Properties

Label 10T16
Order \(160\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_2^4 : C_5) : C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $16$
Group :  $(C_2^4 : C_5) : C_2$
CHM label :  $1/2[2^{5}]D(5)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

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

Group invariants

Order:  $160=2^{5} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [160, 234]
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