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Properties

Label 20T5
Order \(20\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $5$
Group :  $F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,17,15)(2,3,18,16)(5,12,14,7)(6,11,13,8)(9,19,10,20), (1,6,10,14,18)(2,5,9,13,17)(3,8,12,15,19)(4,7,11,16,20)
$|\Aut(F/K)|$:  $20$

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 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

5T3, 10T4

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,20)( 4,19)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 3, 9, 7)( 2, 4,10, 8)( 5,16, 6,15)(11,14,19,17)(12,13,20,18)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 4,17,15)( 2, 3,18,16)( 5,12,14, 7)( 6,11,13, 8)( 9,19,10,20)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,10,14,18)( 2, 5, 9,13,17)( 3, 8,12,15,19)( 4, 7,11,16,20)$

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