# Properties

 Label 10T29 Degree $10$ Order $640$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $((C_2^4 : C_5):C_4)\times C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 29);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $29$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $((C_2^4 : C_5):C_4)\times C_2$ CHM label: $[2^{5}]F(5)$ 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: (5,10), (1,7,9,3)(2,4,8,6), (1,3,5,7,9)(2,4,6,8,10) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$
$320$:  $(C_2^4 : C_5):C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 5: $F_5$

## Low degree siblings

10T29, 20T129, 20T131 x 2, 20T132, 20T133, 20T134, 20T135, 20T137 x 2, 20T140, 32T34608 x 2, 40T460, 40T462, 40T473, 40T474, 40T475, 40T476, 40T487, 40T488, 40T489, 40T490, 40T557, 40T558 x 2, 40T561, 40T562, 40T563, 40T564, 40T565, 40T566, 40T567 x 2, 40T576, 40T577, 40T578, 40T579, 40T586

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 5,10)$ $2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 2, 7)( 5,10)$ $2, 2, 2, 1, 1, 1, 1$ $10$ $2$ $( 2, 7)( 4, 9)( 5,10)$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $2, 2, 2, 2, 2$ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ $5, 5$ $64$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $10$ $64$ $10$ $( 1, 3,10, 2, 4, 6, 8, 5, 7, 9)$ $2, 2, 2, 2, 1, 1$ $20$ $2$ $(1,9)(2,8)(3,7)(4,6)$ $2, 2, 2, 2, 2$ $20$ $2$ $( 1, 9)( 2, 8)( 3, 7)( 4, 6)( 5,10)$ $4, 2, 2, 1, 1$ $40$ $4$ $(1,9)(2,8,7,3)(4,6)$ $4, 2, 2, 2$ $40$ $4$ $( 1, 9)( 2, 8, 7, 3)( 4, 6)( 5,10)$ $4, 4, 1, 1$ $20$ $4$ $(1,4,6,9)(2,8,7,3)$ $4, 4, 2$ $20$ $4$ $( 1, 4, 6, 9)( 2, 8, 7, 3)( 5,10)$ $4, 4, 1, 1$ $40$ $4$ $(1,7,9,3)(2,4,8,6)$ $4, 4, 2$ $40$ $4$ $( 1, 7, 9, 3)( 2, 4, 8, 6)( 5,10)$ $8, 1, 1$ $40$ $8$ $(1,2,4,8,6,7,9,3)$ $8, 2$ $40$ $8$ $( 1, 2, 4, 8, 6, 7, 9, 3)( 5,10)$ $4, 4, 1, 1$ $40$ $4$ $(1,3,9,7)(2,6,8,4)$ $4, 4, 2$ $40$ $4$ $( 1, 3, 9, 7)( 2, 6, 8, 4)( 5,10)$ $8, 1, 1$ $40$ $8$ $(1,3,9,2,6,8,4,7)$ $8, 2$ $40$ $8$ $( 1, 3, 9, 2, 6, 8, 4, 7)( 5,10)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $640=2^{7} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 640.21536 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);