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Properties

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

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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 TypeSizeOrderRepresentative
$ 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);