# Properties

 Label 10T32 Degree $10$ Order $720$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $S_{6}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 32);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $32$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_{6}$ CHM label: $S_{6}(10)=L(10):2$ 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: (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7), (3,6)(4,7)(5,8) 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

6T16 x 2, 12T183 x 2, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1, 1, 1$ $15$ $2$ $( 4, 8)( 5, 7)( 9,10)$ $2, 2, 2, 2, 1, 1$ $45$ $2$ $( 3, 6)( 4, 5)( 7, 8)( 9,10)$ $2, 2, 2, 1, 1, 1, 1$ $15$ $2$ $( 3, 9)( 4, 5)( 6,10)$ $4, 4, 1, 1$ $90$ $4$ $( 3, 9, 6,10)( 4, 8, 5, 7)$ $6, 3, 1$ $120$ $6$ $( 2, 3, 5, 8, 4, 9)( 6, 7,10)$ $3, 3, 3, 1$ $40$ $3$ $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$ $6, 3, 1$ $120$ $6$ $( 2, 3, 6)( 4,10, 7, 8, 9, 5)$ $3, 3, 3, 1$ $40$ $3$ $( 2, 4, 5)( 3, 9, 8)( 6, 7,10)$ $4, 4, 2$ $90$ $4$ $( 1, 2)( 3, 9, 6,10)( 4, 7, 5, 8)$ $5, 5$ $144$ $5$ $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $720=2^{4} \cdot 3^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 720.763 magma: IdentifyGroup(G);
 Character table:  2 4 4 4 4 3 1 1 1 1 3 . 3 2 1 . 1 . 1 2 1 2 . . 5 1 . . . . . . . . . 1 1a 2a 2b 2c 4a 6a 3a 6b 3b 4b 5a 2P 1a 1a 1a 1a 2b 3b 3a 3a 3b 2b 5a 3P 1a 2a 2b 2c 4a 2c 1a 2a 1a 4b 5a 5P 1a 2a 2b 2c 4a 6a 3a 6b 3b 4b 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 -1 1 X.3 5 -3 1 1 -1 1 2 . -1 -1 . X.4 5 3 1 -1 -1 -1 2 . -1 1 . X.5 5 -1 1 3 -1 . -1 -1 2 1 . X.6 5 1 1 -3 -1 . -1 1 2 -1 . X.7 9 -3 1 -3 1 . . . . 1 -1 X.8 9 3 1 3 1 . . . . -1 -1 X.9 10 -2 -2 2 . -1 1 1 1 . . X.10 10 2 -2 -2 . 1 1 -1 1 . . X.11 16 . . . . . -2 . -2 . 1 

magma: CharacterTable(G);