# Properties

 Label 10T16 Degree $10$ Order $160$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $(C_2^4 : C_5) : C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 16);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $16$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $(C_2^4 : C_5) : C_2$ CHM label: $1/2[2^{5}]D(5)$ 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: (1,9)(2,8)(3,7)(4,6)(5,10), (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 5: $D_{5}$

## Low degree siblings

10T15 x 3, 10T16 x 2, 16T415, 20T38 x 6, 20T39, 20T43 x 3, 20T45 x 3, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 4, 9)( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 3, 8)( 5,10)$ $4, 2, 2, 1, 1$ $20$ $4$ $( 2, 5, 7,10)( 3, 4)( 8, 9)$ $4, 2, 2, 1, 1$ $20$ $4$ $( 2, 5)( 3, 4, 8, 9)( 7,10)$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $2, 2, 2, 2, 2$ $20$ $2$ $( 1, 2)( 3, 5)( 4, 9)( 6, 7)( 8,10)$ $5, 5$ $32$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$ $4, 4, 2$ $20$ $4$ $( 1, 2, 6, 7)( 3, 5, 8,10)( 4, 9)$ $5, 5$ $32$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $160=2^{5} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 160.234 magma: IdentifyGroup(G);
 Character table:  2 5 5 5 3 3 5 3 . 3 . 5 1 . . . . . . 1 . 1 1a 2a 2b 4a 4b 2c 2d 5a 4c 5b 2P 1a 1a 1a 2b 2a 1a 1a 5b 2c 5a 3P 1a 2a 2b 4a 4b 2c 2d 5b 4c 5a 5P 1a 2a 2b 4a 4b 2c 2d 1a 4c 1a 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 2 2 2 . . 2 . A . *A X.4 2 2 2 . . 2 . *A . A X.5 5 -3 1 -1 1 1 1 . -1 . X.6 5 -3 1 1 -1 1 -1 . 1 . X.7 5 1 -3 -1 1 1 -1 . 1 . X.8 5 1 -3 1 -1 1 1 . -1 . X.9 5 1 1 -1 -1 -3 1 . 1 . X.10 5 1 1 1 1 -3 -1 . -1 . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 

magma: CharacterTable(G);