Properties

Label 10T4
Degree $10$
Order $20$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5$

Related objects

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Group action invariants

Degree $n$:  $10$
Transitive number $t$:  $4$
Group:  $F_5$
CHM label:  $1/2[F(5)]2$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,3,5,7,9)(2,4,6,8,10), (1,2,9,8)(3,6,7,4)(5,10)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

5T3, 20T5

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

Group invariants

Order:  $20=2^{2} \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [20, 3]
Character table:   
     2  2  2  2  2  .
     5  1  .  .  .  1

       1a 2a 4a 4b 5a
    2P 1a 1a 2a 2a 5a
    3P 1a 2a 4b 4a 5a
    5P 1a 2a 4a 4b 1a

X.1     1  1  1  1  1
X.2     1  1 -1 -1  1
X.3     1 -1  A -A  1
X.4     1 -1 -A  A  1
X.5     4  .  .  . -1

A = -E(4)
  = -Sqrt(-1) = -i