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Properties

Label 20T89
Order \(360\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_6$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $89$
Group :  $A_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,14)(2,6,13)(3,15,18)(4,16,17)(7,12,20)(8,11,19), (1,19,6)(2,20,5)(3,17,12)(4,18,11)(9,16,14)(10,15,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $\PSL(2,9)$

Low degree siblings

6T15 x 2, 10T26, 15T20 x 2, 30T88 x 2, 36T555, 40T304, 45T49

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, 1, 1, 1, 1 $ $45$ $2$ $( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)(17,19)(18,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $40$ $3$ $( 3, 5,11)( 4, 6,12)( 7,17,13)( 8,18,14)( 9,15,19)(10,16,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $40$ $3$ $( 3, 8,10)( 4, 7, 9)( 5,18,16)( 6,17,15)(11,14,20)(12,13,19)$
$ 4, 4, 4, 4, 2, 2 $ $90$ $4$ $( 1, 2)( 3, 4)( 5,17,11,19)( 6,18,12,20)( 7,16, 9,14)( 8,15,10,13)$
$ 5, 5, 5, 5 $ $72$ $5$ $( 1, 3, 7,17,16)( 2, 4, 8,18,15)( 5,13,10,11,20)( 6,14, 9,12,19)$
$ 5, 5, 5, 5 $ $72$ $5$ $( 1, 3, 7,12,14)( 2, 4, 8,11,13)( 5,20,15,10,18)( 6,19,16, 9,17)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 118]
Character table:    
     2  3  3  .  .  2  .  .
     3  2  .  2  2  .  .  .
     5  1  .  .  .  .  1  1

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

X.1     1  1  1  1  1  1  1
X.2     5  1  2 -1 -1  .  .
X.3     5  1 -1  2 -1  .  .
X.4     8  . -1 -1  .  A *A
X.5     8  . -1 -1  . *A  A
X.6     9  1  .  .  1 -1 -1
X.7    10 -2  1  1  .  .  .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5