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Properties

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

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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 TypeSizeOrderRepresentative
$ 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);