6, 7)$ $2, 2, 2, 2, 2$ $10$ $2$ $( 1, 2)( 3, 8)( 4, 9)( 5,10)( 6, 7)$ $4, 4, 2$ $30$ $4$ $( 1, 2, 3, 4)( 5,10)( 6, 7, 8, 9)$ $5, 5$ $24$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

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 1 3 1 2 2 . 3 1 1 . 1 1 . . 5 1 . . . . . 1 1a 3a 2a 6a 2b 4a 5a 2P 1a 3a 1a 3a 1a 2a 5a 3P 1a 1a 2a 2b 2b 4a 5a 5P 1a 3a 2a 6a 2b 4a 1a X.1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 X.3 4 1 . 1 -2 . -1 X.4 4 1 . -1 2 . -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);