Properties

Label 5T3
Order \(20\)
n \(5\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $F_5$

Related objects

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Group action invariants

Degree $n$ :  $5$
Transitive number $t$ :  $3$
Group :  $F_5$
CHM label :  $F(5) = 5:4$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5), (1,2,4,3)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

10T4, 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 $ $1$ $1$ $()$
$ 4, 1 $ $5$ $4$ $(2,3,5,4)$
$ 4, 1 $ $5$ $4$ $(2,4,5,3)$
$ 2, 2, 1 $ $5$ $2$ $(2,5)(3,4)$
$ 5 $ $4$ $5$ $(1,2,3,4,5)$

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

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

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