Properties

Label 20T2
Order \(20\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:C_4$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $2$
Group :  $C_5:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,2,20)(3,17,4,18)(5,15,6,16)(7,14,8,13)(9,12,10,11), (1,12,2,11)(3,10,4,9)(5,7,6,8)(13,19,14,20)(15,17,16,18)
$|\Aut(F/K)|$:  $20$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 10: $D_5$

Low degree siblings

There are no siblings 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 3, 2, 4)( 5,19, 6,20)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 4, 2, 3)( 5,20, 6,19)( 7,18, 8,17)( 9,16,10,15)(11,14,12,13)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 5, 9,13,18)( 2, 6,10,14,17)( 3, 7,12,15,19)( 4, 8,11,16,20)$
$ 10, 10 $ $2$ $10$ $( 1, 6, 9,14,18, 2, 5,10,13,17)( 3, 8,12,16,19, 4, 7,11,15,20)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 9,18, 5,13)( 2,10,17, 6,14)( 3,12,19, 7,15)( 4,11,20, 8,16)$
$ 10, 10 $ $2$ $10$ $( 1,10,18, 6,13, 2, 9,17, 5,14)( 3,11,19, 8,15, 4,12,20, 7,16)$

Group invariants

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

       1a 2a 4a 4b 5a 10a 5b 10b
    2P 1a 1a 2a 2a 5b  5b 5a  5a
    3P 1a 2a 4b 4a 5b 10b 5a 10a
    5P 1a 2a 4a 4b 1a  2a 1a  2a
    7P 1a 2a 4b 4a 5b 10b 5a 10a

X.1     1  1  1  1  1   1  1   1
X.2     1  1 -1 -1  1   1  1   1
X.3     1 -1  A -A  1  -1  1  -1
X.4     1 -1 -A  A  1  -1  1  -1
X.5     2 -2  .  .  B  -B *B -*B
X.6     2 -2  .  . *B -*B  B  -B
X.7     2  2  .  .  B   B *B  *B
X.8     2  2  .  . *B  *B  B   B

A = -E(4)
  = -Sqrt(-1) = -i
B = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5